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## G = C2×Q8.F5order 320 = 26·5

### Direct product of C2 and Q8.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×Q8.F5
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C2×D5⋊C8 — C2×Q8.F5
 Lower central C5 — C10 — C2×Q8.F5
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8.F5
G = < a,b,c,d,e | a2=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 794 in 266 conjugacy classes, 140 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, C2×C8○D4, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×D20, Q82D5, Q8×C10, C2×D5⋊C8, C2×C4.F5, Q8.F5, C2×Q82D5, C2×Q8.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C8○D4, C23×C4, C2×F5, C2×C8○D4, C22×F5, Q8.F5, C23×F5, C2×Q8.F5

Smallest permutation representation of C2×Q8.F5
On 160 points
Generators in S160
(1 111)(2 112)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 159)(50 160)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 125)(58 126)(59 127)(60 128)(61 121)(62 122)(63 123)(64 124)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)
(1 109 5 105)(2 110 6 106)(3 111 7 107)(4 112 8 108)(9 119 13 115)(10 120 14 116)(11 113 15 117)(12 114 16 118)(17 67 21 71)(18 68 22 72)(19 69 23 65)(20 70 24 66)(25 135 29 131)(26 136 30 132)(27 129 31 133)(28 130 32 134)(33 137 37 141)(34 138 38 142)(35 139 39 143)(36 140 40 144)(41 95 45 91)(42 96 46 92)(43 89 47 93)(44 90 48 94)(49 153 53 157)(50 154 54 158)(51 155 55 159)(52 156 56 160)(57 87 61 83)(58 88 62 84)(59 81 63 85)(60 82 64 86)(73 123 77 127)(74 124 78 128)(75 125 79 121)(76 126 80 122)(97 145 101 149)(98 146 102 150)(99 147 103 151)(100 148 104 152)
(1 155 5 159)(2 156 6 160)(3 157 7 153)(4 158 8 154)(9 71 13 67)(10 72 14 68)(11 65 15 69)(12 66 16 70)(17 119 21 115)(18 120 22 116)(19 113 23 117)(20 114 24 118)(25 125 29 121)(26 126 30 122)(27 127 31 123)(28 128 32 124)(33 59 37 63)(34 60 38 64)(35 61 39 57)(36 62 40 58)(41 97 45 101)(42 98 46 102)(43 99 47 103)(44 100 48 104)(49 111 53 107)(50 112 54 108)(51 105 55 109)(52 106 56 110)(73 129 77 133)(74 130 78 134)(75 131 79 135)(76 132 80 136)(81 137 85 141)(82 138 86 142)(83 139 87 143)(84 140 88 144)(89 151 93 147)(90 152 94 148)(91 145 95 149)(92 146 96 150)
(1 97 35 133 19)(2 134 98 20 36)(3 21 135 37 99)(4 38 22 100 136)(5 101 39 129 23)(6 130 102 24 40)(7 17 131 33 103)(8 34 18 104 132)(9 125 85 93 49)(10 94 126 50 86)(11 51 95 87 127)(12 88 52 128 96)(13 121 81 89 53)(14 90 122 54 82)(15 55 91 83 123)(16 84 56 124 92)(25 137 151 107 67)(26 108 138 68 152)(27 69 109 145 139)(28 146 70 140 110)(29 141 147 111 71)(30 112 142 72 148)(31 65 105 149 143)(32 150 66 144 106)(41 57 77 117 159)(42 118 58 160 78)(43 153 119 79 59)(44 80 154 60 120)(45 61 73 113 155)(46 114 62 156 74)(47 157 115 75 63)(48 76 158 64 116)
(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)| (1,111)(2,112)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,125)(58,126)(59,127)(60,128)(61,121)(62,122)(63,123)(64,124)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,109,5,105)(2,110,6,106)(3,111,7,107)(4,112,8,108)(9,119,13,115)(10,120,14,116)(11,113,15,117)(12,114,16,118)(17,67,21,71)(18,68,22,72)(19,69,23,65)(20,70,24,66)(25,135,29,131)(26,136,30,132)(27,129,31,133)(28,130,32,134)(33,137,37,141)(34,138,38,142)(35,139,39,143)(36,140,40,144)(41,95,45,91)(42,96,46,92)(43,89,47,93)(44,90,48,94)(49,153,53,157)(50,154,54,158)(51,155,55,159)(52,156,56,160)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(97,145,101,149)(98,146,102,150)(99,147,103,151)(100,148,104,152), (1,155,5,159)(2,156,6,160)(3,157,7,153)(4,158,8,154)(9,71,13,67)(10,72,14,68)(11,65,15,69)(12,66,16,70)(17,119,21,115)(18,120,22,116)(19,113,23,117)(20,114,24,118)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(41,97,45,101)(42,98,46,102)(43,99,47,103)(44,100,48,104)(49,111,53,107)(50,112,54,108)(51,105,55,109)(52,106,56,110)(73,129,77,133)(74,130,78,134)(75,131,79,135)(76,132,80,136)(81,137,85,141)(82,138,86,142)(83,139,87,143)(84,140,88,144)(89,151,93,147)(90,152,94,148)(91,145,95,149)(92,146,96,150), (1,97,35,133,19)(2,134,98,20,36)(3,21,135,37,99)(4,38,22,100,136)(5,101,39,129,23)(6,130,102,24,40)(7,17,131,33,103)(8,34,18,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,107,67)(26,108,138,68,152)(27,69,109,145,139)(28,146,70,140,110)(29,141,147,111,71)(30,112,142,72,148)(31,65,105,149,143)(32,150,66,144,106)(41,57,77,117,159)(42,118,58,160,78)(43,153,119,79,59)(44,80,154,60,120)(45,61,73,113,155)(46,114,62,156,74)(47,157,115,75,63)(48,76,158,64,116), (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( (1,111)(2,112)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,125)(58,126)(59,127)(60,128)(61,121)(62,122)(63,123)(64,124)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,109,5,105)(2,110,6,106)(3,111,7,107)(4,112,8,108)(9,119,13,115)(10,120,14,116)(11,113,15,117)(12,114,16,118)(17,67,21,71)(18,68,22,72)(19,69,23,65)(20,70,24,66)(25,135,29,131)(26,136,30,132)(27,129,31,133)(28,130,32,134)(33,137,37,141)(34,138,38,142)(35,139,39,143)(36,140,40,144)(41,95,45,91)(42,96,46,92)(43,89,47,93)(44,90,48,94)(49,153,53,157)(50,154,54,158)(51,155,55,159)(52,156,56,160)(57,87,61,83)(58,88,62,84)(59,81,63,85)(60,82,64,86)(73,123,77,127)(74,124,78,128)(75,125,79,121)(76,126,80,122)(97,145,101,149)(98,146,102,150)(99,147,103,151)(100,148,104,152), (1,155,5,159)(2,156,6,160)(3,157,7,153)(4,158,8,154)(9,71,13,67)(10,72,14,68)(11,65,15,69)(12,66,16,70)(17,119,21,115)(18,120,22,116)(19,113,23,117)(20,114,24,118)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(33,59,37,63)(34,60,38,64)(35,61,39,57)(36,62,40,58)(41,97,45,101)(42,98,46,102)(43,99,47,103)(44,100,48,104)(49,111,53,107)(50,112,54,108)(51,105,55,109)(52,106,56,110)(73,129,77,133)(74,130,78,134)(75,131,79,135)(76,132,80,136)(81,137,85,141)(82,138,86,142)(83,139,87,143)(84,140,88,144)(89,151,93,147)(90,152,94,148)(91,145,95,149)(92,146,96,150), (1,97,35,133,19)(2,134,98,20,36)(3,21,135,37,99)(4,38,22,100,136)(5,101,39,129,23)(6,130,102,24,40)(7,17,131,33,103)(8,34,18,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,107,67)(26,108,138,68,152)(27,69,109,145,139)(28,146,70,140,110)(29,141,147,111,71)(30,112,142,72,148)(31,65,105,149,143)(32,150,66,144,106)(41,57,77,117,159)(42,118,58,160,78)(43,153,119,79,59)(44,80,154,60,120)(45,61,73,113,155)(46,114,62,156,74)(47,157,115,75,63)(48,76,158,64,116), (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([[(1,111),(2,112),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,159),(50,160),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,125),(58,126),(59,127),(60,128),(61,121),(62,122),(63,123),(64,124),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,97),(72,98),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(129,137),(130,138),(131,139),(132,140),(133,141),(134,142),(135,143),(136,144)], [(1,109,5,105),(2,110,6,106),(3,111,7,107),(4,112,8,108),(9,119,13,115),(10,120,14,116),(11,113,15,117),(12,114,16,118),(17,67,21,71),(18,68,22,72),(19,69,23,65),(20,70,24,66),(25,135,29,131),(26,136,30,132),(27,129,31,133),(28,130,32,134),(33,137,37,141),(34,138,38,142),(35,139,39,143),(36,140,40,144),(41,95,45,91),(42,96,46,92),(43,89,47,93),(44,90,48,94),(49,153,53,157),(50,154,54,158),(51,155,55,159),(52,156,56,160),(57,87,61,83),(58,88,62,84),(59,81,63,85),(60,82,64,86),(73,123,77,127),(74,124,78,128),(75,125,79,121),(76,126,80,122),(97,145,101,149),(98,146,102,150),(99,147,103,151),(100,148,104,152)], [(1,155,5,159),(2,156,6,160),(3,157,7,153),(4,158,8,154),(9,71,13,67),(10,72,14,68),(11,65,15,69),(12,66,16,70),(17,119,21,115),(18,120,22,116),(19,113,23,117),(20,114,24,118),(25,125,29,121),(26,126,30,122),(27,127,31,123),(28,128,32,124),(33,59,37,63),(34,60,38,64),(35,61,39,57),(36,62,40,58),(41,97,45,101),(42,98,46,102),(43,99,47,103),(44,100,48,104),(49,111,53,107),(50,112,54,108),(51,105,55,109),(52,106,56,110),(73,129,77,133),(74,130,78,134),(75,131,79,135),(76,132,80,136),(81,137,85,141),(82,138,86,142),(83,139,87,143),(84,140,88,144),(89,151,93,147),(90,152,94,148),(91,145,95,149),(92,146,96,150)], [(1,97,35,133,19),(2,134,98,20,36),(3,21,135,37,99),(4,38,22,100,136),(5,101,39,129,23),(6,130,102,24,40),(7,17,131,33,103),(8,34,18,104,132),(9,125,85,93,49),(10,94,126,50,86),(11,51,95,87,127),(12,88,52,128,96),(13,121,81,89,53),(14,90,122,54,82),(15,55,91,83,123),(16,84,56,124,92),(25,137,151,107,67),(26,108,138,68,152),(27,69,109,145,139),(28,146,70,140,110),(29,141,147,111,71),(30,112,142,72,148),(31,65,105,149,143),(32,150,66,144,106),(41,57,77,117,159),(42,118,58,160,78),(43,153,119,79,59),(44,80,154,60,120),(45,61,73,113,155),(46,114,62,156,74),(47,157,115,75,63),(48,76,158,64,116)], [(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)]])

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 8 type + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8○D4 F5 C2×F5 C2×F5 Q8.F5 kernel C2×Q8.F5 C2×D5⋊C8 C2×C4.F5 Q8.F5 C2×Q8⋊2D5 C2×D20 Q8⋊2D5 Q8×C10 C10 C2×Q8 C2×C4 Q8 C2 # reps 1 3 3 8 1 6 8 2 8 1 3 4 2

Matrix representation of C2×Q8.F5 in GL8(𝔽41)

 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 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
,
 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 32 32 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
,
 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 39 0 0 0 0 0 0 1 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
,
 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 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 38 0 22 22 0 0 0 0 22 22 0 38 0 0 0 0 19 16 19 0 0 0 0 0 3 25 25 3

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,32,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,38,22,19,3,0,0,0,0,0,22,16,25,0,0,0,0,22,0,19,25,0,0,0,0,22,38,0,3] >;

C2×Q8.F5 in GAP, Magma, Sage, TeX

C_2\times Q_8.F_5
% in TeX

G:=Group("C2xQ8.F5");
// GroupNames label

G:=SmallGroup(320,1597);
// by ID

G=gap.SmallGroup(320,1597);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,297,136,102,6278,818]);
// Polycyclic

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

׿
×
𝔽