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## G = C2×D10.12D4order 320 = 26·5

### Direct product of C2 and D10.12D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×D10.12D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D10.12D4
 Lower central C5 — C2×C10 — C2×D10.12D4
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×D10.12D4
G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=b5d-1 >

Subgroups: 1214 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×10], C22, C22 [×6], C22 [×26], C5, C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×2], C23 [×16], D5 [×4], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×11], C2×D4 [×8], C24, C24, Dic5 [×6], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C22.D4, C10.D4 [×4], C4⋊Dic5 [×4], D10⋊C4 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5 [×3], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, D10.12D4 [×8], C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, D5×C22×C4, C22×C5⋊D4, C2×D10.12D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C2×C22.D4, C4○D20 [×2], D4×D5 [×2], D42D5 [×2], C23×D5, D10.12D4 [×4], C2×C4○D20, C2×D4×D5, C2×D42D5, C2×D10.12D4

Smallest permutation representation of C2×D10.12D4
On 160 points
Generators in S160
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 71)(19 72)(20 73)(21 100)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 122)(32 123)(33 124)(34 125)(35 126)(36 127)(37 128)(38 129)(39 130)(40 121)(41 120)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 141)(61 140)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(81 160)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)
(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)
(1 104)(2 103)(3 102)(4 101)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 80)(20 79)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 100)(29 99)(30 98)(31 122)(32 121)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 112)(42 111)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 147)(52 146)(53 145)(54 144)(55 143)(56 142)(57 141)(58 150)(59 149)(60 148)(61 137)(62 136)(63 135)(64 134)(65 133)(66 132)(67 131)(68 140)(69 139)(70 138)(81 152)(82 151)(83 160)(84 159)(85 158)(86 157)(87 156)(88 155)(89 154)(90 153)
(1 16 30 154)(2 17 21 155)(3 18 22 156)(4 19 23 157)(5 20 24 158)(6 11 25 159)(7 12 26 160)(8 13 27 151)(9 14 28 152)(10 15 29 153)(31 147 47 131)(32 148 48 132)(33 149 49 133)(34 150 50 134)(35 141 41 135)(36 142 42 136)(37 143 43 137)(38 144 44 138)(39 145 45 139)(40 146 46 140)(51 111 67 127)(52 112 68 128)(53 113 69 129)(54 114 70 130)(55 115 61 121)(56 116 62 122)(57 117 63 123)(58 118 64 124)(59 119 65 125)(60 120 66 126)(71 91 87 107)(72 92 88 108)(73 93 89 109)(74 94 90 110)(75 95 81 101)(76 96 82 102)(77 97 83 103)(78 98 84 104)(79 99 85 105)(80 100 86 106)
(1 59 6 54)(2 60 7 55)(3 51 8 56)(4 52 9 57)(5 53 10 58)(11 125 16 130)(12 126 17 121)(13 127 18 122)(14 128 19 123)(15 129 20 124)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)(41 86 46 81)(42 87 47 82)(43 88 48 83)(44 89 49 84)(45 90 50 85)(91 136 96 131)(92 137 97 132)(93 138 98 133)(94 139 99 134)(95 140 100 135)(101 146 106 141)(102 147 107 142)(103 148 108 143)(104 149 109 144)(105 150 110 145)(111 156 116 151)(112 157 117 152)(113 158 118 153)(114 159 119 154)(115 160 120 155)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 10 10 10 10 2 2 2 2 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C4○D20 D4×D5 D4⋊2D5 kernel C2×D10.12D4 D10.12D4 C2×C10.D4 C2×C4⋊Dic5 C2×D10⋊C4 C2×C23.D5 C10×C22⋊C4 D5×C22×C4 C22×C5⋊D4 C22×D5 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C22 C22 C22 # reps 1 8 1 1 1 1 1 1 1 4 2 8 8 4 2 16 4 4

Matrix representation of C2×D10.12D4 in GL6(𝔽41)

 40 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 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 34 0 0 0 0 7 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 7 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 0 32 0 0 0 0 32 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 23 0 0 0 0 9 9
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 9 9

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,23,9],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9] >;

C2×D10.12D4 in GAP, Magma, Sage, TeX

C_2\times D_{10}._{12}D_4
% in TeX

G:=Group("C2xD10.12D4");
// GroupNames label

G:=SmallGroup(320,1160);
// by ID

G=gap.SmallGroup(320,1160);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,675,297,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^4=1,e^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=b^5*d^-1>;
// generators/relations

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