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## G = C2×C8.F5order 320 = 26·5

### Direct product of C2 and C8.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C8.F5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — C2×C8.F5
 Lower central C5 — C10 — C2×C8.F5
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C2×C8.F5
G = < a,b,c,d | a2=b8=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=c3 >

Subgroups: 250 in 90 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, D10, C2×C10, C2×C16, M5(2), C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, C8.F5, C2×C5⋊C16, D5×C2×C8, C2×C8.F5
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, M5(2), C22×C8, C2×F5, C2×M5(2), D5⋊C8, C22×F5, C8.F5, C2×D5⋊C8, C2×C8.F5

Smallest permutation representation of C2×C8.F5
On 160 points
Generators in S160
(1 140)(2 141)(3 142)(4 143)(5 144)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 112)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 159)(34 160)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 95)(50 96)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 113)(78 114)(79 115)(80 116)
(1 134 5 138 9 142 13 130)(2 143 6 131 10 135 14 139)(3 136 7 140 11 144 15 132)(4 129 8 133 12 137 16 141)(17 153 21 157 25 145 29 149)(18 146 22 150 26 154 30 158)(19 155 23 159 27 147 31 151)(20 148 24 152 28 156 32 160)(33 106 37 110 41 98 45 102)(34 99 38 103 42 107 46 111)(35 108 39 112 43 100 47 104)(36 101 40 105 44 109 48 97)(49 126 53 114 57 118 61 122)(50 119 54 123 58 127 62 115)(51 128 55 116 59 120 63 124)(52 121 56 125 60 113 64 117)(65 82 69 86 73 90 77 94)(66 91 70 95 74 83 78 87)(67 84 71 88 75 92 79 96)(68 93 72 81 76 85 80 89)
(1 60 29 45 67)(2 46 61 68 30)(3 69 47 31 62)(4 32 70 63 48)(5 64 17 33 71)(6 34 49 72 18)(7 73 35 19 50)(8 20 74 51 36)(9 52 21 37 75)(10 38 53 76 22)(11 77 39 23 54)(12 24 78 55 40)(13 56 25 41 79)(14 42 57 80 26)(15 65 43 27 58)(16 28 66 59 44)(81 146 131 99 126)(82 100 147 127 132)(83 128 101 133 148)(84 134 113 149 102)(85 150 135 103 114)(86 104 151 115 136)(87 116 105 137 152)(88 138 117 153 106)(89 154 139 107 118)(90 108 155 119 140)(91 120 109 141 156)(92 142 121 157 110)(93 158 143 111 122)(94 112 159 123 144)(95 124 97 129 160)(96 130 125 145 98)
(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)| (1,140)(2,141)(3,142)(4,143)(5,144)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,112)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,159)(34,160)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,95)(50,96)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,113)(78,114)(79,115)(80,116), (1,134,5,138,9,142,13,130)(2,143,6,131,10,135,14,139)(3,136,7,140,11,144,15,132)(4,129,8,133,12,137,16,141)(17,153,21,157,25,145,29,149)(18,146,22,150,26,154,30,158)(19,155,23,159,27,147,31,151)(20,148,24,152,28,156,32,160)(33,106,37,110,41,98,45,102)(34,99,38,103,42,107,46,111)(35,108,39,112,43,100,47,104)(36,101,40,105,44,109,48,97)(49,126,53,114,57,118,61,122)(50,119,54,123,58,127,62,115)(51,128,55,116,59,120,63,124)(52,121,56,125,60,113,64,117)(65,82,69,86,73,90,77,94)(66,91,70,95,74,83,78,87)(67,84,71,88,75,92,79,96)(68,93,72,81,76,85,80,89), (1,60,29,45,67)(2,46,61,68,30)(3,69,47,31,62)(4,32,70,63,48)(5,64,17,33,71)(6,34,49,72,18)(7,73,35,19,50)(8,20,74,51,36)(9,52,21,37,75)(10,38,53,76,22)(11,77,39,23,54)(12,24,78,55,40)(13,56,25,41,79)(14,42,57,80,26)(15,65,43,27,58)(16,28,66,59,44)(81,146,131,99,126)(82,100,147,127,132)(83,128,101,133,148)(84,134,113,149,102)(85,150,135,103,114)(86,104,151,115,136)(87,116,105,137,152)(88,138,117,153,106)(89,154,139,107,118)(90,108,155,119,140)(91,120,109,141,156)(92,142,121,157,110)(93,158,143,111,122)(94,112,159,123,144)(95,124,97,129,160)(96,130,125,145,98), (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( (1,140)(2,141)(3,142)(4,143)(5,144)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,112)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,159)(34,160)(35,145)(36,146)(37,147)(38,148)(39,149)(40,150)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,95)(50,96)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,113)(78,114)(79,115)(80,116), (1,134,5,138,9,142,13,130)(2,143,6,131,10,135,14,139)(3,136,7,140,11,144,15,132)(4,129,8,133,12,137,16,141)(17,153,21,157,25,145,29,149)(18,146,22,150,26,154,30,158)(19,155,23,159,27,147,31,151)(20,148,24,152,28,156,32,160)(33,106,37,110,41,98,45,102)(34,99,38,103,42,107,46,111)(35,108,39,112,43,100,47,104)(36,101,40,105,44,109,48,97)(49,126,53,114,57,118,61,122)(50,119,54,123,58,127,62,115)(51,128,55,116,59,120,63,124)(52,121,56,125,60,113,64,117)(65,82,69,86,73,90,77,94)(66,91,70,95,74,83,78,87)(67,84,71,88,75,92,79,96)(68,93,72,81,76,85,80,89), (1,60,29,45,67)(2,46,61,68,30)(3,69,47,31,62)(4,32,70,63,48)(5,64,17,33,71)(6,34,49,72,18)(7,73,35,19,50)(8,20,74,51,36)(9,52,21,37,75)(10,38,53,76,22)(11,77,39,23,54)(12,24,78,55,40)(13,56,25,41,79)(14,42,57,80,26)(15,65,43,27,58)(16,28,66,59,44)(81,146,131,99,126)(82,100,147,127,132)(83,128,101,133,148)(84,134,113,149,102)(85,150,135,103,114)(86,104,151,115,136)(87,116,105,137,152)(88,138,117,153,106)(89,154,139,107,118)(90,108,155,119,140)(91,120,109,141,156)(92,142,121,157,110)(93,158,143,111,122)(94,112,159,123,144)(95,124,97,129,160)(96,130,125,145,98), (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([[(1,140),(2,141),(3,142),(4,143),(5,144),(6,129),(7,130),(8,131),(9,132),(10,133),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,112),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,159),(34,160),(35,145),(36,146),(37,147),(38,148),(39,149),(40,150),(41,151),(42,152),(43,153),(44,154),(45,155),(46,156),(47,157),(48,158),(49,95),(50,96),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,113),(78,114),(79,115),(80,116)], [(1,134,5,138,9,142,13,130),(2,143,6,131,10,135,14,139),(3,136,7,140,11,144,15,132),(4,129,8,133,12,137,16,141),(17,153,21,157,25,145,29,149),(18,146,22,150,26,154,30,158),(19,155,23,159,27,147,31,151),(20,148,24,152,28,156,32,160),(33,106,37,110,41,98,45,102),(34,99,38,103,42,107,46,111),(35,108,39,112,43,100,47,104),(36,101,40,105,44,109,48,97),(49,126,53,114,57,118,61,122),(50,119,54,123,58,127,62,115),(51,128,55,116,59,120,63,124),(52,121,56,125,60,113,64,117),(65,82,69,86,73,90,77,94),(66,91,70,95,74,83,78,87),(67,84,71,88,75,92,79,96),(68,93,72,81,76,85,80,89)], [(1,60,29,45,67),(2,46,61,68,30),(3,69,47,31,62),(4,32,70,63,48),(5,64,17,33,71),(6,34,49,72,18),(7,73,35,19,50),(8,20,74,51,36),(9,52,21,37,75),(10,38,53,76,22),(11,77,39,23,54),(12,24,78,55,40),(13,56,25,41,79),(14,42,57,80,26),(15,65,43,27,58),(16,28,66,59,44),(81,146,131,99,126),(82,100,147,127,132),(83,128,101,133,148),(84,134,113,149,102),(85,150,135,103,114),(86,104,151,115,136),(87,116,105,137,152),(88,138,117,153,106),(89,154,139,107,118),(90,108,155,119,140),(91,120,109,141,156),(92,142,121,157,110),(93,158,143,111,122),(94,112,159,123,144),(95,124,97,129,160),(96,130,125,145,98)], [(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)]])

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A 8B 8C 8D 8E ··· 8L 10A 10B 10C 16A ··· 16P 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 5 8 8 8 8 8 ··· 8 10 10 10 16 ··· 16 20 20 20 20 40 ··· 40 size 1 1 1 1 10 10 1 1 1 1 10 10 4 2 2 2 2 5 ··· 5 4 4 4 10 ··· 10 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C8 C8 M5(2) F5 C2×F5 C2×F5 D5⋊C8 D5⋊C8 C8.F5 kernel C2×C8.F5 C8.F5 C2×C5⋊C16 D5×C2×C8 C8×D5 C2×C40 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 C10 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 2 2 8 4 4 8 1 2 1 2 2 8

Matrix representation of C2×C8.F5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 233 0 0 0 0 0 63 8 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0 1 0 0 0 240 0 0 1 0 0 240 0 0 0
,
 19 1 0 0 0 0 129 222 0 0 0 0 0 0 173 68 3 0 0 0 176 68 0 173 0 0 173 0 68 176 0 0 0 3 68 173

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[233,63,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[19,129,0,0,0,0,1,222,0,0,0,0,0,0,173,176,173,0,0,0,68,68,0,3,0,0,3,0,68,68,0,0,0,173,176,173] >;

C2×C8.F5 in GAP, Magma, Sage, TeX

C_2\times C_8.F_5
% in TeX

G:=Group("C2xC8.F5");
// GroupNames label

G:=SmallGroup(320,1052);
// by ID

G=gap.SmallGroup(320,1052);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=c^3>;
// generators/relations

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