Copied to
clipboard

G = C232D20order 320 = 26·5

1st semidirect product of C23 and D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C232D20, C24.15D10, (C2×C20)⋊6D4, (C2×Dic5)⋊5D4, (C22×D5)⋊4D4, (C22×C10)⋊7D4, (C22×D20)⋊3C2, C52(C232D4), C10.33C22≀C2, C2.6(C20⋊D4), C2.7(C207D4), (C22×C4).35D10, C22.242(D4×D5), C2.8(C23⋊D10), C10.59(C4⋊D4), C10.13(C41D4), C22.126(C2×D20), C2.34(C22⋊D20), C2.34(D10⋊D4), (C23×C10).43C22, (C22×C20).61C22, (C23×D5).16C22, C23.372(C22×D5), C10.10C4232C2, C22.100(C4○D20), (C22×C10).334C23, (C22×Dic5).46C22, (C2×C4)⋊3(C5⋊D4), (C2×C22⋊C4)⋊8D5, (C22×C5⋊D4)⋊1C2, (C2×D10⋊C4)⋊8C2, (C10×C22⋊C4)⋊11C2, (C2×C10).325(C2×D4), (C2×C10).80(C4○D4), C22.128(C2×C5⋊D4), SmallGroup(320,587)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C232D20
C1C5C10C2×C10C22×C10C23×D5C22×D20 — C232D20
C5C22×C10 — C232D20
C1C23C2×C22⋊C4

Generators and relations for C232D20
 G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1526 in 322 conjugacy classes, 67 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×7], C22 [×3], C22 [×4], C22 [×30], C5, C2×C4 [×2], C2×C4 [×13], D4 [×24], C23, C23 [×2], C23 [×22], D5 [×4], C10 [×3], C10 [×4], C10 [×2], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×2], C2×D4 [×18], C24, C24 [×2], Dic5 [×4], C20 [×3], D10 [×20], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C22×D4 [×3], D20 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×5], C22×D5 [×4], C22×D5 [×12], C22×C10, C22×C10 [×2], C22×C10 [×6], C232D4, D10⋊C4 [×4], C5×C22⋊C4 [×2], C2×D20 [×6], C22×Dic5 [×2], C2×C5⋊D4 [×12], C22×C20 [×2], C23×D5 [×2], C23×C10, C10.10C42, C2×D10⋊C4 [×2], C10×C22⋊C4, C22×D20, C22×C5⋊D4 [×2], C232D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, D5, C2×D4 [×6], C4○D4, D10 [×3], C22≀C2 [×3], C4⋊D4 [×3], C41D4, D20 [×2], C5⋊D4 [×2], C22×D5, C232D4, C2×D20, C4○D20, D4×D5 [×4], C2×C5⋊D4, C22⋊D20 [×2], D10⋊D4 [×2], C207D4, C23⋊D10, C20⋊D4, C232D20

Smallest permutation representation of C232D20
On 160 points
Generators in S160
(1 58)(2 77)(3 60)(4 79)(5 42)(6 61)(7 44)(8 63)(9 46)(10 65)(11 48)(12 67)(13 50)(14 69)(15 52)(16 71)(17 54)(18 73)(19 56)(20 75)(21 152)(22 139)(23 154)(24 121)(25 156)(26 123)(27 158)(28 125)(29 160)(30 127)(31 142)(32 129)(33 144)(34 131)(35 146)(36 133)(37 148)(38 135)(39 150)(40 137)(41 108)(43 110)(45 112)(47 114)(49 116)(51 118)(53 120)(55 102)(57 104)(59 106)(62 111)(64 113)(66 115)(68 117)(70 119)(72 101)(74 103)(76 105)(78 107)(80 109)(81 153)(82 140)(83 155)(84 122)(85 157)(86 124)(87 159)(88 126)(89 141)(90 128)(91 143)(92 130)(93 145)(94 132)(95 147)(96 134)(97 149)(98 136)(99 151)(100 138)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 92)(42 93)(43 94)(44 95)(45 96)(46 97)(47 98)(48 99)(49 100)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 121)(120 122)
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 101)(18 102)(19 103)(20 104)(21 100)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 151)(138 152)(139 153)(140 154)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 85)(42 84)(43 83)(44 82)(45 81)(46 100)(47 99)(48 98)(49 97)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 140)(134 139)(135 138)(136 137)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H5A5B10A···10N10O···10V20A···20P
order12···2222222444444445510···1010···1020···20
size11···14420202020444420202020222···24···44···4

62 irreducible representations

dim111111222222222224
type+++++++++++++++
imageC1C2C2C2C2C2D4D4D4D4D5C4○D4D10D10C5⋊D4D20C4○D20D4×D5
kernelC232D20C10.10C42C2×D10⋊C4C10×C22⋊C4C22×D20C22×C5⋊D4C2×Dic5C2×C20C22×D5C22×C10C2×C22⋊C4C2×C10C22×C4C24C2×C4C23C22C22
# reps112112424222428888

Matrix representation of C232D20 in GL6(𝔽41)

010000
100000
00403900
000100
0000241
00004017
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
0040000
0004000
000010
000001
,
4000000
0400000
001200
00404000
0000911
00003014
,
4000000
010000
001200
0004000
0000911
00003032

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,2,40,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;

C232D20 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2D_{20}
% in TeX

G:=Group("C2^3:2D20");
// GroupNames label

G:=SmallGroup(320,587);
// by ID

G=gap.SmallGroup(320,587);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽