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## G = Dic5.5D8order 320 = 26·5

### 1st non-split extension by Dic5 of D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5.5D8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C20⋊D4 — Dic5.5D8
 Lower central C5 — C10 — C2×C20 — Dic5.5D8
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for Dic5.5D8
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a5b, dcd=a5c-1 >

Subgroups: 590 in 118 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×4], D4 [×8], Q8 [×2], C23 [×2], D5, C10 [×3], C10, C42, C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, C2×D4, C2×D4 [×3], C2×Q8, Dic5 [×4], Dic5, C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C4×C8, D4⋊C4, D4⋊C4 [×3], C41D4, C4⋊Q8, C52C8, C40, Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C4.4D8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, C2×C5⋊D4 [×2], D4×C10, D206C4, C8×Dic5, D205C4, D4⋊Dic5, C5×D4⋊C4, C20⋊Q8, C20⋊D4, Dic5.5D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], SD16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C2×D8, C2×SD16, C22×D5, C4.4D8, C4○D20, D4×D5, D42D5, Dic5.5D4, D5×D8, D5×SD16, Dic5.5D8

Smallest permutation representation of Dic5.5D8
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 98 6 93)(2 97 7 92)(3 96 8 91)(4 95 9 100)(5 94 10 99)(11 61 16 66)(12 70 17 65)(13 69 18 64)(14 68 19 63)(15 67 20 62)(21 105 26 110)(22 104 27 109)(23 103 28 108)(24 102 29 107)(25 101 30 106)(31 125 36 130)(32 124 37 129)(33 123 38 128)(34 122 39 127)(35 121 40 126)(41 115 46 120)(42 114 47 119)(43 113 48 118)(44 112 49 117)(45 111 50 116)(51 160 56 155)(52 159 57 154)(53 158 58 153)(54 157 59 152)(55 156 60 151)(71 140 76 135)(72 139 77 134)(73 138 78 133)(74 137 79 132)(75 136 80 131)(81 150 86 145)(82 149 87 144)(83 148 88 143)(84 147 89 142)(85 146 90 141)
(1 153 33 148 28 19 43 138)(2 154 34 149 29 20 44 139)(3 155 35 150 30 11 45 140)(4 156 36 141 21 12 46 131)(5 157 37 142 22 13 47 132)(6 158 38 143 23 14 48 133)(7 159 39 144 24 15 49 134)(8 160 40 145 25 16 50 135)(9 151 31 146 26 17 41 136)(10 152 32 147 27 18 42 137)(51 121 86 106 61 111 76 96)(52 122 87 107 62 112 77 97)(53 123 88 108 63 113 78 98)(54 124 89 109 64 114 79 99)(55 125 90 110 65 115 80 100)(56 126 81 101 66 116 71 91)(57 127 82 102 67 117 72 92)(58 128 83 103 68 118 73 93)(59 129 84 104 69 119 74 94)(60 130 85 105 70 120 75 95)
(1 98)(2 99)(3 100)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 D8 SD16 C4○D4 D10 D10 D10 C4○D20 D4⋊2D5 D4×D5 D5×D8 D5×SD16 kernel Dic5.5D8 D20⋊6C4 C8×Dic5 D20⋊5C4 D4⋊Dic5 C5×D4⋊C4 C20⋊Q8 C20⋊D4 C2×Dic5 D4⋊C4 Dic5 Dic5 C20 C4⋊C4 C2×C8 C2×D4 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 4 4 4 2 2 2 8 2 2 4 4

Matrix representation of Dic5.5D8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 1 0 0 0 0 33 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 39 0 0 0 0 1 40 0 0 0 0 0 0 34 1 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 11 0 0 0 0 15 11 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 12 12 0 0 0 0 29 12
,
 40 2 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 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,15,0,0,0,0,11,11,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,29,0,0,0,0,12,12],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

Dic5.5D8 in GAP, Magma, Sage, TeX

{\rm Dic}_5._5D_8
% in TeX

G:=Group("Dic5.5D8");
// GroupNames label

G:=SmallGroup(320,387);
// by ID

G=gap.SmallGroup(320,387);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,422,135,100,570,297,136,12550]);
// Polycyclic

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

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