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## G = Q8⋊Dic5⋊C2order 320 = 26·5

### 2nd semidirect product of Q8⋊Dic5 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Q8⋊Dic5⋊C2
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — C4.Dic10 — Q8⋊Dic5⋊C2
 Lower central C5 — C10 — C2×C20 — Q8⋊Dic5⋊C2
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Q8⋊Dic5⋊C2
G = < a,b,c,d | a2=b8=1, c10=a, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab3, dcd-1=ab4c9 >

Subgroups: 414 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×2], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C4×C8, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C4.4D4, C42.C2, C52C8, C40, D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C42.78C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C40, C2×D20, Q8×C10, D206C4, C8×Dic5, D205C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, C20.23D4, Q8⋊Dic5⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C4○D8 [×2], C22×D5, C42.78C22, C4○D20, D4×D5, D42D5, Dic5.5D4, SD163D5, Q8.D10, Q8⋊Dic5⋊C2

Smallest permutation representation of Q8⋊Dic5⋊C2
On 160 points
Generators in S160
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 39 147 85 56 114 139 76)(2 96 140 30 57 67 148 105)(3 21 149 87 58 116 121 78)(4 98 122 32 59 69 150 107)(5 23 151 89 60 118 123 80)(6 100 124 34 41 71 152 109)(7 25 153 91 42 120 125 62)(8 82 126 36 43 73 154 111)(9 27 155 93 44 102 127 64)(10 84 128 38 45 75 156 113)(11 29 157 95 46 104 129 66)(12 86 130 40 47 77 158 115)(13 31 159 97 48 106 131 68)(14 88 132 22 49 79 160 117)(15 33 141 99 50 108 133 70)(16 90 134 24 51 61 142 119)(17 35 143 81 52 110 135 72)(18 92 136 26 53 63 144 101)(19 37 145 83 54 112 137 74)(20 94 138 28 55 65 146 103)
(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 70 46 89)(2 98 47 79)(3 68 48 87)(4 96 49 77)(5 66 50 85)(6 94 51 75)(7 64 52 83)(8 92 53 73)(9 62 54 81)(10 90 55 71)(11 80 56 99)(12 88 57 69)(13 78 58 97)(14 86 59 67)(15 76 60 95)(16 84 41 65)(17 74 42 93)(18 82 43 63)(19 72 44 91)(20 100 45 61)(21 149 106 131)(22 140 107 158)(23 147 108 129)(24 138 109 156)(25 145 110 127)(26 136 111 154)(27 143 112 125)(28 134 113 152)(29 141 114 123)(30 132 115 150)(31 159 116 121)(32 130 117 148)(33 157 118 139)(34 128 119 146)(35 155 120 137)(36 126 101 144)(37 153 102 135)(38 124 103 142)(39 151 104 133)(40 122 105 160)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 D4⋊2D5 D4×D5 SD16⋊3D5 Q8.D10 kernel Q8⋊Dic5⋊C2 D20⋊6C4 C8×Dic5 D20⋊5C4 Q8⋊Dic5 C5×Q8⋊C4 C4.Dic10 C20.23D4 C2×Dic5 Q8⋊C4 C20 C4⋊C4 C2×C8 C2×Q8 C10 C4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 4 2 2 2 8 8 2 2 4 4

Matrix representation of Q8⋊Dic5⋊C2 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 26 15 0 0 26 26
,
 25 16 0 0 25 39 0 0 0 0 12 12 0 0 12 29
,
 28 28 0 0 32 13 0 0 0 0 15 26 0 0 26 26
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,26,26,0,0,15,26],[25,25,0,0,16,39,0,0,0,0,12,12,0,0,12,29],[28,32,0,0,28,13,0,0,0,0,15,26,0,0,26,26] >;

Q8⋊Dic5⋊C2 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm Dic}_5\rtimes C_2
% in TeX

G:=Group("Q8:Dic5:C2");
// GroupNames label

G:=SmallGroup(320,427);
// by ID

G=gap.SmallGroup(320,427);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,120,1094,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^10=a,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^3,d*c*d^-1=a*b^4*c^9>;
// generators/relations

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