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G = C2×Q8.F5order 320 = 26·5

Direct product of C2 and Q8.F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q8.F5, Dic5.19C24, C5⋊C8.3C23, C102(C8○D4), D5⋊C87C22, (Q8×C10).9C4, Q8.11(C2×F5), (C2×Q8).10F5, C4.F59C22, D20.11(C2×C4), (C2×D20).14C4, Q82D5.2C4, C4.28(C22×F5), C2.12(C23×F5), C20.28(C22×C4), C10.11(C23×C4), (C4×D5).51C23, D10.4(C22×C4), C22.58(C22×F5), Q82D5.15C22, Dic5.47(C22×C4), (C2×Dic5).364C23, C52(C2×C8○D4), (C2×D5⋊C8)⋊6C2, (C2×C4.F5)⋊7C2, (C2×C4).92(C2×F5), (C2×C20).72(C2×C4), (C4×D5).33(C2×C4), (C2×C5⋊C8).45C22, (C5×Q8).11(C2×C4), (C2×C4×D5).218C22, (C2×Q82D5).13C2, (C22×D5).62(C2×C4), (C2×C10).101(C22×C4), SmallGroup(320,1597)

Series: Derived Chief Lower central Upper central

C1C10 — C2×Q8.F5
C1C5C10Dic5C5⋊C8C2×C5⋊C8C2×D5⋊C8 — C2×Q8.F5
C5C10 — C2×Q8.F5
C1C22C2×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 [×2], C2 [×6], C4 [×6], C4 [×2], C22, C22 [×12], C5, C8 [×8], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10, C10 [×2], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×6], D10 [×6], D10 [×6], C2×C10, C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C5⋊C8 [×8], C4×D5 [×12], D20 [×12], C2×Dic5, C2×C20 [×3], C5×Q8 [×4], C22×D5 [×3], C2×C8○D4, D5⋊C8 [×12], C4.F5 [×12], C2×C5⋊C8, C2×C5⋊C8 [×3], C2×C4×D5 [×3], C2×D20 [×3], Q82D5 [×8], Q8×C10, C2×D5⋊C8 [×3], C2×C4.F5 [×3], Q8.F5 [×8], C2×Q82D5, C2×Q8.F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, F5, C8○D4 [×2], C23×C4, C2×F5 [×7], C2×C8○D4, C22×F5 [×7], Q8.F5 [×2], C23×F5, C2×Q8.F5

Smallest permutation representation of C2×Q8.F5
On 160 points
Generators in S160
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 124)(34 125)(35 126)(36 127)(37 128)(38 121)(39 122)(40 123)(49 159)(50 160)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(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)(105 149)(106 150)(107 151)(108 152)(109 145)(110 146)(111 147)(112 148)(129 137)(130 138)(131 139)(132 140)(133 141)(134 142)(135 143)(136 144)
(1 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 119 13 115)(10 120 14 116)(11 113 15 117)(12 114 16 118)(17 138 21 142)(18 139 22 143)(19 140 23 144)(20 141 24 137)(25 135 29 131)(26 136 30 132)(27 129 31 133)(28 130 32 134)(33 86 37 82)(34 87 38 83)(35 88 39 84)(36 81 40 85)(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)(65 111 69 107)(66 112 70 108)(67 105 71 109)(68 106 72 110)(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 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 125 29 121)(26 126 30 122)(27 127 31 123)(28 128 32 124)(41 97 45 101)(42 98 46 102)(43 99 47 103)(44 100 48 104)(49 61 53 57)(50 62 54 58)(51 63 55 59)(52 64 56 60)(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)(105 115 109 119)(106 116 110 120)(107 117 111 113)(108 118 112 114)
(1 97 18 133 111)(2 134 98 112 19)(3 105 135 20 99)(4 21 106 100 136)(5 101 22 129 107)(6 130 102 108 23)(7 109 131 24 103)(8 17 110 104 132)(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 57 67)(26 58 138 68 152)(27 69 59 145 139)(28 146 70 140 60)(29 141 147 61 71)(30 62 142 72 148)(31 65 63 149 143)(32 150 66 144 64)(33 116 48 76 158)(34 77 117 159 41)(35 160 78 42 118)(36 43 153 119 79)(37 120 44 80 154)(38 73 113 155 45)(39 156 74 46 114)(40 47 157 115 75)
(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)| (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,124)(34,125)(35,126)(36,127)(37,128)(38,121)(39,122)(40,123)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(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)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,119,13,115)(10,120,14,116)(11,113,15,117)(12,114,16,118)(17,138,21,142)(18,139,22,143)(19,140,23,144)(20,141,24,137)(25,135,29,131)(26,136,30,132)(27,129,31,133)(28,130,32,134)(33,86,37,82)(34,87,38,83)(35,88,39,84)(36,81,40,85)(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)(65,111,69,107)(66,112,70,108)(67,105,71,109)(68,106,72,110)(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)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(41,97,45,101)(42,98,46,102)(43,99,47,103)(44,100,48,104)(49,61,53,57)(50,62,54,58)(51,63,55,59)(52,64,56,60)(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)(105,115,109,119)(106,116,110,120)(107,117,111,113)(108,118,112,114), (1,97,18,133,111)(2,134,98,112,19)(3,105,135,20,99)(4,21,106,100,136)(5,101,22,129,107)(6,130,102,108,23)(7,109,131,24,103)(8,17,110,104,132)(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)(25,137,151,57,67)(26,58,138,68,152)(27,69,59,145,139)(28,146,70,140,60)(29,141,147,61,71)(30,62,142,72,148)(31,65,63,149,143)(32,150,66,144,64)(33,116,48,76,158)(34,77,117,159,41)(35,160,78,42,118)(36,43,153,119,79)(37,120,44,80,154)(38,73,113,155,45)(39,156,74,46,114)(40,47,157,115,75), (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( (1,61)(2,62)(3,63)(4,64)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,124)(34,125)(35,126)(36,127)(37,128)(38,121)(39,122)(40,123)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(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)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144), (1,59,5,63)(2,60,6,64)(3,61,7,57)(4,62,8,58)(9,119,13,115)(10,120,14,116)(11,113,15,117)(12,114,16,118)(17,138,21,142)(18,139,22,143)(19,140,23,144)(20,141,24,137)(25,135,29,131)(26,136,30,132)(27,129,31,133)(28,130,32,134)(33,86,37,82)(34,87,38,83)(35,88,39,84)(36,81,40,85)(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)(65,111,69,107)(66,112,70,108)(67,105,71,109)(68,106,72,110)(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)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,125,29,121)(26,126,30,122)(27,127,31,123)(28,128,32,124)(41,97,45,101)(42,98,46,102)(43,99,47,103)(44,100,48,104)(49,61,53,57)(50,62,54,58)(51,63,55,59)(52,64,56,60)(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)(105,115,109,119)(106,116,110,120)(107,117,111,113)(108,118,112,114), (1,97,18,133,111)(2,134,98,112,19)(3,105,135,20,99)(4,21,106,100,136)(5,101,22,129,107)(6,130,102,108,23)(7,109,131,24,103)(8,17,110,104,132)(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)(25,137,151,57,67)(26,58,138,68,152)(27,69,59,145,139)(28,146,70,140,60)(29,141,147,61,71)(30,62,142,72,148)(31,65,63,149,143)(32,150,66,144,64)(33,116,48,76,158)(34,77,117,159,41)(35,160,78,42,118)(36,43,153,119,79)(37,120,44,80,154)(38,73,113,155,45)(39,156,74,46,114)(40,47,157,115,75), (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([(1,61),(2,62),(3,63),(4,64),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27),(33,124),(34,125),(35,126),(36,127),(37,128),(38,121),(39,122),(40,123),(49,159),(50,160),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(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),(105,149),(106,150),(107,151),(108,152),(109,145),(110,146),(111,147),(112,148),(129,137),(130,138),(131,139),(132,140),(133,141),(134,142),(135,143),(136,144)], [(1,59,5,63),(2,60,6,64),(3,61,7,57),(4,62,8,58),(9,119,13,115),(10,120,14,116),(11,113,15,117),(12,114,16,118),(17,138,21,142),(18,139,22,143),(19,140,23,144),(20,141,24,137),(25,135,29,131),(26,136,30,132),(27,129,31,133),(28,130,32,134),(33,86,37,82),(34,87,38,83),(35,88,39,84),(36,81,40,85),(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),(65,111,69,107),(66,112,70,108),(67,105,71,109),(68,106,72,110),(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),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,125,29,121),(26,126,30,122),(27,127,31,123),(28,128,32,124),(41,97,45,101),(42,98,46,102),(43,99,47,103),(44,100,48,104),(49,61,53,57),(50,62,54,58),(51,63,55,59),(52,64,56,60),(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),(105,115,109,119),(106,116,110,120),(107,117,111,113),(108,118,112,114)], [(1,97,18,133,111),(2,134,98,112,19),(3,105,135,20,99),(4,21,106,100,136),(5,101,22,129,107),(6,130,102,108,23),(7,109,131,24,103),(8,17,110,104,132),(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),(25,137,151,57,67),(26,58,138,68,152),(27,69,59,145,139),(28,146,70,140,60),(29,141,147,61,71),(30,62,142,72,148),(31,65,63,149,143),(32,150,66,144,64),(33,116,48,76,158),(34,77,117,159,41),(35,160,78,42,118),(36,43,153,119,79),(37,120,44,80,154),(38,73,113,155,45),(39,156,74,46,114),(40,47,157,115,75)], [(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 2A2B2C2D···2I4A···4F4G4H4I4J 5 8A···8H8I···8T10A10B10C20A···20F
order12222···24···4444458···88···810101020···20
size111110···102···2555545···510···104448···8

50 irreducible representations

dim1111111124448
type+++++++++
imageC1C2C2C2C2C4C4C4C8○D4F5C2×F5C2×F5Q8.F5
kernelC2×Q8.F5C2×D5⋊C8C2×C4.F5Q8.F5C2×Q82D5C2×D20Q82D5Q8×C10C10C2×Q8C2×C4Q8C2
# reps1338168281342

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

10000000
01000000
004000000
000400000
00001000
00000100
00000010
00000001
,
320000000
09000000
00900000
0032320000
00001000
00000100
00000010
00000001
,
01000000
400000000
0040390000
00110000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040404040
00001000
00000100
00000010
,
30000000
03000000
00300000
00030000
00003802222
00002222038
00001916190
0000325253

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

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