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G = C5×C22.36C24  order 320 = 26·5

Direct product of C5 and C22.36C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22.36C24, C10.1572+ (1+4), C10.1152- (1+4), C4⋊Q8⋊11C10, (D4×C20)⋊42C2, (C4×D4)⋊13C10, (Q8×C20)⋊30C2, (C4×Q8)⋊10C10, C22⋊Q8⋊9C10, C4.4D4⋊9C10, C4⋊D4.9C10, C42⋊2C2⋊3C10, C42.40(C2×C10), C42⋊C2⋊13C10, C20.278(C4○D4), (C2×C10).362C24, (C4×C20).281C22, (C2×C20).671C23, C22.D4⋊7C10, C2.9(C5×2+ (1+4)), C2.7(C5×2- (1+4)), (D4×C10).219C22, C22.36(C23×C10), (C22×C10).97C23, C23.14(C22×C10), (Q8×C10).182C22, (C22×C20).450C22, (C5×C4⋊Q8)⋊32C2, C4.22(C5×C4○D4), C4⋊C4.70(C2×C10), C2.19(C10×C4○D4), (C5×C22⋊Q8)⋊36C2, (C2×D4).33(C2×C10), C10.238(C2×C4○D4), (C5×C4.4D4)⋊29C2, (C5×C4⋊D4).19C2, C22⋊C4.4(C2×C10), (C2×Q8).26(C2×C10), (C5×C42⋊2C2)⋊14C2, (C5×C42⋊C2)⋊34C2, (C5×C4⋊C4).249C22, (C22×C4).62(C2×C10), (C2×C4).29(C22×C10), (C5×C22.D4)⋊26C2, (C5×C22⋊C4).150C22, SmallGroup(320,1544)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.36C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C4.4D4 — C5×C22.36C24
Lower central
C1 — C22 — C5×C22.36C24
Upper central
C1 — C2×C10 — C5×C22.36C24

Subgroups: 322 in 216 conjugacy classes, 146 normal (62 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×4], Q8 [×4], C23, C23 [×2], C10 [×3], C10 [×3], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C20 [×2], C20 [×11], C2×C10, C2×C10 [×9], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C42⋊2C2 [×2], C4⋊Q8, C2×C20 [×6], C2×C20 [×6], C2×C20 [×4], C5×D4 [×4], C5×Q8 [×4], C22×C10, C22×C10 [×2], C22.36C24, C4×C20 [×2], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C22⋊C4 [×10], C5×C4⋊C4 [×4], C5×C4⋊C4 [×6], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, Q8×C10 [×2], C5×C42⋊C2, D4×C20, Q8×C20, C5×C4⋊D4, C5×C22⋊Q8, C5×C22⋊Q8 [×2], C5×C22.D4 [×2], C5×C4.4D4, C5×C4.4D4 [×2], C5×C42⋊2C2 [×2], C5×C4⋊Q8, C5×C22.36C24

Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×2], C24, C2×C10 [×35], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×C10 [×15], C22.36C24, C5×C4○D4 [×2], C23×C10, C10×C4○D4, C5×2+ (1+4), C5×2- (1+4), C5×C22.36C24

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Smallest permutation representation
►On 160 points
Generators in S160
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)(81 82 83 84 85)(86 87 88 89 90)(91 92 93 94 95)(96 97 98 99 100)(101 102 103 104 105)(106 107 108 109 110)(111 112 113 114 115)(116 117 118 119 120)(121 122 123 124 125)(126 127 128 129 130)(131 132 133 134 135)(136 137 138 139 140)(141 142 143 144 145)(146 147 148 149 150)(151 152 153 154 155)(156 157 158 159 160)

(1 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 156)(12 157)(13 158)(14 159)(15 160)(21 28)(22 29)(23 30)(24 26)(25 27)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)(76 95)(77 91)(78 92)(79 93)(80 94)(81 88)(82 89)(83 90)(84 86)(85 87)(96 115)(97 111)(98 112)(99 113)(100 114)(101 108)(102 109)(103 110)(104 106)(105 107)(116 135)(117 131)(118 132)(119 133)(120 134)(121 128)(122 129)(123 130)(124 126)(125 127)(136 155)(137 151)(138 152)(139 153)(140 154)(141 148)(142 149)(143 150)(144 146)(145 147)

(1 30)(2 26)(3 27)(4 28)(5 29)(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)(76 90)(77 86)(78 87)(79 88)(80 89)(81 93)(82 94)(83 95)(84 91)(85 92)(96 110)(97 106)(98 107)(99 108)(100 109)(101 113)(102 114)(103 115)(104 111)(105 112)(116 130)(117 126)(118 127)(119 128)(120 129)(121 133)(122 134)(123 135)(124 131)(125 132)(136 150)(137 146)(138 147)(139 148)(140 149)(141 153)(142 154)(143 155)(144 151)(145 152)

(1 96)(2 97)(3 98)(4 99)(5 100)(6 90)(7 86)(8 87)(9 88)(10 89)(11 95)(12 91)(13 92)(14 93)(15 94)(16 83)(17 84)(18 85)(19 81)(20 82)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

(1 63 23 56)(2 64 24 57)(3 65 25 58)(4 61 21 59)(5 62 22 60)(6 123 11 116)(7 124 12 117)(8 125 13 118)(9 121 14 119)(10 122 15 120)(16 130 156 135)(17 126 157 131)(18 127 158 132)(19 128 159 133)(20 129 160 134)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)(36 83 43 76)(37 84 44 77)(38 85 45 78)(39 81 41 79)(40 82 42 80)(46 91 51 86)(47 92 52 87)(48 93 53 88)(49 94 54 89)(50 95 55 90)(96 150 103 155)(97 146 104 151)(98 147 105 152)(99 148 101 153)(100 149 102 154)(106 137 111 144)(107 138 112 145)(108 139 113 141)(109 140 114 142)(110 136 115 143)

(6 11)(7 12)(8 13)(9 14)(10 15)(16 156)(17 157)(18 158)(19 159)(20 160)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)(76 95)(77 91)(78 92)(79 93)(80 94)(81 88)(82 89)(83 90)(84 86)(85 87)(96 110)(97 106)(98 107)(99 108)(100 109)(101 113)(102 114)(103 115)(104 111)(105 112)(116 130)(117 126)(118 127)(119 128)(120 129)(121 133)(122 134)(123 135)(124 131)(125 132)(136 143)(137 144)(138 145)(139 141)(140 142)(146 151)(147 152)(148 153)(149 154)(150 155)

(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 143 16 150)(7 144 17 146)(8 145 18 147)(9 141 19 148)(10 142 20 149)(11 136 156 155)(12 137 157 151)(13 138 158 152)(14 139 159 153)(15 140 160 154)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)(56 95 75 76)(57 91 71 77)(58 92 72 78)(59 93 73 79)(60 94 74 80)(61 88 68 81)(62 89 69 82)(63 90 70 83)(64 86 66 84)(65 87 67 85)(96 116 115 135)(97 117 111 131)(98 118 112 132)(99 119 113 133)(100 120 114 134)(101 121 108 128)(102 122 109 129)(103 123 110 130)(104 124 106 126)(105 125 107 127)
 
G:=sub<Sym(160)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,115)(97,111)(98,112)(99,113)(100,114)(101,108)(102,109)(103,110)(104,106)(105,107)(116,135)(117,131)(118,132)(119,133)(120,134)(121,128)(122,129)(123,130)(124,126)(125,127)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147), (1,30)(2,26)(3,27)(4,28)(5,29)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)(76,90)(77,86)(78,87)(79,88)(80,89)(81,93)(82,94)(83,95)(84,91)(85,92)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152), (1,96)(2,97)(3,98)(4,99)(5,100)(6,90)(7,86)(8,87)(9,88)(10,89)(11,95)(12,91)(13,92)(14,93)(15,94)(16,83)(17,84)(18,85)(19,81)(20,82)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,63,23,56)(2,64,24,57)(3,65,25,58)(4,61,21,59)(5,62,22,60)(6,123,11,116)(7,124,12,117)(8,125,13,118)(9,121,14,119)(10,122,15,120)(16,130,156,135)(17,126,157,131)(18,127,158,132)(19,128,159,133)(20,129,160,134)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)(36,83,43,76)(37,84,44,77)(38,85,45,78)(39,81,41,79)(40,82,42,80)(46,91,51,86)(47,92,52,87)(48,93,53,88)(49,94,54,89)(50,95,55,90)(96,150,103,155)(97,146,104,151)(98,147,105,152)(99,148,101,153)(100,149,102,154)(106,137,111,144)(107,138,112,145)(108,139,113,141)(109,140,114,142)(110,136,115,143), (6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,143)(137,144)(138,145)(139,141)(140,142)(146,151)(147,152)(148,153)(149,154)(150,155), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,88,68,81)(62,89,69,82)(63,90,70,83)(64,86,66,84)(65,87,67,85)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,108,128)(102,122,109,129)(103,123,110,130)(104,124,106,126)(105,125,107,127)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100)(101,102,103,104,105)(106,107,108,109,110)(111,112,113,114,115)(116,117,118,119,120)(121,122,123,124,125)(126,127,128,129,130)(131,132,133,134,135)(136,137,138,139,140)(141,142,143,144,145)(146,147,148,149,150)(151,152,153,154,155)(156,157,158,159,160), (1,35)(2,31)(3,32)(4,33)(5,34)(6,16)(7,17)(8,18)(9,19)(10,20)(11,156)(12,157)(13,158)(14,159)(15,160)(21,28)(22,29)(23,30)(24,26)(25,27)(36,55)(37,51)(38,52)(39,53)(40,54)(41,48)(42,49)(43,50)(44,46)(45,47)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,115)(97,111)(98,112)(99,113)(100,114)(101,108)(102,109)(103,110)(104,106)(105,107)(116,135)(117,131)(118,132)(119,133)(120,134)(121,128)(122,129)(123,130)(124,126)(125,127)(136,155)(137,151)(138,152)(139,153)(140,154)(141,148)(142,149)(143,150)(144,146)(145,147), (1,30)(2,26)(3,27)(4,28)(5,29)(6,156)(7,157)(8,158)(9,159)(10,160)(11,16)(12,17)(13,18)(14,19)(15,20)(21,33)(22,34)(23,35)(24,31)(25,32)(36,50)(37,46)(38,47)(39,48)(40,49)(41,53)(42,54)(43,55)(44,51)(45,52)(56,70)(57,66)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,71)(65,72)(76,90)(77,86)(78,87)(79,88)(80,89)(81,93)(82,94)(83,95)(84,91)(85,92)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,150)(137,146)(138,147)(139,148)(140,149)(141,153)(142,154)(143,155)(144,151)(145,152), (1,96)(2,97)(3,98)(4,99)(5,100)(6,90)(7,86)(8,87)(9,88)(10,89)(11,95)(12,91)(13,92)(14,93)(15,94)(16,83)(17,84)(18,85)(19,81)(20,82)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,63,23,56)(2,64,24,57)(3,65,25,58)(4,61,21,59)(5,62,22,60)(6,123,11,116)(7,124,12,117)(8,125,13,118)(9,121,14,119)(10,122,15,120)(16,130,156,135)(17,126,157,131)(18,127,158,132)(19,128,159,133)(20,129,160,134)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70)(36,83,43,76)(37,84,44,77)(38,85,45,78)(39,81,41,79)(40,82,42,80)(46,91,51,86)(47,92,52,87)(48,93,53,88)(49,94,54,89)(50,95,55,90)(96,150,103,155)(97,146,104,151)(98,147,105,152)(99,148,101,153)(100,149,102,154)(106,137,111,144)(107,138,112,145)(108,139,113,141)(109,140,114,142)(110,136,115,143), (6,11)(7,12)(8,13)(9,14)(10,15)(16,156)(17,157)(18,158)(19,159)(20,160)(56,75)(57,71)(58,72)(59,73)(60,74)(61,68)(62,69)(63,70)(64,66)(65,67)(76,95)(77,91)(78,92)(79,93)(80,94)(81,88)(82,89)(83,90)(84,86)(85,87)(96,110)(97,106)(98,107)(99,108)(100,109)(101,113)(102,114)(103,115)(104,111)(105,112)(116,130)(117,126)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,131)(125,132)(136,143)(137,144)(138,145)(139,141)(140,142)(146,151)(147,152)(148,153)(149,154)(150,155), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,88,68,81)(62,89,69,82)(63,90,70,83)(64,86,66,84)(65,87,67,85)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,108,128)(102,122,109,129)(103,123,110,130)(104,124,106,126)(105,125,107,127) );
 
G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100),(101,102,103,104,105),(106,107,108,109,110),(111,112,113,114,115),(116,117,118,119,120),(121,122,123,124,125),(126,127,128,129,130),(131,132,133,134,135),(136,137,138,139,140),(141,142,143,144,145),(146,147,148,149,150),(151,152,153,154,155),(156,157,158,159,160)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,16),(7,17),(8,18),(9,19),(10,20),(11,156),(12,157),(13,158),(14,159),(15,160),(21,28),(22,29),(23,30),(24,26),(25,27),(36,55),(37,51),(38,52),(39,53),(40,54),(41,48),(42,49),(43,50),(44,46),(45,47),(56,75),(57,71),(58,72),(59,73),(60,74),(61,68),(62,69),(63,70),(64,66),(65,67),(76,95),(77,91),(78,92),(79,93),(80,94),(81,88),(82,89),(83,90),(84,86),(85,87),(96,115),(97,111),(98,112),(99,113),(100,114),(101,108),(102,109),(103,110),(104,106),(105,107),(116,135),(117,131),(118,132),(119,133),(120,134),(121,128),(122,129),(123,130),(124,126),(125,127),(136,155),(137,151),(138,152),(139,153),(140,154),(141,148),(142,149),(143,150),(144,146),(145,147)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,156),(7,157),(8,158),(9,159),(10,160),(11,16),(12,17),(13,18),(14,19),(15,20),(21,33),(22,34),(23,35),(24,31),(25,32),(36,50),(37,46),(38,47),(39,48),(40,49),(41,53),(42,54),(43,55),(44,51),(45,52),(56,70),(57,66),(58,67),(59,68),(60,69),(61,73),(62,74),(63,75),(64,71),(65,72),(76,90),(77,86),(78,87),(79,88),(80,89),(81,93),(82,94),(83,95),(84,91),(85,92),(96,110),(97,106),(98,107),(99,108),(100,109),(101,113),(102,114),(103,115),(104,111),(105,112),(116,130),(117,126),(118,127),(119,128),(120,129),(121,133),(122,134),(123,135),(124,131),(125,132),(136,150),(137,146),(138,147),(139,148),(140,149),(141,153),(142,154),(143,155),(144,151),(145,152)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,90),(7,86),(8,87),(9,88),(10,89),(11,95),(12,91),(13,92),(14,93),(15,94),(16,83),(17,84),(18,85),(19,81),(20,82),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,63,23,56),(2,64,24,57),(3,65,25,58),(4,61,21,59),(5,62,22,60),(6,123,11,116),(7,124,12,117),(8,125,13,118),(9,121,14,119),(10,122,15,120),(16,130,156,135),(17,126,157,131),(18,127,158,132),(19,128,159,133),(20,129,160,134),(26,71,31,66),(27,72,32,67),(28,73,33,68),(29,74,34,69),(30,75,35,70),(36,83,43,76),(37,84,44,77),(38,85,45,78),(39,81,41,79),(40,82,42,80),(46,91,51,86),(47,92,52,87),(48,93,53,88),(49,94,54,89),(50,95,55,90),(96,150,103,155),(97,146,104,151),(98,147,105,152),(99,148,101,153),(100,149,102,154),(106,137,111,144),(107,138,112,145),(108,139,113,141),(109,140,114,142),(110,136,115,143)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,156),(17,157),(18,158),(19,159),(20,160),(56,75),(57,71),(58,72),(59,73),(60,74),(61,68),(62,69),(63,70),(64,66),(65,67),(76,95),(77,91),(78,92),(79,93),(80,94),(81,88),(82,89),(83,90),(84,86),(85,87),(96,110),(97,106),(98,107),(99,108),(100,109),(101,113),(102,114),(103,115),(104,111),(105,112),(116,130),(117,126),(118,127),(119,128),(120,129),(121,133),(122,134),(123,135),(124,131),(125,132),(136,143),(137,144),(138,145),(139,141),(140,142),(146,151),(147,152),(148,153),(149,154),(150,155)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,143,16,150),(7,144,17,146),(8,145,18,147),(9,141,19,148),(10,142,20,149),(11,136,156,155),(12,137,157,151),(13,138,158,152),(14,139,159,153),(15,140,160,154),(21,48,28,41),(22,49,29,42),(23,50,30,43),(24,46,26,44),(25,47,27,45),(56,95,75,76),(57,91,71,77),(58,92,72,78),(59,93,73,79),(60,94,74,80),(61,88,68,81),(62,89,69,82),(63,90,70,83),(64,86,66,84),(65,87,67,85),(96,116,115,135),(97,117,111,131),(98,118,112,132),(99,119,113,133),(100,120,114,134),(101,121,108,128),(102,122,109,129),(103,123,110,130),(104,124,106,126),(105,125,107,127)])
 

Matrix representation ►G ⊆ GL6(𝔽41)

3700000
0370000
0010000
0001000
0000100
0000010
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
3390000
4380000
0004000
0040000
000001
000010
,
3200000
0320000
000010
000001
0040000
0004000
,
100000
3400000
001000
000100
0000400
0000040
,
100000
010000
000100
0040000
000001
0000400

G:=sub<GL(6,GF(41))| [37,0,0,0,0,0,0,37,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,4,0,0,0,0,39,38,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0] >;
 

110 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G···4O5A5B5C5D10A···10L10M···10X20A···20X20Y···20BH
order12222224···44···4555510···1010···1020···2020···20
size11114442···24···411111···14···42···24···4

110 irreducible representations

dim11111111111111111111224444
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C10C10C4○D4C5×C4○D42+ (1+4)2- (1+4)C5×2+ (1+4)C5×2- (1+4)
kernelC5×C22.36C24C5×C42⋊C2D4×C20Q8×C20C5×C4⋊D4C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C42⋊2C2C5×C4⋊Q8C22.36C24C42⋊C2C4×D4C4×Q8C4⋊D4C22⋊Q8C22.D4C4.4D4C42⋊2C2C4⋊Q8C20C4C10C10C2C2
# reps11111323214444412812844161144

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{36}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.36C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1544);
 
// by ID
 
G=gap.SmallGroup(320,1544);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,891,2467,304]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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