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G = C4×C8⋊D5order 320 = 26·5

Direct product of C4 and C8⋊D5

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

Aliases: C4×C8⋊D5, C2011M4(2), D10.4C42, C42.283D10, Dic5.4C42, (C4×C8)⋊13D5, C812(C4×D5), C4035(C2×C4), (C4×C40)⋊20C2, C54(C4×M4(2)), C408C430C2, C2.7(D5×C42), (C2×C8).323D10, C10.26(C2×C42), (C4×Dic5).16C4, (D5×C42).12C2, (C4×C20).339C22, C20.182(C22×C4), (C2×C20).805C23, (C2×C40).405C22, C10.37(C2×M4(2)), (C4×Dic5).295C22, C4.97(C2×C4×D5), (C2×C4×D5).17C4, (C4×C52C8)⋊20C2, C52C823(C2×C4), C2.2(C2×C8⋊D5), C22.37(C2×C4×D5), (C4×D5).55(C2×C4), (C2×C4).174(C4×D5), (C2×C20).420(C2×C4), (C2×C8⋊D5).17C2, (C2×C4×D5).337C22, (C22×D5).95(C2×C4), (C2×C4).747(C22×D5), (C2×C10).161(C22×C4), (C2×C52C8).299C22, (C2×Dic5).135(C2×C4), SmallGroup(320,314)

Series: Derived Chief Lower central Upper central

C1C10 — C4×C8⋊D5
C1C5C10C20C2×C20C2×C4×D5D5×C42 — C4×C8⋊D5
C5C10 — C4×C8⋊D5
C1C42C4×C8

Generators and relations for C4×C8⋊D5
 G = < a,b,c,d | a4=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 398 in 142 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×6], C4 [×4], C22, C22 [×4], C5, C8 [×4], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, D5 [×2], C10, C10 [×2], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], M4(2) [×8], C22×C4 [×3], Dic5 [×2], Dic5 [×2], C20 [×6], D10 [×2], D10 [×2], C2×C10, C4×C8, C4×C8, C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C52C8 [×4], C40 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×M4(2), C8⋊D5 [×8], C2×C52C8 [×2], C4×Dic5, C4×Dic5 [×2], C4×C20, C2×C40 [×2], C2×C4×D5, C2×C4×D5 [×2], C4×C52C8, C408C4 [×2], C4×C40, D5×C42, C2×C8⋊D5 [×2], C4×C8⋊D5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], M4(2) [×4], C22×C4 [×3], D10 [×3], C2×C42, C2×M4(2) [×2], C4×D5 [×6], C22×D5, C4×M4(2), C8⋊D5 [×4], C2×C4×D5 [×3], D5×C42, C2×C8⋊D5 [×2], C4×C8⋊D5

Smallest permutation representation of C4×C8⋊D5
On 160 points
Generators in S160
(1 18 121 142)(2 19 122 143)(3 20 123 144)(4 21 124 137)(5 22 125 138)(6 23 126 139)(7 24 127 140)(8 17 128 141)(9 48 86 129)(10 41 87 130)(11 42 88 131)(12 43 81 132)(13 44 82 133)(14 45 83 134)(15 46 84 135)(16 47 85 136)(25 111 102 50)(26 112 103 51)(27 105 104 52)(28 106 97 53)(29 107 98 54)(30 108 99 55)(31 109 100 56)(32 110 101 49)(33 62 145 113)(34 63 146 114)(35 64 147 115)(36 57 148 116)(37 58 149 117)(38 59 150 118)(39 60 151 119)(40 61 152 120)(65 94 153 73)(66 95 154 74)(67 96 155 75)(68 89 156 76)(69 90 157 77)(70 91 158 78)(71 92 159 79)(72 93 160 80)
(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 35 81 107 73)(2 36 82 108 74)(3 37 83 109 75)(4 38 84 110 76)(5 39 85 111 77)(6 40 86 112 78)(7 33 87 105 79)(8 34 88 106 80)(9 51 91 126 152)(10 52 92 127 145)(11 53 93 128 146)(12 54 94 121 147)(13 55 95 122 148)(14 56 96 123 149)(15 49 89 124 150)(16 50 90 125 151)(17 63 131 97 72)(18 64 132 98 65)(19 57 133 99 66)(20 58 134 100 67)(21 59 135 101 68)(22 60 136 102 69)(23 61 129 103 70)(24 62 130 104 71)(25 157 138 119 47)(26 158 139 120 48)(27 159 140 113 41)(28 160 141 114 42)(29 153 142 115 43)(30 154 143 116 44)(31 155 144 117 45)(32 156 137 118 46)
(1 94)(2 91)(3 96)(4 93)(5 90)(6 95)(7 92)(8 89)(9 82)(10 87)(11 84)(12 81)(13 86)(14 83)(15 88)(16 85)(17 156)(18 153)(19 158)(20 155)(21 160)(22 157)(23 154)(24 159)(25 60)(26 57)(27 62)(28 59)(29 64)(30 61)(31 58)(32 63)(33 52)(34 49)(35 54)(36 51)(37 56)(38 53)(39 50)(40 55)(41 130)(42 135)(43 132)(44 129)(45 134)(46 131)(47 136)(48 133)(65 142)(66 139)(67 144)(68 141)(69 138)(70 143)(71 140)(72 137)(73 121)(74 126)(75 123)(76 128)(77 125)(78 122)(79 127)(80 124)(97 118)(98 115)(99 120)(100 117)(101 114)(102 119)(103 116)(104 113)(105 145)(106 150)(107 147)(108 152)(109 149)(110 146)(111 151)(112 148)

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

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

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

104 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R5A5B8A···8H8I···8P10A···10F20A···20X40A···40AF
order1222224···44···4558···88···810···1020···2040···40
size111110101···110···10222···210···102···22···22···2

104 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4D5M4(2)D10D10C4×D5C4×D5C8⋊D5
kernelC4×C8⋊D5C4×C52C8C408C4C4×C40D5×C42C2×C8⋊D5C8⋊D5C4×Dic5C2×C4×D5C4×C8C20C42C2×C8C8C2×C4C4
# reps1121121644282416832

Matrix representation of C4×C8⋊D5 in GL4(𝔽41) generated by

9000
0900
0090
0009
,
1000
0100
001018
002031
,
64000
1000
00740
00840
,
35100
6600
00035
00340
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,10,20,0,0,18,31],[6,1,0,0,40,0,0,0,0,0,7,8,0,0,40,40],[35,6,0,0,1,6,0,0,0,0,0,34,0,0,35,0] >;

C4×C8⋊D5 in GAP, Magma, Sage, TeX

C_4\times C_8\rtimes D_5
% in TeX

G:=Group("C4xC8:D5");
// GroupNames label

G:=SmallGroup(320,314);
// by ID

G=gap.SmallGroup(320,314);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,58,136,12550]);
// Polycyclic

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

׿
×
𝔽