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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 [×6], C4 [×12], C22 [×7], C5, C2×C4 [×18], Q8 [×16], C23, C10, C10 [×6], C22×C4 [×3], C2×Q8 [×12], C20 [×12], C2×C10 [×7], C22×Q8, C2×C20 [×18], C5×Q8 [×16], C22×C10, C22×C20 [×3], Q8×C10 [×12], Q8×C2×C10
Quotients: C1, C2 [×15], C22 [×35], C5, Q8 [×4], C23 [×15], C10 [×15], C2×Q8 [×6], C24, C2×C10 [×35], C22×Q8, C5×Q8 [×4], C22×C10 [×15], Q8×C10 [×6], 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)(11 138)(12 139)(13 140)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 39)(22 40)(23 31)(24 32)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 81)(60 82)(61 79)(62 80)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 121)(100 122)(101 119)(102 120)(103 111)(104 112)(105 113)(106 114)(107 115)(108 116)(109 117)(110 118)(141 159)(142 160)(143 151)(144 152)(145 153)(146 154)(147 155)(148 156)(149 157)(150 158)
(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 80)(2 88 28 71)(3 89 29 72)(4 90 30 73)(5 81 21 74)(6 82 22 75)(7 83 23 76)(8 84 24 77)(9 85 25 78)(10 86 26 79)(11 93 153 110)(12 94 154 101)(13 95 155 102)(14 96 156 103)(15 97 157 104)(16 98 158 105)(17 99 159 106)(18 100 160 107)(19 91 151 108)(20 92 152 109)(31 68 48 51)(32 69 49 52)(33 70 50 53)(34 61 41 54)(35 62 42 55)(36 63 43 56)(37 64 44 57)(38 65 45 58)(39 66 46 59)(40 67 47 60)(111 131 128 148)(112 132 129 149)(113 133 130 150)(114 134 121 141)(115 135 122 142)(116 136 123 143)(117 137 124 144)(118 138 125 145)(119 139 126 146)(120 140 127 147)
(1 127 27 120)(2 128 28 111)(3 129 29 112)(4 130 30 113)(5 121 21 114)(6 122 22 115)(7 123 23 116)(8 124 24 117)(9 125 25 118)(10 126 26 119)(11 70 153 53)(12 61 154 54)(13 62 155 55)(14 63 156 56)(15 64 157 57)(16 65 158 58)(17 66 159 59)(18 67 160 60)(19 68 151 51)(20 69 152 52)(31 108 48 91)(32 109 49 92)(33 110 50 93)(34 101 41 94)(35 102 42 95)(36 103 43 96)(37 104 44 97)(38 105 45 98)(39 106 46 99)(40 107 47 100)(71 148 88 131)(72 149 89 132)(73 150 90 133)(74 141 81 134)(75 142 82 135)(76 143 83 136)(77 144 84 137)(78 145 85 138)(79 146 86 139)(80 147 87 140)

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

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

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

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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