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## G = C5×C22.M4(2)  order 320 = 26·5

### Direct product of C5 and C22.M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22.M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C20 — C5×C22⋊C8 — C5×C22.M4(2)
 Lower central C1 — C2 — C22 — C5×C22.M4(2)
 Upper central C1 — C2×C10 — C22×C20 — C5×C22.M4(2)

Generators and relations for C5×C22.M4(2)
G = < a,b,c,d,e | a5=b2=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd5 >

Subgroups: 138 in 78 conjugacy classes, 38 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×4], C2×C4 [×5], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4 [×3], C20 [×5], C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C4⋊C4, C40 [×2], C2×C20 [×4], C2×C20 [×5], C22×C10, C22.M4(2), C5×C4⋊C4 [×2], C2×C40 [×2], C22×C20 [×3], C5×C22⋊C8 [×2], C10×C4⋊C4, C5×C22.M4(2)
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C8 [×2], C2×C4, D4 [×2], C10 [×3], C22⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C22⋊C8, C23⋊C4, C4.10D4, C40 [×2], C2×C20, C5×D4 [×2], C22.M4(2), C5×C22⋊C4, C2×C40, C5×M4(2), C5×C22⋊C8, C5×C23⋊C4, C5×C4.10D4, C5×C22.M4(2)

Smallest permutation representation of C5×C22.M4(2)
On 160 points
Generators in S160
(1 122 155 35 147)(2 123 156 36 148)(3 124 157 37 149)(4 125 158 38 150)(5 126 159 39 151)(6 127 160 40 152)(7 128 153 33 145)(8 121 154 34 146)(9 113 137 17 129)(10 114 138 18 130)(11 115 139 19 131)(12 116 140 20 132)(13 117 141 21 133)(14 118 142 22 134)(15 119 143 23 135)(16 120 144 24 136)(25 41 91 49 87)(26 42 92 50 88)(27 43 93 51 81)(28 44 94 52 82)(29 45 95 53 83)(30 46 96 54 84)(31 47 89 55 85)(32 48 90 56 86)(57 73 107 65 99)(58 74 108 66 100)(59 75 109 67 101)(60 76 110 68 102)(61 77 111 69 103)(62 78 112 70 104)(63 79 105 71 97)(64 80 106 72 98)
(2 54)(4 56)(6 50)(8 52)(10 97)(12 99)(14 101)(16 103)(18 105)(20 107)(22 109)(24 111)(26 160)(28 154)(30 156)(32 158)(34 44)(36 46)(38 48)(40 42)(57 116)(59 118)(61 120)(63 114)(65 132)(67 134)(69 136)(71 130)(73 140)(75 142)(77 144)(79 138)(82 121)(84 123)(86 125)(88 127)(90 150)(92 152)(94 146)(96 148)
(1 53)(2 54)(3 55)(4 56)(5 49)(6 50)(7 51)(8 52)(9 104)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 112)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 159)(26 160)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)(57 116)(58 117)(59 118)(60 119)(61 120)(62 113)(63 114)(64 115)(65 132)(66 133)(67 134)(68 135)(69 136)(70 129)(71 130)(72 131)(73 140)(74 141)(75 142)(76 143)(77 144)(78 137)(79 138)(80 139)(81 128)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 149)(90 150)(91 151)(92 152)(93 145)(94 146)(95 147)(96 148)
(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 64 53 115)(2 120 54 61)(3 117 55 58)(4 63 56 114)(5 60 49 119)(6 116 50 57)(7 113 51 62)(8 59 52 118)(9 93 104 145)(10 150 97 90)(11 147 98 95)(12 92 99 152)(13 89 100 149)(14 146 101 94)(15 151 102 91)(16 96 103 148)(17 27 112 153)(18 158 105 32)(19 155 106 29)(20 26 107 160)(21 31 108 157)(22 154 109 28)(23 159 110 25)(24 30 111 156)(33 129 43 70)(34 67 44 134)(35 72 45 131)(36 136 46 69)(37 133 47 66)(38 71 48 130)(39 68 41 135)(40 132 42 65)(73 127 140 88)(74 124 141 85)(75 82 142 121)(76 87 143 126)(77 123 144 84)(78 128 137 81)(79 86 138 125)(80 83 139 122)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AF 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 1 1 1 1 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + - image C1 C2 C2 C4 C5 C8 C10 C10 C20 C40 D4 M4(2) C5×D4 C5×M4(2) C23⋊C4 C4.10D4 C5×C23⋊C4 C5×C4.10D4 kernel C5×C22.M4(2) C5×C22⋊C8 C10×C4⋊C4 C22×C20 C22.M4(2) C2×C20 C22⋊C8 C2×C4⋊C4 C22×C4 C2×C4 C2×C20 C2×C10 C2×C4 C22 C10 C10 C2 C2 # reps 1 2 1 4 4 8 8 4 16 32 2 2 8 8 1 1 4 4

Matrix representation of C5×C22.M4(2) in GL6(𝔽41)

 16 0 0 0 0 0 0 16 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
,
 1 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 38 39 40 0 0 0 34 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 40 0 0 0 0 0 0 40
,
 32 7 0 0 0 0 34 9 0 0 0 0 0 0 14 23 23 0 0 0 30 40 40 9 0 0 11 19 28 32 0 0 17 22 22 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 1 32 0 0 0 0 40 18 9 0 0 0 19 0 0 32

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,38,34,0,0,0,1,39,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,34,0,0,0,0,7,9,0,0,0,0,0,0,14,30,11,17,0,0,23,40,19,22,0,0,23,40,28,22,0,0,0,9,32,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,1,40,19,0,0,0,32,18,0,0,0,0,0,9,0,0,0,0,0,0,32] >;

C5×C22.M4(2) in GAP, Magma, Sage, TeX

C_5\times C_2^2.M_4(2)
% in TeX

G:=Group("C5xC2^2.M4(2)");
// GroupNames label

G:=SmallGroup(320,129);
// by ID

G=gap.SmallGroup(320,129);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,2803,2111,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=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=b*c*d^5>;
// generators/relations

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