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G = C2×C22.D20  order 320 = 26·5

Direct product of C2 and C22.D20

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

Aliases: C2×C22.D20, C23.51D20, C24.55D10, C22⋊C4⋊42D10, C10.7(C22×D4), C2.9(C22×D20), (C2×C10).36C24, C4⋊Dic5⋊52C22, (C23×Dic5)⋊4C2, C22.66(C2×D20), (C2×C20).129C23, (C22×C4).171D10, (C22×C10).117D4, (C2×Dic5).9C23, D10⋊C4⋊48C22, (C22×D5).8C23, C22.75(C23×D5), C10⋊2(C22.D4), (C23×C10).62C22, (C22×C20).72C22, (C23×D5).33C22, C23.147(C22×D5), C22.69(D4⋊2D5), (C22×C10).126C23, (C22×Dic5)⋊42C22, (C2×C4⋊Dic5)⋊20C2, (C2×C22⋊C4)⋊15D5, C10.68(C2×C4○D4), (C2×C10).48(C2×D4), (C10×C22⋊C4)⋊14C2, C5⋊2(C2×C22.D4), C2.11(C2×D4⋊2D5), (C2×D10⋊C4)⋊19C2, (C5×C22⋊C4)⋊47C22, (C2×C4).135(C22×D5), (C2×C5⋊D4).99C22, (C22×C5⋊D4).12C2, (C2×C10).168(C4○D4), SmallGroup(320,1164)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×C22.D20
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×C5⋊D4 — C2×C22.D20
Lower central
C5 — C2×C10 — C2×C22.D20

Subgroups: 1214 in 342 conjugacy classes, 127 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C5, C2×C4 [×4], C2×C4 [×24], D4 [×8], C23, C23 [×6], C23 [×12], D5 [×2], C10, C10 [×6], C10 [×4], 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 [×10], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×Dic5 [×6], C2×Dic5 [×14], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×4], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22.D4, C4⋊Dic5 [×8], D10⋊C4 [×8], C5×C22⋊C4 [×4], C22×Dic5, C22×Dic5 [×6], C22×Dic5 [×4], C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×C10, C22.D20 [×8], C2×C4⋊Dic5 [×2], C2×D10⋊C4 [×2], C10×C22⋊C4, C23×Dic5, C22×C5⋊D4, C2×C22.D20

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], D20 [×4], C22×D5 [×7], C2×C22.D4, C2×D20 [×6], D4⋊2D5 [×4], C23×D5, C22.D20 [×4], C22×D20, C2×D4⋊2D5 [×2], C2×C22.D20

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
010000
100000
001000
000100
000010
000001
,
4000000
0400000
001000
000100
000010
000001
,
0320000
900000
00344000
001000
00001139
00001627
,
0320000
3200000
00344000
007700
00003216
0000369

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,11,16,0,0,0,0,39,27],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,32,36,0,0,0,0,16,9] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4L4M4N5A5B10A···10N10O···10V20A···20P
order12···222222244444···4445510···1010···1020···20
size11···122222020444410···102020222···24···44···4

68 irreducible representations

dim111111122222224
type+++++++++++++-
imageC1C2C2C2C2C2C2D4D5C4○D4D10D10D10D20D4⋊2D5
kernelC2×C22.D20C22.D20C2×C4⋊Dic5C2×D10⋊C4C10×C22⋊C4C23×Dic5C22×C5⋊D4C22×C10C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C23C22
# reps1822111428842168

In GAP, Magma, Sage, TeX 

C_2\times C_2^2.D_{20}
 
% in TeX
 
G:=Group("C2xC2^2.D20");
 
// GroupNames label
 
G:=SmallGroup(320,1164);
 
// by ID
 
G=gap.SmallGroup(320,1164);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,297,192,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
 
// generators/relations
 

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