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## G = Dic10⋊24D4order 320 = 26·5

### 2nd semidirect product of Dic10 and D4 acting through Inn(Dic10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊24D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — D20⋊8C4 — Dic10⋊24D4
 Lower central C5 — C2×C10 — Dic10⋊24D4
 Upper central C1 — C22 — C4×D4

Generators and relations for Dic1024D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 1270 in 312 conjugacy classes, 107 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×18], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×24], Q8 [×4], C23 [×2], C23 [×4], D5 [×4], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×14], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×2], C20 [×4], C20 [×3], D10 [×12], C2×C10, C2×C10 [×6], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4 [×6], C41D4 [×3], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×8], D20 [×10], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×D5 [×4], C22×C10 [×2], Q86D4, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×4], C2×D20 [×6], C4○D20 [×8], C2×C5⋊D4 [×8], C22×C20 [×2], D4×C10, C4×Dic10, C204D4, D10⋊D4 [×4], D208C4 [×2], C207D4 [×2], C20⋊D4 [×2], D4×C20, C2×C4○D20 [×2], Dic1024D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C22×D5 [×7], Q86D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D48D10, Dic1024D4

Smallest permutation representation of Dic1024D4
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)
(1 122 11 132)(2 121 12 131)(3 140 13 130)(4 139 14 129)(5 138 15 128)(6 137 16 127)(7 136 17 126)(8 135 18 125)(9 134 19 124)(10 133 20 123)(21 102 31 112)(22 101 32 111)(23 120 33 110)(24 119 34 109)(25 118 35 108)(26 117 36 107)(27 116 37 106)(28 115 38 105)(29 114 39 104)(30 113 40 103)(41 141 51 151)(42 160 52 150)(43 159 53 149)(44 158 54 148)(45 157 55 147)(46 156 56 146)(47 155 57 145)(48 154 58 144)(49 153 59 143)(50 152 60 142)(61 81 71 91)(62 100 72 90)(63 99 73 89)(64 98 74 88)(65 97 75 87)(66 96 76 86)(67 95 77 85)(68 94 78 84)(69 93 79 83)(70 92 80 82)
(1 100 28 141)(2 81 29 142)(3 82 30 143)(4 83 31 144)(5 84 32 145)(6 85 33 146)(7 86 34 147)(8 87 35 148)(9 88 36 149)(10 89 37 150)(11 90 38 151)(12 91 39 152)(13 92 40 153)(14 93 21 154)(15 94 22 155)(16 95 23 156)(17 96 24 157)(18 97 25 158)(19 98 26 159)(20 99 27 160)(41 132 62 105)(42 133 63 106)(43 134 64 107)(44 135 65 108)(45 136 66 109)(46 137 67 110)(47 138 68 111)(48 139 69 112)(49 140 70 113)(50 121 71 114)(51 122 72 115)(52 123 73 116)(53 124 74 117)(54 125 75 118)(55 126 76 119)(56 127 77 120)(57 128 78 101)(58 129 79 102)(59 130 80 103)(60 131 61 104)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 20 20 20 20 2 ··· 2 4 10 10 10 10 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 D10 C4○D20 2+ 1+4 D4×D5 D4⋊8D10 kernel Dic10⋊24D4 C4×Dic10 C20⋊4D4 D10⋊D4 D20⋊8C4 C20⋊7D4 C20⋊D4 D4×C20 C2×C4○D20 Dic10 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C10 C4 C2 # reps 1 1 1 4 2 2 2 1 2 4 2 4 2 4 2 4 2 16 1 4 4

Matrix representation of Dic1024D4 in GL6(𝔽41)

 0 1 0 0 0 0 40 6 0 0 0 0 0 0 7 20 0 0 0 0 18 34 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 35 1 0 0 0 0 0 0 22 16 0 0 0 0 3 19 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 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 7 36 0 0 0 0 10 34
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 19 25 0 0 0 0 2 22 0 0 0 0 0 0 1 0 0 0 0 0 11 40

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,7,18,0,0,0,0,20,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,22,3,0,0,0,0,16,19,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,0,0,0,36,34],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,19,2,0,0,0,0,25,22,0,0,0,0,0,0,1,11,0,0,0,0,0,40] >;

Dic1024D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{24}D_4
% in TeX

G:=Group("Dic10:24D4");
// GroupNames label

G:=SmallGroup(320,1225);
// by ID

G=gap.SmallGroup(320,1225);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,387,100,675,570,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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