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G = C2×D10⋊Q8order 320 = 26·5

Direct product of C2 and D10⋊Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D10⋊Q8, C4⋊C439D10, D101(C2×Q8), (C22×D5)⋊4Q8, C102(C22⋊Q8), C22.33(Q8×D5), (C2×C10).52C24, Dic5.82(C2×D4), C10.44(C22×D4), C22.134(D4×D5), C10.24(C22×Q8), (C2×C20).580C23, (C2×Dic5).246D4, (C22×Dic10)⋊7C2, (C22×C4).178D10, C22.86(C23×D5), (C2×Dic10)⋊50C22, C22.77(C4○D20), C10.D461C22, C23.329(C22×D5), D10⋊C4.92C22, (C22×C10).401C23, (C22×C20).434C22, (C2×Dic5).200C23, (C22×D5).168C23, (C23×D5).113C22, (C22×Dic5).81C22, C2.7(C2×Q8×D5), C2.17(C2×D4×D5), (C2×C4⋊C4)⋊17D5, C52(C2×C22⋊Q8), (C10×C4⋊C4)⋊14C2, (C5×C4⋊C4)⋊47C22, C10.21(C2×C4○D4), C2.23(C2×C4○D20), (C2×C10).93(C2×Q8), (D5×C22×C4).26C2, (C2×C10).390(C2×D4), (C2×C4×D5).371C22, (C2×C10.D4)⋊44C2, (C2×C4).142(C22×D5), (C2×D10⋊C4).20C2, (C2×C10).107(C4○D4), SmallGroup(320,1180)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×D10⋊Q8
C1C5C10C2×C10C22×D5C23×D5D5×C22×C4 — C2×D10⋊Q8
C5C2×C10 — C2×D10⋊Q8
C1C23C2×C4⋊C4

Generators and relations for C2×D10⋊Q8
 G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b3c, ece-1=b8c, ede-1=d-1 >

Subgroups: 1134 in 322 conjugacy classes, 127 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22, C22 [×6], C22 [×16], C5, C2×C4 [×6], C2×C4 [×28], Q8 [×8], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×11], C2×Q8 [×8], C24, Dic5 [×4], Dic5 [×4], C20 [×6], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C4×D5 [×8], C2×Dic5 [×10], C2×Dic5 [×4], C2×C20 [×6], C2×C20 [×6], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊Q8, C10.D4 [×8], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5 [×3], C22×C20 [×3], C23×D5, D10⋊Q8 [×8], C2×C10.D4 [×2], C2×D10⋊C4 [×2], C10×C4⋊C4, C22×Dic10, D5×C22×C4, C2×D10⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C22×D5 [×7], C2×C22⋊Q8, C4○D20 [×2], D4×D5 [×2], Q8×D5 [×2], C23×D5, D10⋊Q8 [×4], C2×C4○D20, C2×D4×D5, C2×Q8×D5, C2×D10⋊Q8

Smallest permutation representation of C2×D10⋊Q8
On 160 points
Generators in S160
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(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 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 21)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 151)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(31 45)(32 44)(33 43)(34 42)(35 41)(36 50)(37 49)(38 48)(39 47)(40 46)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(71 85)(72 84)(73 83)(74 82)(75 81)(76 90)(77 89)(78 88)(79 87)(80 86)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(111 125)(112 124)(113 123)(114 122)(115 121)(116 130)(117 129)(118 128)(119 127)(120 126)(131 150)(132 149)(133 148)(134 147)(135 146)(136 145)(137 144)(138 143)(139 142)(140 141)
(1 45 27 32)(2 44 28 31)(3 43 29 40)(4 42 30 39)(5 41 21 38)(6 50 22 37)(7 49 23 36)(8 48 24 35)(9 47 25 34)(10 46 26 33)(11 145 152 132)(12 144 153 131)(13 143 154 140)(14 142 155 139)(15 141 156 138)(16 150 157 137)(17 149 158 136)(18 148 159 135)(19 147 160 134)(20 146 151 133)(51 86 64 73)(52 85 65 72)(53 84 66 71)(54 83 67 80)(55 82 68 79)(56 81 69 78)(57 90 70 77)(58 89 61 76)(59 88 62 75)(60 87 63 74)(91 118 104 121)(92 117 105 130)(93 116 106 129)(94 115 107 128)(95 114 108 127)(96 113 109 126)(97 112 110 125)(98 111 101 124)(99 120 102 123)(100 119 103 122)
(1 65 27 52)(2 64 28 51)(3 63 29 60)(4 62 30 59)(5 61 21 58)(6 70 22 57)(7 69 23 56)(8 68 24 55)(9 67 25 54)(10 66 26 53)(11 112 152 125)(12 111 153 124)(13 120 154 123)(14 119 155 122)(15 118 156 121)(16 117 157 130)(17 116 158 129)(18 115 159 128)(19 114 160 127)(20 113 151 126)(31 73 44 86)(32 72 45 85)(33 71 46 84)(34 80 47 83)(35 79 48 82)(36 78 49 81)(37 77 50 90)(38 76 41 89)(39 75 42 88)(40 74 43 87)(91 138 104 141)(92 137 105 150)(93 136 106 149)(94 135 107 148)(95 134 108 147)(96 133 109 146)(97 132 110 145)(98 131 101 144)(99 140 102 143)(100 139 103 142)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10N20A···20X
order12···2222244444444444444445510···1020···20
size11···110101010222244441010101020202020222···24···4

68 irreducible representations

dim1111111222222244
type++++++++-++++-
imageC1C2C2C2C2C2C2D4Q8D5C4○D4D10D10C4○D20D4×D5Q8×D5
kernelC2×D10⋊Q8D10⋊Q8C2×C10.D4C2×D10⋊C4C10×C4⋊C4C22×Dic10D5×C22×C4C2×Dic5C22×D5C2×C4⋊C4C2×C10C4⋊C4C22×C4C22C22C22
# reps18221114424861644

Matrix representation of C2×D10⋊Q8 in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
170000
34340000
0040000
0004000
0000400
0000040
,
170000
0400000
0040000
001100
000010
0000040
,
2410000
38170000
0025900
00171600
0000040
0000400
,
900000
19320000
009000
00323200
000010
000001

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,34,0,0,0,0,7,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,7,40,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[24,38,0,0,0,0,1,17,0,0,0,0,0,0,25,17,0,0,0,0,9,16,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[9,19,0,0,0,0,0,32,0,0,0,0,0,0,9,32,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×D10⋊Q8 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes Q_8
% in TeX

G:=Group("C2xD10:Q8");
// GroupNames label

G:=SmallGroup(320,1180);
// by ID

G=gap.SmallGroup(320,1180);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1571,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^3*c,e*c*e^-1=b^8*c,e*d*e^-1=d^-1>;
// generators/relations

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