Copied to
clipboard

## G = C4×D5⋊C8order 320 = 26·5

### Direct product of C4 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C4×D5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C4×D5⋊C8
 Lower central C5 — C4×D5⋊C8
 Upper central C1 — C42

Generators and relations for C4×D5⋊C8
G = < a,b,c,d | a4=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 426 in 162 conjugacy classes, 96 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C4×C8, C2×C42, C22×C8, C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C2×C4×C8, C4×Dic5, C4×Dic5, C4×C20, D5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C4×C5⋊C8, D5×C42, C2×D5⋊C8, C4×D5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, F5, C4×C8, C2×C42, C22×C8, C2×F5, C2×C4×C8, D5⋊C8, C4×F5, C22×F5, C2×D5⋊C8, C2×C4×F5, C4×D5⋊C8

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4X 5 8A ··· 8AF 10A 10B 10C 20A ··· 20L order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 8 ··· 8 10 10 10 20 ··· 20 size 1 1 1 1 5 5 5 5 1 ··· 1 5 ··· 5 4 5 ··· 5 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 F5 C2×F5 D5⋊C8 C4×F5 kernel C4×D5⋊C8 C4×C5⋊C8 D5×C42 C2×D5⋊C8 C4×Dic5 C4×C20 D5⋊C8 C2×C4×D5 C4×D5 C42 C2×C4 C4 C4 # reps 1 4 1 2 2 2 16 4 32 1 3 8 4

Matrix representation of C4×D5⋊C8 in GL5(𝔽41)

 1 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32
,
 1 0 0 0 0 0 0 40 0 0 0 1 34 0 0 0 0 0 40 7 0 0 0 34 7
,
 40 0 0 0 0 0 34 7 0 0 0 40 7 0 0 0 0 0 7 40 0 0 0 7 34
,
 27 0 0 0 0 0 0 0 9 0 0 0 0 0 9 0 32 22 0 0 0 0 9 0 0

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,40,34,0,0,0,0,0,40,34,0,0,0,7,7],[40,0,0,0,0,0,34,40,0,0,0,7,7,0,0,0,0,0,7,7,0,0,0,40,34],[27,0,0,0,0,0,0,0,32,0,0,0,0,22,9,0,9,0,0,0,0,0,9,0,0] >;

C4×D5⋊C8 in GAP, Magma, Sage, TeX

C_4\times D_5\rtimes C_8
% in TeX

G:=Group("C4xD5:C8");
// GroupNames label

G:=SmallGroup(320,1013);
// by ID

G=gap.SmallGroup(320,1013);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

׿
×
𝔽