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G = C8×C5⋊D4order 320 = 26·5

Direct product of C8 and C5⋊D4

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

Aliases: C8×C5⋊D4, C4031D4, C58(C8×D4), D106(C2×C8), (C22×C8)⋊2D5, C222(C8×D5), Dic53(C2×C8), (C22×C40)⋊15C2, (C8×Dic5)⋊26C2, C10.107(C4×D4), (C2×C8).345D10, C20.435(C2×D4), D101C842C2, C23.33(C4×D5), C10.41(C8○D4), C10.44(C22×C8), C20.8Q843C2, C23.D5.22C4, D10⋊C4.30C4, C20.251(C4○D4), C4.135(C4○D20), C20.55D432C2, (C2×C20).859C23, (C2×C40).353C22, C10.D4.30C4, (C22×C4).401D10, C2.5(D20.3C4), (C22×C20).560C22, (C4×Dic5).313C22, (D5×C2×C8)⋊24C2, C2.20(D5×C2×C8), C2.3(C4×C5⋊D4), (C2×C10)⋊11(C2×C8), (C2×C4).93(C4×D5), C22.61(C2×C4×D5), (C2×C5⋊D4).27C4, (C4×C5⋊D4).19C2, C4.125(C2×C5⋊D4), (C2×C20).382(C2×C4), (C2×C4×D5).353C22, (C22×D5).80(C2×C4), (C2×C4).801(C22×D5), (C22×C10).162(C2×C4), (C2×C10).230(C22×C4), (C2×C52C8).328C22, (C2×Dic5).111(C2×C4), SmallGroup(320,736)

Series: Derived Chief Lower central Upper central

C1C10 — C8×C5⋊D4
C1C5C10C20C2×C20C2×C4×D5C4×C5⋊D4 — C8×C5⋊D4
C5C10 — C8×C5⋊D4
C1C2×C8C22×C8

Generators and relations for C8×C5⋊D4
 G = < a,b,c,d | a8=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 382 in 134 conjugacy classes, 63 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C22×C8, C52C8 [×2], C40 [×2], C40, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C8×D4, C8×D5 [×2], C2×C52C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C2×C40 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D101C8, C20.55D4, D5×C2×C8, C4×C5⋊D4, C22×C40, C8×C5⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C22×C8, C8○D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C8×D4, C8×D5 [×2], C2×C4×D5, C4○D20, C2×C5⋊D4, D5×C2×C8, D20.3C4, C4×C5⋊D4, C8×C5⋊D4

Smallest permutation representation of C8×C5⋊D4
On 160 points
Generators in S160
(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 54 121 107 73)(2 55 122 108 74)(3 56 123 109 75)(4 49 124 110 76)(5 50 125 111 77)(6 51 126 112 78)(7 52 127 105 79)(8 53 128 106 80)(9 91 86 152 34)(10 92 87 145 35)(11 93 88 146 36)(12 94 81 147 37)(13 95 82 148 38)(14 96 83 149 39)(15 89 84 150 40)(16 90 85 151 33)(17 48 116 144 72)(18 41 117 137 65)(19 42 118 138 66)(20 43 119 139 67)(21 44 120 140 68)(22 45 113 141 69)(23 46 114 142 70)(24 47 115 143 71)(25 157 131 103 62)(26 158 132 104 63)(27 159 133 97 64)(28 160 134 98 57)(29 153 135 99 58)(30 154 136 100 59)(31 155 129 101 60)(32 156 130 102 61)
(1 131 81 22)(2 132 82 23)(3 133 83 24)(4 134 84 17)(5 135 85 18)(6 136 86 19)(7 129 87 20)(8 130 88 21)(9 118 112 59)(10 119 105 60)(11 120 106 61)(12 113 107 62)(13 114 108 63)(14 115 109 64)(15 116 110 57)(16 117 111 58)(25 37 141 121)(26 38 142 122)(27 39 143 123)(28 40 144 124)(29 33 137 125)(30 34 138 126)(31 35 139 127)(32 36 140 128)(41 77 99 90)(42 78 100 91)(43 79 101 92)(44 80 102 93)(45 73 103 94)(46 74 104 95)(47 75 97 96)(48 76 98 89)(49 160 150 72)(50 153 151 65)(51 154 152 66)(52 155 145 67)(53 156 146 68)(54 157 147 69)(55 158 148 70)(56 159 149 71)
(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 134)(18 135)(19 136)(20 129)(21 130)(22 131)(23 132)(24 133)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)(57 144)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(89 150)(90 151)(91 152)(92 145)(93 146)(94 147)(95 148)(96 149)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)

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

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

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

104 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L5A5B8A···8H8I8J8K8L8M···8T10A···10N20A···20P40A···40AF
order122222224444444···4558···888888···810···1020···2040···40
size111122101011112210···10221···1222210···102···22···22···2

104 irreducible representations

dim1111111111111222222222222
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4C8D4D5C4○D4D10D10C8○D4C5⋊D4C4×D5C4×D5C4○D20C8×D5D20.3C4
kernelC8×C5⋊D4C8×Dic5C20.8Q8D101C8C20.55D4D5×C2×C8C4×C5⋊D4C22×C40C10.D4D10⋊C4C23.D5C2×C5⋊D4C5⋊D4C40C22×C8C20C2×C8C22×C4C10C8C2×C4C23C4C22C2
# reps1111111122221622242484481616

Matrix representation of C8×C5⋊D4 in GL3(𝔽41) generated by

300
0380
0038
,
100
0740
0840
,
100
0318
0438
,
4000
0341
0347
G:=sub<GL(3,GF(41))| [3,0,0,0,38,0,0,0,38],[1,0,0,0,7,8,0,40,40],[1,0,0,0,3,4,0,18,38],[40,0,0,0,34,34,0,1,7] >;

C8×C5⋊D4 in GAP, Magma, Sage, TeX

C_8\times C_5\rtimes D_4
% in TeX

G:=Group("C8xC5:D4");
// GroupNames label

G:=SmallGroup(320,736);
// by ID

G=gap.SmallGroup(320,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,58,136,12550]);
// Polycyclic

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

׿
×
𝔽