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G = C5×C22.33C24  order 320 = 26·5

Direct product of C5 and C22.33C24

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

Aliases: C5×C22.33C24, C10.1552+ (1+4), C10.1132- (1+4), (D4×C20)⋊40C2, (C4×D4)⋊11C10, C22⋊Q8⋊8C10, C4⋊D4.8C10, C42.C2⋊4C10, C42⋊2C2⋊2C10, C42.37(C2×C10), (C2×C10).359C24, (C2×C20).668C23, (C4×C20).278C22, C22.D4⋊5C10, C2.7(C5×2+ (1+4)), C2.5(C5×2- (1+4)), (D4×C10).217C22, C23.38(C22×C10), (C22×C20).38C22, (C22×C10).94C23, C22.33(C23×C10), (Q8×C10).181C22, (C2×C4⋊C4)⋊19C10, (C10×C4⋊C4)⋊46C2, C4⋊C4.68(C2×C10), C2.16(C10×C4○D4), (C5×C22⋊Q8)⋊35C2, C22.5(C5×C4○D4), (C2×D4).31(C2×C10), C10.235(C2×C4○D4), (C5×C4⋊D4).18C2, (C2×Q8).25(C2×C10), (C5×C42.C2)⋊21C2, (C5×C42⋊2C2)⋊13C2, (C2×C10).53(C4○D4), (C5×C4⋊C4).392C22, C22⋊C4.15(C2×C10), (C22×C4).59(C2×C10), (C2×C4).26(C22×C10), (C5×C22.D4)⋊24C2, (C5×C22⋊C4).85C22, SmallGroup(320,1541)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.33C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C42⋊2C2 — C5×C22.33C24
Lower central
C1 — C22 — C5×C22.33C24
Upper central
C1 — C2×C10 — C5×C22.33C24

Subgroups: 322 in 218 conjugacy classes, 146 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×10], C2×C4 [×6], D4 [×5], Q8, C23, C23 [×2], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C20 [×12], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C2×C20 [×2], C2×C20 [×10], C2×C20 [×6], C5×D4 [×5], C5×Q8, C22×C10, C22×C10 [×2], C22.33C24, C4×C20 [×2], C5×C22⋊C4 [×10], C5×C4⋊C4 [×2], C5×C4⋊C4 [×12], C22×C20 [×3], C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C10×C4⋊C4, D4×C20 [×2], C5×C4⋊D4, C5×C22⋊Q8, C5×C22⋊Q8 [×2], C5×C22.D4 [×4], C5×C42.C2 [×2], C5×C42⋊2C2 [×2], C5×C22.33C24

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], C22.33C24, C5×C4○D4 [×2], C23×C10, C10×C4○D4, C5×2+ (1+4), C5×2- (1+4), C5×C22.33C24

Generators and relations
 G = < a,b,c,d,e,f,g | a5=b2=c2=d2=g2=1, e2=c, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef-1=be=eb, bf=fb, bg=gb, fdf-1=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)

(6 156)(7 157)(8 158)(9 159)(10 160)(11 16)(12 17)(13 18)(14 19)(15 20)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)(76 83)(77 84)(78 85)(79 81)(80 82)(86 91)(87 92)(88 93)(89 94)(90 95)(96 115)(97 111)(98 112)(99 113)(100 114)(101 108)(102 109)(103 110)(104 106)(105 107)(136 143)(137 144)(138 145)(139 141)(140 142)(146 151)(147 152)(148 153)(149 154)(150 155)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

100000
010000
0037000
0003700
0000370
0000037
,
100000
010000
0040000
0004000
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
100000
1400000
001000
0004000
000010
0000040
,
900000
090000
00024032
00240320
00032017
00320170
,
1390000
0400000
000010
000001
0040000
0004000
,
100000
010000
000100
001000
000001
000010

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[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],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,24,0,32,0,0,24,0,32,0,0,0,0,32,0,17,0,0,32,0,17,0],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

110 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N5A5B5C5D10A···10L10M···10T10U···10AB20A···20P20Q···20BD
order1222222244444···4555510···1010···1010···1020···2020···20
size1111224422224···411111···12···24···42···24···4

110 irreducible representations

dim1111111111111111224444
type+++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C4○D4C5×C4○D42+ (1+4)2- (1+4)C5×2+ (1+4)C5×2- (1+4)
kernelC5×C22.33C24C10×C4⋊C4D4×C20C5×C4⋊D4C5×C22⋊Q8C5×C22.D4C5×C42.C2C5×C42⋊2C2C22.33C24C2×C4⋊C4C4×D4C4⋊D4C22⋊Q8C22.D4C42.C2C42⋊2C2C2×C10C22C10C10C2C2
# reps1121342244841216884161144

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{33}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.33C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1541);
 
// by ID
 
G=gap.SmallGroup(320,1541);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,891,436,2467]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=g^2=1,e^2=c,f^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,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=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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