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## G = Q8×C2×C10order 160 = 25·5

### Direct product of C2×C10 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C10
 Chief series C1 — C2 — C10 — C20 — C5×Q8 — Q8×C10 — Q8×C2×C10
 Lower central C1 — C2 — Q8×C2×C10
 Upper central C1 — C22×C10 — Q8×C2×C10

Generators and relations for Q8×C2×C10
G = < a,b,c,d | a2=b10=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 156, all normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, Q8, C23, C10, C10, C22×C4, C2×Q8, C20, C2×C10, C22×Q8, C2×C20, C5×Q8, C22×C10, C22×C20, Q8×C10, Q8×C2×C10
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C24, C2×C10, C22×Q8, C5×Q8, C22×C10, Q8×C10, C23×C10, Q8×C2×C10

Smallest permutation representation of Q8×C2×C10
Regular action on 160 points
Generators in S160
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 41)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 131)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(51 89)(52 90)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)(91 129)(92 130)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)(109 111)(110 112)(141 153)(142 154)(143 155)(144 156)(145 157)(146 158)(147 159)(148 160)(149 151)(150 152)
(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 87 27 74)(2 88 28 75)(3 89 29 76)(4 90 30 77)(5 81 21 78)(6 82 22 79)(7 83 23 80)(8 84 24 71)(9 85 25 72)(10 86 26 73)(11 97 157 110)(12 98 158 101)(13 99 159 102)(14 100 160 103)(15 91 151 104)(16 92 152 105)(17 93 153 106)(18 94 154 107)(19 95 155 108)(20 96 156 109)(31 64 44 51)(32 65 45 52)(33 66 46 53)(34 67 47 54)(35 68 48 55)(36 69 49 56)(37 70 50 57)(38 61 41 58)(39 62 42 59)(40 63 43 60)(111 131 124 144)(112 132 125 145)(113 133 126 146)(114 134 127 147)(115 135 128 148)(116 136 129 149)(117 137 130 150)(118 138 121 141)(119 139 122 142)(120 140 123 143)
(1 127 27 114)(2 128 28 115)(3 129 29 116)(4 130 30 117)(5 121 21 118)(6 122 22 119)(7 123 23 120)(8 124 24 111)(9 125 25 112)(10 126 26 113)(11 70 157 57)(12 61 158 58)(13 62 159 59)(14 63 160 60)(15 64 151 51)(16 65 152 52)(17 66 153 53)(18 67 154 54)(19 68 155 55)(20 69 156 56)(31 104 44 91)(32 105 45 92)(33 106 46 93)(34 107 47 94)(35 108 48 95)(36 109 49 96)(37 110 50 97)(38 101 41 98)(39 102 42 99)(40 103 43 100)(71 144 84 131)(72 145 85 132)(73 146 86 133)(74 147 87 134)(75 148 88 135)(76 149 89 136)(77 150 90 137)(78 141 81 138)(79 142 82 139)(80 143 83 140)

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

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

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

Q8×C2×C10 is a maximal subgroup of
(Q8×C10)⋊16C4  (C5×Q8)⋊13D4  (C2×C10)⋊8Q16  C10.C22≀C2  (Q8×C10)⋊17C4  (C22×D5)⋊Q8  C10.422- 1+4  C10.442- 1+4  C10.452- 1+4

100 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 5A 5B 5C 5D 10A ··· 10AB 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C5 C10 C10 Q8 C5×Q8 kernel Q8×C2×C10 C22×C20 Q8×C10 C22×Q8 C22×C4 C2×Q8 C2×C10 C22 # reps 1 3 12 4 12 48 4 16

Matrix representation of Q8×C2×C10 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 40 0 0 0 0 40 39 0 0 1 1
,
 1 0 0 0 0 1 0 0 0 0 18 1 0 0 3 23
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,40,0,0,0,0,40,1,0,0,39,1],[1,0,0,0,0,1,0,0,0,0,18,3,0,0,1,23] >;

Q8×C2×C10 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{10}
% in TeX

G:=Group("Q8xC2xC10");
// GroupNames label

G:=SmallGroup(160,230);
// by ID

G=gap.SmallGroup(160,230);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,480,985,487]);
// Polycyclic

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

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