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G = C4.(D4×D5)  order 320 = 26·5

28th non-split extension by C4 of D4×D5 acting via D4×D5/C22×D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C85D4, C54(C8⋊D4), C4⋊C4.62D10, C4⋊D4.7D5, C4.173(D4×D5), (C2×C20).74D4, (C2×D4).42D10, C20.151(C2×D4), C10.D837C2, D4⋊Dic518C2, C10.Q1636C2, (C22×C10).88D4, C20.186(C4○D4), C4.62(D42D5), C20.48D425C2, C10.96(C4⋊D4), C10.92(C8⋊C22), (C2×C20).361C23, (D4×C10).58C22, (C22×C4).123D10, C23.25(C5⋊D4), C4⋊Dic5.145C22, C2.17(Dic5⋊D4), C2.13(D4.D10), C2.13(D4.9D10), C10.115(C8.C22), (C22×C20).165C22, (C2×Dic10).108C22, (C2×D4.D5)⋊11C2, (C5×C4⋊D4).6C2, (C2×C10).492(C2×D4), (C2×C4).52(C5⋊D4), (C2×C4.Dic5)⋊12C2, (C5×C4⋊C4).109C22, (C2×C4).461(C22×D5), C22.167(C2×C5⋊D4), (C2×C52C8).111C22, SmallGroup(320,669)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4.(D4×D5)
C1C5C10C20C2×C20C2×Dic10C20.48D4 — C4.(D4×D5)
C5C10C2×C20 — C4.(D4×D5)
C1C22C22×C4C4⋊D4

Generators and relations for C4.(D4×D5)
 G = < a,b,c,d,e | a4=b4=c2=d5=1, e2=a, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=ab, cd=dc, ece-1=a2c, ede-1=d-1 >

Subgroups: 406 in 120 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×3], C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8 [×2], C52C8, Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×3], C5×D4 [×4], C22×C10, C22×C10, C8⋊D4, C2×C52C8 [×2], C4.Dic5 [×2], C10.D4, C4⋊Dic5, D4.D5 [×2], C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, D4×C10, C10.D8, C10.Q16, D4⋊Dic5, C2×C4.Dic5, C20.48D4, C2×D4.D5, C5×C4⋊D4, C4.(D4×D5)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, C5⋊D4 [×2], C22×D5, C8⋊D4, D4×D5, D42D5, C2×C5⋊D4, D4.D10, Dic5⋊D4, D4.9D10, C4.(D4×D5)

Smallest permutation representation of C4.(D4×D5)
On 160 points
Generators in S160
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(1 104 46 10)(2 103 47 9)(3 102 48 16)(4 101 41 15)(5 100 42 14)(6 99 43 13)(7 98 44 12)(8 97 45 11)(17 127 140 57)(18 126 141 64)(19 125 142 63)(20 124 143 62)(21 123 144 61)(22 122 137 60)(23 121 138 59)(24 128 139 58)(25 39 159 78)(26 38 160 77)(27 37 153 76)(28 36 154 75)(29 35 155 74)(30 34 156 73)(31 33 157 80)(32 40 158 79)(49 131 115 69)(50 130 116 68)(51 129 117 67)(52 136 118 66)(53 135 119 65)(54 134 120 72)(55 133 113 71)(56 132 114 70)(81 149 92 110)(82 148 93 109)(83 147 94 108)(84 146 95 107)(85 145 96 106)(86 152 89 105)(87 151 90 112)(88 150 91 111)
(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)(17 121)(18 126)(19 123)(20 128)(21 125)(22 122)(23 127)(24 124)(25 39)(26 36)(27 33)(28 38)(29 35)(30 40)(31 37)(32 34)(41 103)(42 100)(43 97)(44 102)(45 99)(46 104)(47 101)(48 98)(49 129)(50 134)(51 131)(52 136)(53 133)(54 130)(55 135)(56 132)(57 138)(58 143)(59 140)(60 137)(61 142)(62 139)(63 144)(64 141)(65 113)(66 118)(67 115)(68 120)(69 117)(70 114)(71 119)(72 116)(73 158)(74 155)(75 160)(76 157)(77 154)(78 159)(79 156)(80 153)(81 108)(82 105)(83 110)(84 107)(85 112)(86 109)(87 106)(88 111)(89 148)(90 145)(91 150)(92 147)(93 152)(94 149)(95 146)(96 151)
(1 66 35 95 64)(2 57 96 36 67)(3 68 37 89 58)(4 59 90 38 69)(5 70 39 91 60)(6 61 92 40 71)(7 72 33 93 62)(8 63 94 34 65)(9 140 145 28 117)(10 118 29 146 141)(11 142 147 30 119)(12 120 31 148 143)(13 144 149 32 113)(14 114 25 150 137)(15 138 151 26 115)(16 116 27 152 139)(17 106 154 51 103)(18 104 52 155 107)(19 108 156 53 97)(20 98 54 157 109)(21 110 158 55 99)(22 100 56 159 111)(23 112 160 49 101)(24 102 50 153 105)(41 121 87 77 131)(42 132 78 88 122)(43 123 81 79 133)(44 134 80 82 124)(45 125 83 73 135)(46 136 74 84 126)(47 127 85 75 129)(48 130 76 86 128)
(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)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222244444455888810···10101010101010101020···2020202020
size1111482248404022202020202···2444488884···48888

44 irreducible representations

dim111111112222222222444444
type++++++++++++++++-+--
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C5⋊D4C8⋊C22C8.C22D4×D5D42D5D4.D10D4.9D10
kernelC4.(D4×D5)C10.D8C10.Q16D4⋊Dic5C2×C4.Dic5C20.48D4C2×D4.D5C5×C4⋊D4C52C8C2×C20C22×C10C4⋊D4C20C4⋊C4C22×C4C2×D4C2×C4C23C10C10C4C4C2C2
# reps111111112112222244112244

Matrix representation of C4.(D4×D5) in GL8(𝔽41)

400000000
040000000
004000000
000400000
00000010
00000001
000040000
000004000
,
10900000
00010000
1804000000
040000000
000000228
0000001339
000022800
0000133900
,
10900000
00010000
004000000
01000000
0000003913
000000282
000022800
0000133900
,
370000000
2310000000
003700000
2302100000
000004000
00001600
000000040
00000016
,
21025110000
12026200000
02420170000
1220000000
000016172524
00003253816
000016171617
0000325325

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,18,0,0,0,0,0,0,0,0,40,0,0,0,0,9,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,28,39,0,0,0,0,2,13,0,0,0,0,0,0,28,39,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,9,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,28,39,0,0,0,0,39,28,0,0,0,0,0,0,13,2,0,0],[37,23,0,23,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37,2,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6],[21,12,0,12,0,0,0,0,0,0,24,20,0,0,0,0,25,26,20,0,0,0,0,0,11,20,17,0,0,0,0,0,0,0,0,0,16,3,16,3,0,0,0,0,17,25,17,25,0,0,0,0,25,38,16,3,0,0,0,0,24,16,17,25] >;

C4.(D4×D5) in GAP, Magma, Sage, TeX

C_4.(D_4\times D_5)
% in TeX

G:=Group("C4.(D4xD5)");
// GroupNames label

G:=SmallGroup(320,669);
// by ID

G=gap.SmallGroup(320,669);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,555,1123,297,136,12550]);
// Polycyclic

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

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