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G = C2×C8.F5order 320 = 26·5

Direct product of C2 and C8.F5

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

Aliases: C2×C8.F5, C101M5(2), C5⋊C161C22, (C4×D5).8C8, C51(C2×M5(2)), C8.32(C2×F5), (C2×C8).19F5, C20.24(C2×C8), C40.39(C2×C4), (C2×C40).23C4, (C8×D5).11C4, C4.19(D5⋊C8), D10.17(C2×C8), C10.9(C22×C8), (C22×D5).8C8, C4.46(C22×F5), C20.86(C22×C4), C52C8.36C23, Dic5.17(C2×C8), (C2×Dic5).13C8, (C8×D5).62C22, C22.13(D5⋊C8), (C2×C5⋊C16)⋊7C2, (C2×C4×D5).46C4, (D5×C2×C8).30C2, C2.10(C2×D5⋊C8), (C2×C10).11(C2×C8), C52C8.52(C2×C4), (C4×D5).92(C2×C4), (C2×C4).164(C2×F5), (C2×C20).173(C2×C4), (C2×C52C8).349C22, SmallGroup(320,1052)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C8.F5
C1C5C10C20C52C8C5⋊C16C2×C5⋊C16 — C2×C8.F5
C5C10 — C2×C8.F5
C1C2×C4C2×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 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], C16 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], M5(2) [×4], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16 [×4], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, C8.F5 [×4], C2×C5⋊C16 [×2], D5×C2×C8, C2×C8.F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C2×F5 [×3], C2×M5(2), D5⋊C8 [×2], C22×F5, C8.F5 [×2], C2×D5⋊C8, C2×C8.F5

Smallest permutation representation of C2×C8.F5
On 160 points
Generators in S160
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 141)(12 142)(13 143)(14 144)(15 129)(16 130)(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)(33 149)(34 150)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 145)(46 146)(47 147)(48 148)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 65)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 113)(93 114)(94 115)(95 116)(96 117)
(1 141 5 129 9 133 13 137)(2 134 6 138 10 142 14 130)(3 143 7 131 11 135 15 139)(4 136 8 140 12 144 16 132)(17 35 21 39 25 43 29 47)(18 44 22 48 26 36 30 40)(19 37 23 41 27 45 31 33)(20 46 24 34 28 38 32 42)(49 123 53 127 57 115 61 119)(50 116 54 120 58 124 62 128)(51 125 55 113 59 117 63 121)(52 118 56 122 60 126 64 114)(65 93 69 81 73 85 77 89)(66 86 70 90 74 94 78 82)(67 95 71 83 75 87 79 91)(68 88 72 92 76 96 80 84)(97 160 101 148 105 152 109 156)(98 153 102 157 106 145 110 149)(99 146 103 150 107 154 111 158)(100 155 104 159 108 147 112 151)
(1 156 122 67 24)(2 68 157 25 123)(3 26 69 124 158)(4 125 27 159 70)(5 160 126 71 28)(6 72 145 29 127)(7 30 73 128 146)(8 113 31 147 74)(9 148 114 75 32)(10 76 149 17 115)(11 18 77 116 150)(12 117 19 151 78)(13 152 118 79 20)(14 80 153 21 119)(15 22 65 120 154)(16 121 23 155 66)(33 112 94 140 59)(34 141 97 60 95)(35 61 142 96 98)(36 81 62 99 143)(37 100 82 144 63)(38 129 101 64 83)(39 49 130 84 102)(40 85 50 103 131)(41 104 86 132 51)(42 133 105 52 87)(43 53 134 88 106)(44 89 54 107 135)(45 108 90 136 55)(46 137 109 56 91)(47 57 138 92 110)(48 93 58 111 139)
(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)| (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,129)(16,130)(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)(33,149)(34,150)(35,151)(36,152)(37,153)(38,154)(39,155)(40,156)(41,157)(42,158)(43,159)(44,160)(45,145)(46,146)(47,147)(48,148)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,65)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,113)(93,114)(94,115)(95,116)(96,117), (1,141,5,129,9,133,13,137)(2,134,6,138,10,142,14,130)(3,143,7,131,11,135,15,139)(4,136,8,140,12,144,16,132)(17,35,21,39,25,43,29,47)(18,44,22,48,26,36,30,40)(19,37,23,41,27,45,31,33)(20,46,24,34,28,38,32,42)(49,123,53,127,57,115,61,119)(50,116,54,120,58,124,62,128)(51,125,55,113,59,117,63,121)(52,118,56,122,60,126,64,114)(65,93,69,81,73,85,77,89)(66,86,70,90,74,94,78,82)(67,95,71,83,75,87,79,91)(68,88,72,92,76,96,80,84)(97,160,101,148,105,152,109,156)(98,153,102,157,106,145,110,149)(99,146,103,150,107,154,111,158)(100,155,104,159,108,147,112,151), (1,156,122,67,24)(2,68,157,25,123)(3,26,69,124,158)(4,125,27,159,70)(5,160,126,71,28)(6,72,145,29,127)(7,30,73,128,146)(8,113,31,147,74)(9,148,114,75,32)(10,76,149,17,115)(11,18,77,116,150)(12,117,19,151,78)(13,152,118,79,20)(14,80,153,21,119)(15,22,65,120,154)(16,121,23,155,66)(33,112,94,140,59)(34,141,97,60,95)(35,61,142,96,98)(36,81,62,99,143)(37,100,82,144,63)(38,129,101,64,83)(39,49,130,84,102)(40,85,50,103,131)(41,104,86,132,51)(42,133,105,52,87)(43,53,134,88,106)(44,89,54,107,135)(45,108,90,136,55)(46,137,109,56,91)(47,57,138,92,110)(48,93,58,111,139), (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( (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,129)(16,130)(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)(33,149)(34,150)(35,151)(36,152)(37,153)(38,154)(39,155)(40,156)(41,157)(42,158)(43,159)(44,160)(45,145)(46,146)(47,147)(48,148)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,65)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)(89,126)(90,127)(91,128)(92,113)(93,114)(94,115)(95,116)(96,117), (1,141,5,129,9,133,13,137)(2,134,6,138,10,142,14,130)(3,143,7,131,11,135,15,139)(4,136,8,140,12,144,16,132)(17,35,21,39,25,43,29,47)(18,44,22,48,26,36,30,40)(19,37,23,41,27,45,31,33)(20,46,24,34,28,38,32,42)(49,123,53,127,57,115,61,119)(50,116,54,120,58,124,62,128)(51,125,55,113,59,117,63,121)(52,118,56,122,60,126,64,114)(65,93,69,81,73,85,77,89)(66,86,70,90,74,94,78,82)(67,95,71,83,75,87,79,91)(68,88,72,92,76,96,80,84)(97,160,101,148,105,152,109,156)(98,153,102,157,106,145,110,149)(99,146,103,150,107,154,111,158)(100,155,104,159,108,147,112,151), (1,156,122,67,24)(2,68,157,25,123)(3,26,69,124,158)(4,125,27,159,70)(5,160,126,71,28)(6,72,145,29,127)(7,30,73,128,146)(8,113,31,147,74)(9,148,114,75,32)(10,76,149,17,115)(11,18,77,116,150)(12,117,19,151,78)(13,152,118,79,20)(14,80,153,21,119)(15,22,65,120,154)(16,121,23,155,66)(33,112,94,140,59)(34,141,97,60,95)(35,61,142,96,98)(36,81,62,99,143)(37,100,82,144,63)(38,129,101,64,83)(39,49,130,84,102)(40,85,50,103,131)(41,104,86,132,51)(42,133,105,52,87)(43,53,134,88,106)(44,89,54,107,135)(45,108,90,136,55)(46,137,109,56,91)(47,57,138,92,110)(48,93,58,111,139), (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([(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,141),(12,142),(13,143),(14,144),(15,129),(16,130),(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),(33,149),(34,150),(35,151),(36,152),(37,153),(38,154),(39,155),(40,156),(41,157),(42,158),(43,159),(44,160),(45,145),(46,146),(47,147),(48,148),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,65),(81,118),(82,119),(83,120),(84,121),(85,122),(86,123),(87,124),(88,125),(89,126),(90,127),(91,128),(92,113),(93,114),(94,115),(95,116),(96,117)], [(1,141,5,129,9,133,13,137),(2,134,6,138,10,142,14,130),(3,143,7,131,11,135,15,139),(4,136,8,140,12,144,16,132),(17,35,21,39,25,43,29,47),(18,44,22,48,26,36,30,40),(19,37,23,41,27,45,31,33),(20,46,24,34,28,38,32,42),(49,123,53,127,57,115,61,119),(50,116,54,120,58,124,62,128),(51,125,55,113,59,117,63,121),(52,118,56,122,60,126,64,114),(65,93,69,81,73,85,77,89),(66,86,70,90,74,94,78,82),(67,95,71,83,75,87,79,91),(68,88,72,92,76,96,80,84),(97,160,101,148,105,152,109,156),(98,153,102,157,106,145,110,149),(99,146,103,150,107,154,111,158),(100,155,104,159,108,147,112,151)], [(1,156,122,67,24),(2,68,157,25,123),(3,26,69,124,158),(4,125,27,159,70),(5,160,126,71,28),(6,72,145,29,127),(7,30,73,128,146),(8,113,31,147,74),(9,148,114,75,32),(10,76,149,17,115),(11,18,77,116,150),(12,117,19,151,78),(13,152,118,79,20),(14,80,153,21,119),(15,22,65,120,154),(16,121,23,155,66),(33,112,94,140,59),(34,141,97,60,95),(35,61,142,96,98),(36,81,62,99,143),(37,100,82,144,63),(38,129,101,64,83),(39,49,130,84,102),(40,85,50,103,131),(41,104,86,132,51),(42,133,105,52,87),(43,53,134,88,106),(44,89,54,107,135),(45,108,90,136,55),(46,137,109,56,91),(47,57,138,92,110),(48,93,58,111,139)], [(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 2A2B2C2D2E4A4B4C4D4E4F 5 8A8B8C8D8E···8L10A10B10C16A···16P20A20B20C20D40A···40H
order122222444444588888···810101016···162020202040···40
size1111101011111010422225···544410···1044444···4

56 irreducible representations

dim11111111112444444
type+++++++
imageC1C2C2C2C4C4C4C8C8C8M5(2)F5C2×F5C2×F5D5⋊C8D5⋊C8C8.F5
kernelC2×C8.F5C8.F5C2×C5⋊C16D5×C2×C8C8×D5C2×C40C2×C4×D5C4×D5C2×Dic5C22×D5C10C2×C8C8C2×C4C4C22C2
# reps14214228448121228

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

100000
010000
00240000
00024000
00002400
00000240
,
23300000
6380000
00240000
00024000
00002400
00000240
,
100000
010000
00240100
00240010
00240001
00240000
,
1910000
1292220000
001736830
00176680173
00173068176
000368173

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

׿
×
𝔽