Copied to
clipboard

## G = (C2×C10).40D8order 320 = 26·5

### 17th non-split extension by C2×C10 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C10).40D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C20⋊7D4 — (C2×C10).40D8
 Lower central C5 — C10 — C2×C20 — (C2×C10).40D8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for (C2×C10).40D8
G = < a,b,c,d | a10=b2=c8=1, d2=a5, ab=ba, cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 462 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C4⋊D4, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×6], C22×D5, C22×C10, C22.D8, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, C10.D8 [×2], D206C4 [×2], C20.55D4, C207D4, C10×C4⋊C4, (C2×C10).40D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×D8, C8.C22, C5⋊D4 [×2], C22×D5, C22.D8, D4⋊D5 [×2], C4○D20 [×2], C2×C5⋊D4, C23.23D10, C2×D4⋊D5, C20.C23, (C2×C10).40D8

Smallest permutation representation of (C2×C10).40D8
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)
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 97)(22 98)(23 99)(24 100)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 121)(38 122)(39 123)(40 124)(41 117)(42 118)(43 119)(44 120)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 141)(58 142)(59 143)(60 144)(61 137)(62 138)(63 139)(64 140)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(81 157)(82 158)(83 159)(84 160)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)
(1 71 45 70 30 85 31 56)(2 80 46 69 21 84 32 55)(3 79 47 68 22 83 33 54)(4 78 48 67 23 82 34 53)(5 77 49 66 24 81 35 52)(6 76 50 65 25 90 36 51)(7 75 41 64 26 89 37 60)(8 74 42 63 27 88 38 59)(9 73 43 62 28 87 39 58)(10 72 44 61 29 86 40 57)(11 116 136 91 151 130 150 105)(12 115 137 100 152 129 141 104)(13 114 138 99 153 128 142 103)(14 113 139 98 154 127 143 102)(15 112 140 97 155 126 144 101)(16 111 131 96 156 125 145 110)(17 120 132 95 157 124 146 109)(18 119 133 94 158 123 147 108)(19 118 134 93 159 122 148 107)(20 117 135 92 160 121 149 106)
(1 131 6 136)(2 132 7 137)(3 133 8 138)(4 134 9 139)(5 135 10 140)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 146 26 141)(22 147 27 142)(23 148 28 143)(24 149 29 144)(25 150 30 145)(31 156 36 151)(32 157 37 152)(33 158 38 153)(34 159 39 154)(35 160 40 155)(51 91 56 96)(52 92 57 97)(53 93 58 98)(54 94 59 99)(55 95 60 100)(61 101 66 106)(62 102 67 107)(63 103 68 108)(64 104 69 109)(65 105 70 110)(71 111 76 116)(72 112 77 117)(73 113 78 118)(74 114 79 119)(75 115 80 120)(81 121 86 126)(82 122 87 127)(83 123 88 128)(84 124 89 129)(85 125 90 130)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 2 4 4 4 ··· 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 40 2 2 4 ··· 4 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D8 D10 D10 C5⋊D4 C5⋊D4 C4○D20 C8.C22 D4⋊D5 C20.C23 kernel (C2×C10).40D8 C10.D8 D20⋊6C4 C20.55D4 C20⋊7D4 C10×C4⋊C4 C2×C20 C22×C10 C2×C4⋊C4 C20 C2×C10 C4⋊C4 C22×C4 C2×C4 C23 C4 C10 C22 C2 # reps 1 2 2 1 1 1 1 1 2 4 4 4 2 4 4 16 1 4 4

Matrix representation of (C2×C10).40D8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 25 1 0 0 0 0 32 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 20 9 0 0 0 0 24 21 0 0 0 0 0 0 7 34 0 0 0 0 1 34 0 0 0 0 0 0 12 12 0 0 0 0 29 12
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 29 29 0 0 0 0 29 12

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,32,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,24,0,0,0,0,9,21,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,0,0,12,29,0,0,0,0,12,12],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,29,29,0,0,0,0,29,12] >;

(C2×C10).40D8 in GAP, Magma, Sage, TeX

(C_2\times C_{10})._{40}D_8
% in TeX

G:=Group("(C2xC10).40D8");
// GroupNames label

G:=SmallGroup(320,594);
// by ID

G=gap.SmallGroup(320,594);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,100,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=1,d^2=a^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽