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## G = (Q8×C10)⋊16C4order 320 = 26·5

### 2nd semidirect product of Q8×C10 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (Q8×C10)⋊16C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — C23.21D10 — (Q8×C10)⋊16C4
 Lower central C5 — C10 — C20 — (Q8×C10)⋊16C4
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (Q8×C10)⋊16C4
G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a5b-1c >

Subgroups: 366 in 150 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×6], C23, C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×6], C2×Q8 [×3], Dic5 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C52C8 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C23.38D4, C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], Q8⋊Dic5 [×4], C2×C4.Dic5, C23.21D10, Q8×C2×C10, (Q8×C10)⋊16C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C8.C22 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.38D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C20.C23 [×2], C2×C23.D5, (Q8×C10)⋊16C4

Smallest permutation representation of (Q8×C10)⋊16C4
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 45 31 20)(2 46 32 11)(3 47 33 12)(4 48 34 13)(5 49 35 14)(6 50 36 15)(7 41 37 16)(8 42 38 17)(9 43 39 18)(10 44 40 19)(21 132 145 157)(22 133 146 158)(23 134 147 159)(24 135 148 160)(25 136 149 151)(26 137 150 152)(27 138 141 153)(28 139 142 154)(29 140 143 155)(30 131 144 156)(51 63 90 78)(52 64 81 79)(53 65 82 80)(54 66 83 71)(55 67 84 72)(56 68 85 73)(57 69 86 74)(58 70 87 75)(59 61 88 76)(60 62 89 77)(91 128 116 104)(92 129 117 105)(93 130 118 106)(94 121 119 107)(95 122 120 108)(96 123 111 109)(97 124 112 110)(98 125 113 101)(99 126 114 102)(100 127 115 103)
(1 72 31 67)(2 73 32 68)(3 74 33 69)(4 75 34 70)(5 76 35 61)(6 77 36 62)(7 78 37 63)(8 79 38 64)(9 80 39 65)(10 71 40 66)(11 56 46 85)(12 57 47 86)(13 58 48 87)(14 59 49 88)(15 60 50 89)(16 51 41 90)(17 52 42 81)(18 53 43 82)(19 54 44 83)(20 55 45 84)(21 92 145 117)(22 93 146 118)(23 94 147 119)(24 95 148 120)(25 96 149 111)(26 97 150 112)(27 98 141 113)(28 99 142 114)(29 100 143 115)(30 91 144 116)(101 153 125 138)(102 154 126 139)(103 155 127 140)(104 156 128 131)(105 157 129 132)(106 158 130 133)(107 159 121 134)(108 160 122 135)(109 151 123 136)(110 152 124 137)
(1 93 36 98)(2 117 37 112)(3 91 38 96)(4 115 39 120)(5 99 40 94)(6 113 31 118)(7 97 32 92)(8 111 33 116)(9 95 34 100)(10 119 35 114)(11 129 41 124)(12 104 42 109)(13 127 43 122)(14 102 44 107)(15 125 45 130)(16 110 46 105)(17 123 47 128)(18 108 48 103)(19 121 49 126)(20 106 50 101)(21 85 26 51)(22 55 27 89)(23 83 28 59)(24 53 29 87)(25 81 30 57)(52 144 86 149)(54 142 88 147)(56 150 90 145)(58 148 82 143)(60 146 84 141)(61 134 71 139)(62 158 72 153)(63 132 73 137)(64 156 74 151)(65 140 75 135)(66 154 76 159)(67 138 77 133)(68 152 78 157)(69 136 79 131)(70 160 80 155)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C2 C4 D4 D4 D5 D10 Dic5 D10 C5⋊D4 C5⋊D4 C8.C22 C20.C23 kernel (Q8×C10)⋊16C4 Q8⋊Dic5 C2×C4.Dic5 C23.21D10 Q8×C2×C10 Q8×C10 C2×C20 C22×C10 C22×Q8 C22×C4 C2×Q8 C2×Q8 C2×C4 C23 C10 C2 # reps 1 4 1 1 1 8 3 1 2 2 8 4 12 4 2 8

Matrix representation of (Q8×C10)⋊16C4 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 38 0 10 0 0 0 0 38 0 10
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 33 22 0 0 0 0 40 8 0 0 0 0 0 0 7 27 0 0 0 0 27 34 0 0 0 0 17 10 14 7 0 0 10 24 7 27
,
 33 22 0 0 0 0 25 8 0 0 0 0 0 0 1 0 39 0 0 0 0 1 0 39 0 0 0 0 40 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,38,0,0,0,0,4,0,38,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[33,40,0,0,0,0,22,8,0,0,0,0,0,0,7,27,17,10,0,0,27,34,10,24,0,0,0,0,14,7,0,0,0,0,7,27],[33,25,0,0,0,0,22,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,39,0,40,0,0,0,0,39,0,40] >;

(Q8×C10)⋊16C4 in GAP, Magma, Sage, TeX

(Q_8\times C_{10})\rtimes_{16}C_4
% in TeX

G:=Group("(Q8xC10):16C4");
// GroupNames label

G:=SmallGroup(320,852);
// by ID

G=gap.SmallGroup(320,852);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^5*b^-1*c>;
// generators/relations

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