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G = C5×C23.7Q8order 320 = 26·5

Direct product of C5 and C23.7Q8

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

Aliases: C5×C23.7Q8, (C22×C4)⋊8C20, (C22×C20)⋊30C4, (C2×C20).511D4, C23.7(C5×Q8), (C23×C4).7C10, C23.21(C5×D4), C2012(C22⋊C4), C22.9(Q8×C10), C24.24(C2×C10), C23.25(C2×C20), (C23×C20).22C2, C22.30(D4×C10), (C22×C10).19Q8, C2.C421C10, (C22×C10).126D4, C10.82(C22⋊Q8), C10.132(C4⋊D4), C22.29(C22×C20), (C23×C10).84C22, C23.53(C22×C10), C10.72(C42⋊C2), (C22×C10).444C23, (C22×C20).573C22, (C2×C4⋊C4)⋊1C10, C2.4(C10×C4⋊C4), C42(C5×C22⋊C4), C221(C5×C4⋊C4), (C10×C4⋊C4)⋊28C2, (C2×C10)⋊7(C4⋊C4), C10.82(C2×C4⋊C4), C2.1(C5×C4⋊D4), (C2×C4).55(C2×C20), C2.1(C5×C22⋊Q8), (C2×C4).116(C5×D4), C2.5(C10×C22⋊C4), (C2×C20).503(C2×C4), (C2×C10).597(C2×D4), (C2×C22⋊C4).3C10, (C10×C22⋊C4).9C2, C2.5(C5×C42⋊C2), (C2×C10).101(C2×Q8), C22.15(C5×C4○D4), C10.133(C2×C22⋊C4), (C5×C2.C42)⋊3C2, (C22×C4).84(C2×C10), (C2×C10).205(C4○D4), (C2×C10).317(C22×C4), (C22×C10).179(C2×C4), SmallGroup(320,881)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C23.7Q8
C1C2C22C23C22×C10C22×C20C10×C4⋊C4 — C5×C23.7Q8
C1C22 — C5×C23.7Q8
C1C22×C10 — C5×C23.7Q8

Generators and relations for C5×C23.7Q8
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 370 in 234 conjugacy classes, 114 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C20 [×4], C20 [×6], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×C20 [×8], C2×C20 [×22], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.7Q8, C5×C22⋊C4 [×4], C5×C4⋊C4 [×4], C22×C20 [×2], C22×C20 [×8], C22×C20 [×4], C23×C10, C5×C2.C42 [×2], C10×C22⋊C4 [×2], C10×C4⋊C4 [×2], C23×C20, C5×C23.7Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×6], Q8 [×2], C23, C10 [×7], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C20 [×4], C2×C10 [×7], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×C20 [×6], C5×D4 [×6], C5×Q8 [×2], C22×C10, C23.7Q8, C5×C22⋊C4 [×4], C5×C4⋊C4 [×4], C22×C20, D4×C10 [×3], Q8×C10, C5×C4○D4 [×2], C10×C22⋊C4, C10×C4⋊C4, C5×C42⋊C2, C5×C4⋊D4 [×2], C5×C22⋊Q8 [×2], C5×C23.7Q8

Smallest permutation representation of C5×C23.7Q8
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 54)(2 55)(3 51)(4 52)(5 53)(6 31)(7 32)(8 33)(9 34)(10 35)(11 28)(12 29)(13 30)(14 26)(15 27)(16 23)(17 24)(18 25)(19 21)(20 22)(36 41)(37 42)(38 43)(39 44)(40 45)(46 70)(47 66)(48 67)(49 68)(50 69)(56 63)(57 64)(58 65)(59 61)(60 62)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 112)(97 113)(98 114)(99 115)(100 111)(101 119)(102 120)(103 116)(104 117)(105 118)(121 127)(122 128)(123 129)(124 130)(125 126)(131 148)(132 149)(133 150)(134 146)(135 147)(136 152)(137 153)(138 154)(139 155)(140 151)(141 159)(142 160)(143 156)(144 157)(145 158)
(1 54)(2 55)(3 51)(4 52)(5 53)(6 22)(7 23)(8 24)(9 25)(10 21)(11 28)(12 29)(13 30)(14 26)(15 27)(16 32)(17 33)(18 34)(19 35)(20 31)(36 41)(37 42)(38 43)(39 44)(40 45)(46 70)(47 66)(48 67)(49 68)(50 69)(56 63)(57 64)(58 65)(59 61)(60 62)(71 80)(72 76)(73 77)(74 78)(75 79)(81 94)(82 95)(83 91)(84 92)(85 93)(86 110)(87 106)(88 107)(89 108)(90 109)(96 103)(97 104)(98 105)(99 101)(100 102)(111 120)(112 116)(113 117)(114 118)(115 119)(121 134)(122 135)(123 131)(124 132)(125 133)(126 150)(127 146)(128 147)(129 148)(130 149)(136 143)(137 144)(138 145)(139 141)(140 142)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 20)(7 16)(8 17)(9 18)(10 19)(11 41)(12 42)(13 43)(14 44)(15 45)(21 35)(22 31)(23 32)(24 33)(25 34)(26 39)(27 40)(28 36)(29 37)(30 38)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 116)(97 117)(98 118)(99 119)(100 120)(101 115)(102 111)(103 112)(104 113)(105 114)(121 146)(122 147)(123 148)(124 149)(125 150)(126 133)(127 134)(128 135)(129 131)(130 132)(136 156)(137 157)(138 158)(139 159)(140 160)(141 155)(142 151)(143 152)(144 153)(145 154)
(1 64 12 81)(2 65 13 82)(3 61 14 83)(4 62 15 84)(5 63 11 85)(6 103 143 125)(7 104 144 121)(8 105 145 122)(9 101 141 123)(10 102 142 124)(16 113 153 146)(17 114 154 147)(18 115 155 148)(19 111 151 149)(20 112 152 150)(21 100 140 132)(22 96 136 133)(23 97 137 134)(24 98 138 135)(25 99 139 131)(26 91 51 59)(27 92 52 60)(28 93 53 56)(29 94 54 57)(30 95 55 58)(31 116 156 126)(32 117 157 127)(33 118 158 128)(34 119 159 129)(35 120 160 130)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 110 70 72)(42 106 66 73)(43 107 67 74)(44 108 68 75)(45 109 69 71)
(1 97 29 121)(2 98 30 122)(3 99 26 123)(4 100 27 124)(5 96 28 125)(6 85 136 56)(7 81 137 57)(8 82 138 58)(9 83 139 59)(10 84 140 60)(11 133 53 103)(12 134 54 104)(13 135 55 105)(14 131 51 101)(15 132 52 102)(16 106 157 77)(17 107 158 78)(18 108 159 79)(19 109 160 80)(20 110 156 76)(21 92 142 62)(22 93 143 63)(23 94 144 64)(24 95 145 65)(25 91 141 61)(31 86 152 72)(32 87 153 73)(33 88 154 74)(34 89 155 75)(35 90 151 71)(36 150 70 116)(37 146 66 117)(38 147 67 118)(39 148 68 119)(40 149 69 120)(41 126 46 112)(42 127 47 113)(43 128 48 114)(44 129 49 115)(45 130 50 111)

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

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

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

140 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B5C5D10A···10AB10AC···10AR20A···20AF20AG···20BL
order12···222224···44···4555510···1010···1020···2020···20
size11···122222···24···411111···12···22···24···4

140 irreducible representations

dim11111111111122222222
type+++++++-
imageC1C2C2C2C2C4C5C10C10C10C10C20D4D4Q8C4○D4C5×D4C5×D4C5×Q8C5×C4○D4
kernelC5×C23.7Q8C5×C2.C42C10×C22⋊C4C10×C4⋊C4C23×C20C22×C20C23.7Q8C2.C42C2×C22⋊C4C2×C4⋊C4C23×C4C22×C4C2×C20C22×C10C22×C10C2×C10C2×C4C23C23C22
# reps12221848884324224168816

Matrix representation of C5×C23.7Q8 in GL5(𝔽41)

10000
010000
001000
000160
000016
,
10000
01000
004000
000400
000040
,
400000
01000
00100
000400
000040
,
10000
040000
004000
00010
00001
,
400000
01000
00100
0004039
00011
,
320000
00100
01000
0001638
000325

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,1,0,0,0,39,1],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,3,0,0,0,38,25] >;

C5×C23.7Q8 in GAP, Magma, Sage, TeX

C_5\times C_2^3._7Q_8
% in TeX

G:=Group("C5xC2^3.7Q8");
// GroupNames label

G:=SmallGroup(320,881);
// by ID

G=gap.SmallGroup(320,881);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽