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## G = (C22×C4).F5order 320 = 26·5

### 1st non-split extension by C22×C4 of F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C22×C4).F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.2F5 — (C22×C4).F5
 Lower central C5 — C10 — C2×C10 — (C22×C4).F5
 Upper central C1 — C22 — C23 — C22×C4

Generators and relations for (C22×C4).F5
G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=c2, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=abc, ede-1=d3 >

Subgroups: 306 in 78 conjugacy classes, 28 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×9], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], Dic5 [×4], C20, C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C4⋊C4, C5⋊C8 [×2], C2×Dic5 [×4], C2×Dic5 [×3], C2×C20 [×2], C22×C10, C22.M4(2), C10.D4 [×2], C2×C5⋊C8 [×2], C22×Dic5 [×2], C22×C20, C23.2F5 [×2], C2×C10.D4, (C22×C4).F5
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.10D4, C2×F5, C22.M4(2), D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, Dic5.D4, C23⋊F5, (C22×C4).F5

Smallest permutation representation of (C22×C4).F5
On 160 points
Generators in S160
(2 78)(4 80)(6 74)(8 76)(9 133)(11 135)(13 129)(15 131)(17 48)(19 42)(21 44)(23 46)(25 152)(27 146)(29 148)(31 150)(34 101)(36 103)(38 97)(40 99)(49 93)(51 95)(53 89)(55 91)(57 123)(59 125)(61 127)(63 121)(66 160)(68 154)(70 156)(72 158)(81 144)(83 138)(85 140)(87 142)(105 118)(107 120)(109 114)(111 116)
(1 77)(2 78)(3 79)(4 80)(5 73)(6 74)(7 75)(8 76)(9 133)(10 134)(11 135)(12 136)(13 129)(14 130)(15 131)(16 132)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 152)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 100)(34 101)(35 102)(36 103)(37 104)(38 97)(39 98)(40 99)(49 93)(50 94)(51 95)(52 96)(53 89)(54 90)(55 91)(56 92)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 121)(64 122)(65 159)(66 160)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(81 144)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(105 118)(106 119)(107 120)(108 113)(109 114)(110 115)(111 116)(112 117)
(1 137 5 141)(2 83 6 87)(3 84 7 88)(4 140 8 144)(9 57 13 61)(10 58 14 62)(11 125 15 121)(12 126 16 122)(17 97 21 101)(18 98 22 102)(19 40 23 36)(20 33 24 37)(25 68 29 72)(26 69 30 65)(27 156 31 160)(28 157 32 153)(34 48 38 44)(35 41 39 45)(42 99 46 103)(43 100 47 104)(49 109 53 105)(50 110 54 106)(51 116 55 120)(52 117 56 113)(59 131 63 135)(60 132 64 136)(66 146 70 150)(67 147 71 151)(73 86 77 82)(74 142 78 138)(75 143 79 139)(76 81 80 85)(89 118 93 114)(90 119 94 115)(91 107 95 111)(92 108 96 112)(123 129 127 133)(124 130 128 134)(145 155 149 159)(148 158 152 154)
(1 100 60 155 54)(2 156 101 55 61)(3 56 157 62 102)(4 63 49 103 158)(5 104 64 159 50)(6 160 97 51 57)(7 52 153 58 98)(8 59 53 99 154)(9 83 31 17 120)(10 18 84 113 32)(11 114 19 25 85)(12 26 115 86 20)(13 87 27 21 116)(14 22 88 117 28)(15 118 23 29 81)(16 30 119 82 24)(33 126 69 90 77)(34 91 127 78 70)(35 79 92 71 128)(36 72 80 121 93)(37 122 65 94 73)(38 95 123 74 66)(39 75 96 67 124)(40 68 76 125 89)(41 139 108 151 134)(42 152 140 135 109)(43 136 145 110 141)(44 111 129 142 146)(45 143 112 147 130)(46 148 144 131 105)(47 132 149 106 137)(48 107 133 138 150)
(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)| (2,78)(4,80)(6,74)(8,76)(9,133)(11,135)(13,129)(15,131)(17,48)(19,42)(21,44)(23,46)(25,152)(27,146)(29,148)(31,150)(34,101)(36,103)(38,97)(40,99)(49,93)(51,95)(53,89)(55,91)(57,123)(59,125)(61,127)(63,121)(66,160)(68,154)(70,156)(72,158)(81,144)(83,138)(85,140)(87,142)(105,118)(107,120)(109,114)(111,116), (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,133)(10,134)(11,135)(12,136)(13,129)(14,130)(15,131)(16,132)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,152)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,100)(34,101)(35,102)(36,103)(37,104)(38,97)(39,98)(40,99)(49,93)(50,94)(51,95)(52,96)(53,89)(54,90)(55,91)(56,92)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,121)(64,122)(65,159)(66,160)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(81,144)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,137,5,141)(2,83,6,87)(3,84,7,88)(4,140,8,144)(9,57,13,61)(10,58,14,62)(11,125,15,121)(12,126,16,122)(17,97,21,101)(18,98,22,102)(19,40,23,36)(20,33,24,37)(25,68,29,72)(26,69,30,65)(27,156,31,160)(28,157,32,153)(34,48,38,44)(35,41,39,45)(42,99,46,103)(43,100,47,104)(49,109,53,105)(50,110,54,106)(51,116,55,120)(52,117,56,113)(59,131,63,135)(60,132,64,136)(66,146,70,150)(67,147,71,151)(73,86,77,82)(74,142,78,138)(75,143,79,139)(76,81,80,85)(89,118,93,114)(90,119,94,115)(91,107,95,111)(92,108,96,112)(123,129,127,133)(124,130,128,134)(145,155,149,159)(148,158,152,154), (1,100,60,155,54)(2,156,101,55,61)(3,56,157,62,102)(4,63,49,103,158)(5,104,64,159,50)(6,160,97,51,57)(7,52,153,58,98)(8,59,53,99,154)(9,83,31,17,120)(10,18,84,113,32)(11,114,19,25,85)(12,26,115,86,20)(13,87,27,21,116)(14,22,88,117,28)(15,118,23,29,81)(16,30,119,82,24)(33,126,69,90,77)(34,91,127,78,70)(35,79,92,71,128)(36,72,80,121,93)(37,122,65,94,73)(38,95,123,74,66)(39,75,96,67,124)(40,68,76,125,89)(41,139,108,151,134)(42,152,140,135,109)(43,136,145,110,141)(44,111,129,142,146)(45,143,112,147,130)(46,148,144,131,105)(47,132,149,106,137)(48,107,133,138,150), (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( (2,78)(4,80)(6,74)(8,76)(9,133)(11,135)(13,129)(15,131)(17,48)(19,42)(21,44)(23,46)(25,152)(27,146)(29,148)(31,150)(34,101)(36,103)(38,97)(40,99)(49,93)(51,95)(53,89)(55,91)(57,123)(59,125)(61,127)(63,121)(66,160)(68,154)(70,156)(72,158)(81,144)(83,138)(85,140)(87,142)(105,118)(107,120)(109,114)(111,116), (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,133)(10,134)(11,135)(12,136)(13,129)(14,130)(15,131)(16,132)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,152)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,100)(34,101)(35,102)(36,103)(37,104)(38,97)(39,98)(40,99)(49,93)(50,94)(51,95)(52,96)(53,89)(54,90)(55,91)(56,92)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,121)(64,122)(65,159)(66,160)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(81,144)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,137,5,141)(2,83,6,87)(3,84,7,88)(4,140,8,144)(9,57,13,61)(10,58,14,62)(11,125,15,121)(12,126,16,122)(17,97,21,101)(18,98,22,102)(19,40,23,36)(20,33,24,37)(25,68,29,72)(26,69,30,65)(27,156,31,160)(28,157,32,153)(34,48,38,44)(35,41,39,45)(42,99,46,103)(43,100,47,104)(49,109,53,105)(50,110,54,106)(51,116,55,120)(52,117,56,113)(59,131,63,135)(60,132,64,136)(66,146,70,150)(67,147,71,151)(73,86,77,82)(74,142,78,138)(75,143,79,139)(76,81,80,85)(89,118,93,114)(90,119,94,115)(91,107,95,111)(92,108,96,112)(123,129,127,133)(124,130,128,134)(145,155,149,159)(148,158,152,154), (1,100,60,155,54)(2,156,101,55,61)(3,56,157,62,102)(4,63,49,103,158)(5,104,64,159,50)(6,160,97,51,57)(7,52,153,58,98)(8,59,53,99,154)(9,83,31,17,120)(10,18,84,113,32)(11,114,19,25,85)(12,26,115,86,20)(13,87,27,21,116)(14,22,88,117,28)(15,118,23,29,81)(16,30,119,82,24)(33,126,69,90,77)(34,91,127,78,70)(35,79,92,71,128)(36,72,80,121,93)(37,122,65,94,73)(38,95,123,74,66)(39,75,96,67,124)(40,68,76,125,89)(41,139,108,151,134)(42,152,140,135,109)(43,136,145,110,141)(44,111,129,142,146)(45,143,112,147,130)(46,148,144,131,105)(47,132,149,106,137)(48,107,133,138,150), (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([(2,78),(4,80),(6,74),(8,76),(9,133),(11,135),(13,129),(15,131),(17,48),(19,42),(21,44),(23,46),(25,152),(27,146),(29,148),(31,150),(34,101),(36,103),(38,97),(40,99),(49,93),(51,95),(53,89),(55,91),(57,123),(59,125),(61,127),(63,121),(66,160),(68,154),(70,156),(72,158),(81,144),(83,138),(85,140),(87,142),(105,118),(107,120),(109,114),(111,116)], [(1,77),(2,78),(3,79),(4,80),(5,73),(6,74),(7,75),(8,76),(9,133),(10,134),(11,135),(12,136),(13,129),(14,130),(15,131),(16,132),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,152),(26,145),(27,146),(28,147),(29,148),(30,149),(31,150),(32,151),(33,100),(34,101),(35,102),(36,103),(37,104),(38,97),(39,98),(40,99),(49,93),(50,94),(51,95),(52,96),(53,89),(54,90),(55,91),(56,92),(57,123),(58,124),(59,125),(60,126),(61,127),(62,128),(63,121),(64,122),(65,159),(66,160),(67,153),(68,154),(69,155),(70,156),(71,157),(72,158),(81,144),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(105,118),(106,119),(107,120),(108,113),(109,114),(110,115),(111,116),(112,117)], [(1,137,5,141),(2,83,6,87),(3,84,7,88),(4,140,8,144),(9,57,13,61),(10,58,14,62),(11,125,15,121),(12,126,16,122),(17,97,21,101),(18,98,22,102),(19,40,23,36),(20,33,24,37),(25,68,29,72),(26,69,30,65),(27,156,31,160),(28,157,32,153),(34,48,38,44),(35,41,39,45),(42,99,46,103),(43,100,47,104),(49,109,53,105),(50,110,54,106),(51,116,55,120),(52,117,56,113),(59,131,63,135),(60,132,64,136),(66,146,70,150),(67,147,71,151),(73,86,77,82),(74,142,78,138),(75,143,79,139),(76,81,80,85),(89,118,93,114),(90,119,94,115),(91,107,95,111),(92,108,96,112),(123,129,127,133),(124,130,128,134),(145,155,149,159),(148,158,152,154)], [(1,100,60,155,54),(2,156,101,55,61),(3,56,157,62,102),(4,63,49,103,158),(5,104,64,159,50),(6,160,97,51,57),(7,52,153,58,98),(8,59,53,99,154),(9,83,31,17,120),(10,18,84,113,32),(11,114,19,25,85),(12,26,115,86,20),(13,87,27,21,116),(14,22,88,117,28),(15,118,23,29,81),(16,30,119,82,24),(33,126,69,90,77),(34,91,127,78,70),(35,79,92,71,128),(36,72,80,121,93),(37,122,65,94,73),(38,95,123,74,66),(39,75,96,67,124),(40,68,76,125,89),(41,139,108,151,134),(42,152,140,135,109),(43,136,145,110,141),(44,111,129,142,146),(45,143,112,147,130),(46,148,144,131,105),(47,132,149,106,137),(48,107,133,138,150)], [(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)])

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5 8A ··· 8H 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 10 10 10 10 20 20 4 20 ··· 20 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + - + + - image C1 C2 C2 C4 C4 C8 D4 M4(2) F5 C23⋊C4 C4.10D4 C2×F5 D5⋊C8 C4.F5 C22⋊F5 Dic5.D4 C23⋊F5 kernel (C22×C4).F5 C23.2F5 C2×C10.D4 C22×Dic5 C22×C20 C2×Dic5 C2×Dic5 C2×C10 C22×C4 C10 C10 C23 C22 C22 C22 C2 C2 # reps 1 2 1 2 2 8 2 2 1 1 1 1 2 2 2 4 4

Matrix representation of (C22×C4).F5 in GL10(𝔽41)

 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 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 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 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 0 0 40 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 3 20 0 0 0 0 0 0 0 0 20 38 0 0 0 0 0 0 0 0 0 0 3 20 0 0 0 0 0 0 0 0 20 38 0 0 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 0 0 40 0 0 0 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 0 1 0 40 0 0 0 0 0 0 0 0 1 40
,
 0 38 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 1 10 4 24 0 0 0 0 0 0 5 34 6 25 0 0 0 0 0 0 7 35 16 29 0 0 0 0 0 0 17 39 40 31

G:=sub<GL(10,GF(41))| [40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,20,38,0,0,0,0,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,20,38,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40],[0,3,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,5,7,17,0,0,0,0,0,0,10,34,35,39,0,0,0,0,0,0,4,6,16,40,0,0,0,0,0,0,24,25,29,31] >;

(C22×C4).F5 in GAP, Magma, Sage, TeX

(C_2^2\times C_4).F_5
% in TeX

G:=Group("(C2^2xC4).F5");
// GroupNames label

G:=SmallGroup(320,252);
// by ID

G=gap.SmallGroup(320,252);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,100,1123,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=c^2,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a*b*c,e*d*e^-1=d^3>;
// generators/relations

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