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G = C5×C22.49C24  order 320 = 26·5

Direct product of C5 and C22.49C24

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

Aliases: C5×C22.49C24, C10.1662+ (1+4), C4⋊Q8⋊16C10, (C4×D4)⋊21C10, (D4×C20)⋊50C2, C4⋊D4⋊16C10, C4.4D4⋊13C10, C42.49(C2×C10), C42⋊C2⋊17C10, C20.326(C4○D4), (C2×C20).964C23, (C2×C10).375C24, (C4×C20).290C22, (D4×C10).222C22, C23.20(C22×C10), C22.49(C23×C10), (Q8×C10).185C22, C2.18(C5×2+ (1+4)), (C22×C20).460C22, (C22×C10).103C23, (C5×C4⋊Q8)⋊37C2, C4.38(C5×C4○D4), C4⋊C4.75(C2×C10), (C5×C4⋊D4)⋊43C2, C2.28(C10×C4○D4), (C2×D4).35(C2×C10), C10.247(C2×C4○D4), (C5×C4.4D4)⋊33C2, (C2×Q8).28(C2×C10), (C5×C42⋊C2)⋊38C2, C22⋊C4.25(C2×C10), (C5×C4⋊C4).409C22, (C22×C4).71(C2×C10), (C2×C4).37(C22×C10), (C5×C22⋊C4).157C22, SmallGroup(320,1557)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×C22.49C24
Chief series
C1 — C2 — C22 — C2×C10 — C22×C10 — C5×C22⋊C4 — C5×C4.4D4 — C5×C22.49C24
Lower central
C1 — C22 — C5×C22.49C24
Upper central
C1 — C2×C10 — C5×C22.49C24

Subgroups: 362 in 236 conjugacy classes, 150 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C5, C2×C4, C2×C4 [×10], C2×C4 [×8], D4 [×8], Q8 [×2], C23 [×4], C10, C10 [×2], C10 [×4], C42, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C20 [×4], C20 [×9], C2×C10, C2×C10 [×12], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, C2×C20, C2×C20 [×10], C2×C20 [×8], C5×D4 [×8], C5×Q8 [×2], C22×C10 [×4], C22.49C24, C4×C20, C4×C20 [×4], C5×C22⋊C4 [×12], C5×C4⋊C4 [×6], C22×C20 [×4], D4×C10 [×6], Q8×C10 [×2], C5×C42⋊C2 [×4], D4×C20 [×2], C5×C4⋊D4 [×4], C5×C4.4D4 [×4], C5×C4⋊Q8, C5×C22.49C24

Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2+ (1+4), C22×C10 [×15], C22.49C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2+ (1+4), C5×C22.49C24

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

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

(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)(6 130 16 123)(7 126 17 124)(8 127 18 125)(9 128 19 121)(10 129 20 122)(11 135 156 116)(12 131 157 117)(13 132 158 118)(14 133 159 119)(15 134 160 120)(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)(96 155 115 136)(97 151 111 137)(98 152 112 138)(99 153 113 139)(100 154 114 140)(101 148 108 141)(102 149 109 142)(103 150 110 143)(104 146 106 144)(105 147 107 145)

(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), (1,156)(2,157)(3,158)(4,159)(5,160)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,23)(17,24)(18,25)(19,21)(20,22)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,153)(54,154)(55,155)(56,123)(57,124)(58,125)(59,121)(60,122)(61,119)(62,120)(63,116)(64,117)(65,118)(66,131)(67,132)(68,133)(69,134)(70,135)(71,126)(72,127)(73,128)(74,129)(75,130)(76,103)(77,104)(78,105)(79,101)(80,102)(81,99)(82,100)(83,96)(84,97)(85,98)(86,111)(87,112)(88,113)(89,114)(90,115)(91,106)(92,107)(93,108)(94,109)(95,110), (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)(6,130,16,123)(7,126,17,124)(8,127,18,125)(9,128,19,121)(10,129,20,122)(11,135,156,116)(12,131,157,117)(13,132,158,118)(14,133,159,119)(15,134,160,120)(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)(96,155,115,136)(97,151,111,137)(98,152,112,138)(99,153,113,139)(100,154,114,140)(101,148,108,141)(102,149,109,142)(103,150,110,143)(104,146,106,144)(105,147,107,145), (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), (1,156)(2,157)(3,158)(4,159)(5,160)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,23)(17,24)(18,25)(19,21)(20,22)(36,136)(37,137)(38,138)(39,139)(40,140)(41,141)(42,142)(43,143)(44,144)(45,145)(46,146)(47,147)(48,148)(49,149)(50,150)(51,151)(52,152)(53,153)(54,154)(55,155)(56,123)(57,124)(58,125)(59,121)(60,122)(61,119)(62,120)(63,116)(64,117)(65,118)(66,131)(67,132)(68,133)(69,134)(70,135)(71,126)(72,127)(73,128)(74,129)(75,130)(76,103)(77,104)(78,105)(79,101)(80,102)(81,99)(82,100)(83,96)(84,97)(85,98)(86,111)(87,112)(88,113)(89,114)(90,115)(91,106)(92,107)(93,108)(94,109)(95,110), (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)(6,130,16,123)(7,126,17,124)(8,127,18,125)(9,128,19,121)(10,129,20,122)(11,135,156,116)(12,131,157,117)(13,132,158,118)(14,133,159,119)(15,134,160,120)(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)(96,155,115,136)(97,151,111,137)(98,152,112,138)(99,153,113,139)(100,154,114,140)(101,148,108,141)(102,149,109,142)(103,150,110,143)(104,146,106,144)(105,147,107,145), (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)], [(1,156),(2,157),(3,158),(4,159),(5,160),(6,30),(7,26),(8,27),(9,28),(10,29),(11,35),(12,31),(13,32),(14,33),(15,34),(16,23),(17,24),(18,25),(19,21),(20,22),(36,136),(37,137),(38,138),(39,139),(40,140),(41,141),(42,142),(43,143),(44,144),(45,145),(46,146),(47,147),(48,148),(49,149),(50,150),(51,151),(52,152),(53,153),(54,154),(55,155),(56,123),(57,124),(58,125),(59,121),(60,122),(61,119),(62,120),(63,116),(64,117),(65,118),(66,131),(67,132),(68,133),(69,134),(70,135),(71,126),(72,127),(73,128),(74,129),(75,130),(76,103),(77,104),(78,105),(79,101),(80,102),(81,99),(82,100),(83,96),(84,97),(85,98),(86,111),(87,112),(88,113),(89,114),(90,115),(91,106),(92,107),(93,108),(94,109),(95,110)], [(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),(6,130,16,123),(7,126,17,124),(8,127,18,125),(9,128,19,121),(10,129,20,122),(11,135,156,116),(12,131,157,117),(13,132,158,118),(14,133,159,119),(15,134,160,120),(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),(96,155,115,136),(97,151,111,137),(98,152,112,138),(99,153,113,139),(100,154,114,140),(101,148,108,141),(102,149,109,142),(103,150,110,143),(104,146,106,144),(105,147,107,145)], [(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 ⊆ GL4(𝔽41) generated by

16000
01600
00370
00037
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
91800
323200
0009
00320
,
1000
404000
00320
00032
,
9000
0900
0001
0010
,
403900
1100
00400
00040
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,32,0,0,18,32,0,0,0,0,0,32,0,0,9,0],[1,40,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,0,0,1,0,0,1,0],[40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40] >;
 

125 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4L4M···4Q5A5B5C5D10A···10L10M···10AB20A···20AV20AW···20BP
order122222224···44···4555510···1010···1020···2020···20
size111144442···24···411111···14···42···24···4

125 irreducible representations

dim1111111111112244
type+++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10C4○D4C5×C4○D42+ (1+4)C5×2+ (1+4)
kernelC5×C22.49C24C5×C42⋊C2D4×C20C5×C4⋊D4C5×C4.4D4C5×C4⋊Q8C22.49C24C42⋊C2C4×D4C4⋊D4C4.4D4C4⋊Q8C20C4C10C2
# reps14244141681616483214

In GAP, Magma, Sage, TeX 

C_5\times C_2^2._{49}C_2^4
 
% in TeX
 
G:=Group("C5xC2^2.49C2^4");
 
// GroupNames label
 
G:=SmallGroup(320,1557);
 
// by ID
 
G=gap.SmallGroup(320,1557);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,436,1242,304]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=1,e^2=c,f^2=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,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g^-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,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀