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G = C5×C22.C42order 320 = 26·5

Direct product of C5 and C22.C42

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C22.C42, M4(2)⋊3C20, C20.84(C4⋊C4), (C2×C20).37Q8, (C2×C20).509D4, (C22×C20).6C4, (C22×C4).3C20, C22.2(C4×C20), (C5×M4(2))⋊15C4, (C2×C10).31C42, C23.23(C2×C20), (C2×M4(2)).8C10, C20.152(C22⋊C4), C10.22(C4.D4), (C10×M4(2)).20C2, C10.18(C4.10D4), (C22×C20).388C22, C10.46(C2.C42), C4.4(C5×C4⋊C4), (C2×C4⋊C4).3C10, (C2×C4).2(C5×Q8), C22.5(C5×C4⋊C4), (C10×C4⋊C4).30C2, (C2×C4).14(C2×C20), (C2×C4).114(C5×D4), C4.20(C5×C22⋊C4), C2.2(C5×C4.D4), (C2×C10).50(C4⋊C4), (C2×C20).354(C2×C4), C2.2(C5×C4.10D4), (C22×C4).18(C2×C10), C22.30(C5×C22⋊C4), C2.8(C5×C2.C42), (C22×C10).176(C2×C4), (C2×C10).196(C22⋊C4), SmallGroup(320,148)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C22.C42
C1C2C22C2×C4C22×C4C22×C20C10×M4(2) — C5×C22.C42
C1C2C22 — C5×C22.C42
C1C2×C10C22×C20 — C5×C22.C42

Generators and relations for C5×C22.C42
 G = < a,b,c,d,e | a5=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 154 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×2], C2×C4⋊C4, C2×M4(2) [×2], C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×C10, C22.C42, C5×C4⋊C4 [×2], C2×C40 [×2], C5×M4(2) [×4], C5×M4(2) [×2], C22×C20, C22×C20 [×2], C10×C4⋊C4, C10×M4(2) [×2], C5×C22.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2.C42, C4.D4, C4.10D4, C2×C20 [×3], C5×D4 [×3], C5×Q8, C22.C42, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×C2.C42, C5×C4.D4, C5×C4.10D4, C5×C22.C42

Smallest permutation representation of C5×C22.C42
On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 46 118 38 90)(10 47 119 39 91)(11 48 120 40 92)(12 41 113 33 93)(13 42 114 34 94)(14 43 115 35 95)(15 44 116 36 96)(16 45 117 37 89)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 80 152)
(1 127)(2 124)(3 121)(4 126)(5 123)(6 128)(7 125)(8 122)(9 156)(10 153)(11 158)(12 155)(13 160)(14 157)(15 154)(16 159)(17 55)(18 52)(19 49)(20 54)(21 51)(22 56)(23 53)(24 50)(25 63)(26 60)(27 57)(28 62)(29 59)(30 64)(31 61)(32 58)(33 67)(34 72)(35 69)(36 66)(37 71)(38 68)(39 65)(40 70)(41 75)(42 80)(43 77)(44 74)(45 79)(46 76)(47 73)(48 78)(81 93)(82 90)(83 95)(84 92)(85 89)(86 94)(87 91)(88 96)(97 135)(98 132)(99 129)(100 134)(101 131)(102 136)(103 133)(104 130)(105 143)(106 140)(107 137)(108 142)(109 139)(110 144)(111 141)(112 138)(113 147)(114 152)(115 149)(116 146)(117 151)(118 148)(119 145)(120 150)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(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 67 127 33)(2 38 128 72)(3 65 121 39)(4 36 122 70)(5 71 123 37)(6 34 124 68)(7 69 125 35)(8 40 126 66)(9 64 160 26)(10 27 153 57)(11 62 154 32)(12 25 155 63)(13 60 156 30)(14 31 157 61)(15 58 158 28)(16 29 159 59)(17 147 55 113)(18 118 56 152)(19 145 49 119)(20 116 50 150)(21 151 51 117)(22 114 52 148)(23 149 53 115)(24 120 54 146)(41 97 75 135)(42 132 76 102)(43 103 77 133)(44 130 78 100)(45 101 79 131)(46 136 80 98)(47 99 73 129)(48 134 74 104)(81 143 93 105)(82 110 94 140)(83 141 95 111)(84 108 96 138)(85 139 89 109)(86 106 90 144)(87 137 91 107)(88 112 92 142)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B5C5D8A···8H10A···10L10M···10T20A···20P20Q···20AF40A···40AF
order1222224444444455558···810···1010···1020···2020···2040···40
size1111222222444411114···41···12···22···24···44···4

110 irreducible representations

dim111111111122224444
type++++-+-
imageC1C2C2C4C4C5C10C10C20C20D4Q8C5×D4C5×Q8C4.D4C4.10D4C5×C4.D4C5×C4.10D4
kernelC5×C22.C42C10×C4⋊C4C10×M4(2)C5×M4(2)C22×C20C22.C42C2×C4⋊C4C2×M4(2)M4(2)C22×C4C2×C20C2×C20C2×C4C2×C4C10C10C2C2
# reps112844483216311241144

Matrix representation of C5×C22.C42 in GL8(𝔽41)

10000000
01000000
001000000
000100000
00001000
00000100
00000010
00000001
,
400000000
040000000
004000000
000400000
000040000
000004000
00000010
00000001
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
3040000000
4011000000
0030160000
0013110000
00000010
00000001
00000100
000040000
,
01000000
400000000
00990000
0023320000
000011100
000013000
0000003040
0000004011

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[30,40,0,0,0,0,0,0,40,11,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,16,11,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,23,0,0,0,0,0,0,9,32,0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,1,30,0,0,0,0,0,0,0,0,30,40,0,0,0,0,0,0,40,11] >;

C5×C22.C42 in GAP, Magma, Sage, TeX

C_5\times C_2^2.C_4^2
% in TeX

G:=Group("C5xC2^2.C4^2");
// GroupNames label

G:=SmallGroup(320,148);
// by ID

G=gap.SmallGroup(320,148);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,3511,172]);
// Polycyclic

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

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