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## G = Dic5⋊5SD16order 320 = 26·5

### 2nd semidirect product of Dic5 and SD16 acting via SD16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5⋊5SD16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C20⋊D4 — Dic5⋊5SD16
 Lower central C5 — C10 — C2×C20 — Dic5⋊5SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for Dic55SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=c3 >

Subgroups: 582 in 128 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C42, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×Q8, C41D4, C2×SD16, C2×SD16, C52C8, C40, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C4⋊SD16, C2×C52C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, Q8⋊D5, C2×C40, C5×SD16, C2×D20, C2×C5⋊D4, D4×C10, Q8×C10, C20.8Q8, D205C4, D4⋊Dic5, C20⋊D4, C2×Q8⋊D5, Q8×Dic5, C10×SD16, Dic55SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, C4⋊SD16, D4×D5, D42D5, C2×C5⋊D4, D5×SD16, D40⋊C2, Dic5⋊D4, Dic55SD16

Smallest permutation representation of Dic55SD16
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)
(1 108 6 103)(2 107 7 102)(3 106 8 101)(4 105 9 110)(5 104 10 109)(11 56 16 51)(12 55 17 60)(13 54 18 59)(14 53 19 58)(15 52 20 57)(21 95 26 100)(22 94 27 99)(23 93 28 98)(24 92 29 97)(25 91 30 96)(31 115 36 120)(32 114 37 119)(33 113 38 118)(34 112 39 117)(35 111 40 116)(41 125 46 130)(42 124 47 129)(43 123 48 128)(44 122 49 127)(45 121 50 126)(61 155 66 160)(62 154 67 159)(63 153 68 158)(64 152 69 157)(65 151 70 156)(71 145 76 150)(72 144 77 149)(73 143 78 148)(74 142 79 147)(75 141 80 146)(81 135 86 140)(82 134 87 139)(83 133 88 138)(84 132 89 137)(85 131 90 136)
(1 148 38 14 28 138 48 158)(2 149 39 15 29 139 49 159)(3 150 40 16 30 140 50 160)(4 141 31 17 21 131 41 151)(5 142 32 18 22 132 42 152)(6 143 33 19 23 133 43 153)(7 144 34 20 24 134 44 154)(8 145 35 11 25 135 45 155)(9 146 36 12 26 136 46 156)(10 147 37 13 27 137 47 157)(51 91 81 121 61 101 71 111)(52 92 82 122 62 102 72 112)(53 93 83 123 63 103 73 113)(54 94 84 124 64 104 74 114)(55 95 85 125 65 105 75 115)(56 96 86 126 66 106 76 116)(57 97 87 127 67 107 77 117)(58 98 88 128 68 108 78 118)(59 99 89 129 69 109 79 119)(60 100 90 130 70 110 80 120)
(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), (1,108,6,103)(2,107,7,102)(3,106,8,101)(4,105,9,110)(5,104,10,109)(11,56,16,51)(12,55,17,60)(13,54,18,59)(14,53,19,58)(15,52,20,57)(21,95,26,100)(22,94,27,99)(23,93,28,98)(24,92,29,97)(25,91,30,96)(31,115,36,120)(32,114,37,119)(33,113,38,118)(34,112,39,117)(35,111,40,116)(41,125,46,130)(42,124,47,129)(43,123,48,128)(44,122,49,127)(45,121,50,126)(61,155,66,160)(62,154,67,159)(63,153,68,158)(64,152,69,157)(65,151,70,156)(71,145,76,150)(72,144,77,149)(73,143,78,148)(74,142,79,147)(75,141,80,146)(81,135,86,140)(82,134,87,139)(83,133,88,138)(84,132,89,137)(85,131,90,136), (1,148,38,14,28,138,48,158)(2,149,39,15,29,139,49,159)(3,150,40,16,30,140,50,160)(4,141,31,17,21,131,41,151)(5,142,32,18,22,132,42,152)(6,143,33,19,23,133,43,153)(7,144,34,20,24,134,44,154)(8,145,35,11,25,135,45,155)(9,146,36,12,26,136,46,156)(10,147,37,13,27,137,47,157)(51,91,81,121,61,101,71,111)(52,92,82,122,62,102,72,112)(53,93,83,123,63,103,73,113)(54,94,84,124,64,104,74,114)(55,95,85,125,65,105,75,115)(56,96,86,126,66,106,76,116)(57,97,87,127,67,107,77,117)(58,98,88,128,68,108,78,118)(59,99,89,129,69,109,79,119)(60,100,90,130,70,110,80,120), (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), (1,108,6,103)(2,107,7,102)(3,106,8,101)(4,105,9,110)(5,104,10,109)(11,56,16,51)(12,55,17,60)(13,54,18,59)(14,53,19,58)(15,52,20,57)(21,95,26,100)(22,94,27,99)(23,93,28,98)(24,92,29,97)(25,91,30,96)(31,115,36,120)(32,114,37,119)(33,113,38,118)(34,112,39,117)(35,111,40,116)(41,125,46,130)(42,124,47,129)(43,123,48,128)(44,122,49,127)(45,121,50,126)(61,155,66,160)(62,154,67,159)(63,153,68,158)(64,152,69,157)(65,151,70,156)(71,145,76,150)(72,144,77,149)(73,143,78,148)(74,142,79,147)(75,141,80,146)(81,135,86,140)(82,134,87,139)(83,133,88,138)(84,132,89,137)(85,131,90,136), (1,148,38,14,28,138,48,158)(2,149,39,15,29,139,49,159)(3,150,40,16,30,140,50,160)(4,141,31,17,21,131,41,151)(5,142,32,18,22,132,42,152)(6,143,33,19,23,133,43,153)(7,144,34,20,24,134,44,154)(8,145,35,11,25,135,45,155)(9,146,36,12,26,136,46,156)(10,147,37,13,27,137,47,157)(51,91,81,121,61,101,71,111)(52,92,82,122,62,102,72,112)(53,93,83,123,63,103,73,113)(54,94,84,124,64,104,74,114)(55,95,85,125,65,105,75,115)(56,96,86,126,66,106,76,116)(57,97,87,127,67,107,77,117)(58,98,88,128,68,108,78,118)(59,99,89,129,69,109,79,119)(60,100,90,130,70,110,80,120), (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)], [(1,108,6,103),(2,107,7,102),(3,106,8,101),(4,105,9,110),(5,104,10,109),(11,56,16,51),(12,55,17,60),(13,54,18,59),(14,53,19,58),(15,52,20,57),(21,95,26,100),(22,94,27,99),(23,93,28,98),(24,92,29,97),(25,91,30,96),(31,115,36,120),(32,114,37,119),(33,113,38,118),(34,112,39,117),(35,111,40,116),(41,125,46,130),(42,124,47,129),(43,123,48,128),(44,122,49,127),(45,121,50,126),(61,155,66,160),(62,154,67,159),(63,153,68,158),(64,152,69,157),(65,151,70,156),(71,145,76,150),(72,144,77,149),(73,143,78,148),(74,142,79,147),(75,141,80,146),(81,135,86,140),(82,134,87,139),(83,133,88,138),(84,132,89,137),(85,131,90,136)], [(1,148,38,14,28,138,48,158),(2,149,39,15,29,139,49,159),(3,150,40,16,30,140,50,160),(4,141,31,17,21,131,41,151),(5,142,32,18,22,132,42,152),(6,143,33,19,23,133,43,153),(7,144,34,20,24,134,44,154),(8,145,35,11,25,135,45,155),(9,146,36,12,26,136,46,156),(10,147,37,13,27,137,47,157),(51,91,81,121,61,101,71,111),(52,92,82,122,62,102,72,112),(53,93,83,123,63,103,73,113),(54,94,84,124,64,104,74,114),(55,95,85,125,65,105,75,115),(56,96,86,126,66,106,76,116),(57,97,87,127,67,107,77,117),(58,98,88,128,68,108,78,118),(59,99,89,129,69,109,79,119),(60,100,90,130,70,110,80,120)], [(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)]])

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 SD16 C4○D4 D10 D10 D10 C5⋊D4 C8⋊C22 D4⋊2D5 D4×D5 D5×SD16 D40⋊C2 kernel Dic5⋊5SD16 C20.8Q8 D20⋊5C4 D4⋊Dic5 C20⋊D4 C2×Q8⋊D5 Q8×Dic5 C10×SD16 C2×Dic5 C5×Q8 C2×SD16 Dic5 C20 C2×C8 C2×D4 C2×Q8 Q8 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 2 4 4

Matrix representation of Dic55SD16 in GL4(𝔽41) generated by

 0 40 0 0 1 7 0 0 0 0 1 0 0 0 0 1
,
 17 40 0 0 3 24 0 0 0 0 1 0 0 0 0 1
,
 24 1 0 0 40 17 0 0 0 0 15 15 0 0 26 15
,
 24 1 0 0 40 17 0 0 0 0 40 0 0 0 0 1
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,15,26,0,0,15,15],[24,40,0,0,1,17,0,0,0,0,40,0,0,0,0,1] >;

Dic55SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("Dic5:5SD16");
// GroupNames label

G:=SmallGroup(320,790);
// by ID

G=gap.SmallGroup(320,790);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,135,184,570,297,136,12550]);
// 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,c*b*c^-1=d*b*d=a^5*b,d*c*d=c^3>;
// generators/relations

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