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G = C5×C2.C42order 160 = 25·5

Direct product of C5 and C2.C42

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

Aliases: C5×C2.C42, C10.11C42, (C2×C4)⋊2C20, (C2×C20)⋊9C4, C2.1(C4×C20), (C2×C10).7Q8, (C2×C10).45D4, C10.17(C4⋊C4), C22.7(C5×D4), C22.2(C5×Q8), (C22×C20).2C2, C22.7(C2×C20), (C22×C4).1C10, C23.12(C2×C10), C10.30(C22⋊C4), (C22×C10).48C22, C2.1(C5×C4⋊C4), C2.1(C5×C22⋊C4), (C2×C10).56(C2×C4), SmallGroup(160,45)

Series: Derived Chief Lower central Upper central

C1C2 — C5×C2.C42
C1C2C22C23C22×C10C22×C20 — C5×C2.C42
C1C2 — C5×C2.C42
C1C22×C10 — C5×C2.C42

Generators and relations for C5×C2.C42
 G = < a,b,c,d | a5=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×6], C4 [×6], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×6], C23, C10, C10 [×6], C22×C4 [×3], C20 [×6], C2×C10, C2×C10 [×6], C2.C42, C2×C20 [×6], C2×C20 [×6], C22×C10, C22×C20 [×3], C5×C2.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2.C42, C2×C20 [×3], C5×D4 [×3], C5×Q8, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×C2.C42

Smallest permutation representation of C5×C2.C42
Regular action 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 12)(2 13)(3 14)(4 15)(5 11)(6 143)(7 144)(8 145)(9 141)(10 142)(16 153)(17 154)(18 155)(19 151)(20 152)(21 140)(22 136)(23 137)(24 138)(25 139)(26 51)(27 52)(28 53)(29 54)(30 55)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(41 70)(42 66)(43 67)(44 68)(45 69)(56 93)(57 94)(58 95)(59 91)(60 92)(61 83)(62 84)(63 85)(64 81)(65 82)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 133)(97 134)(98 135)(99 131)(100 132)(101 123)(102 124)(103 125)(104 121)(105 122)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 146 66 121)(2 147 67 122)(3 148 68 123)(4 149 69 124)(5 150 70 125)(6 85 20 110)(7 81 16 106)(8 82 17 107)(9 83 18 108)(10 84 19 109)(11 112 41 103)(12 113 42 104)(13 114 43 105)(14 115 44 101)(15 111 45 102)(21 92 35 90)(22 93 31 86)(23 94 32 87)(24 95 33 88)(25 91 34 89)(26 119 39 99)(27 120 40 100)(28 116 36 96)(29 117 37 97)(30 118 38 98)(46 133 53 126)(47 134 54 127)(48 135 55 128)(49 131 51 129)(50 132 52 130)(56 156 76 136)(57 157 77 137)(58 158 78 138)(59 159 79 139)(60 160 80 140)(61 155 75 141)(62 151 71 142)(63 152 72 143)(64 153 73 144)(65 154 74 145)
(1 57 29 81)(2 58 30 82)(3 59 26 83)(4 60 27 84)(5 56 28 85)(6 103 136 133)(7 104 137 134)(8 105 138 135)(9 101 139 131)(10 102 140 132)(11 93 53 63)(12 94 54 64)(13 95 55 65)(14 91 51 61)(15 92 52 62)(16 113 157 127)(17 114 158 128)(18 115 159 129)(19 111 160 130)(20 112 156 126)(21 100 142 124)(22 96 143 125)(23 97 144 121)(24 98 145 122)(25 99 141 123)(31 116 152 150)(32 117 153 146)(33 118 154 147)(34 119 155 148)(35 120 151 149)(36 110 70 76)(37 106 66 77)(38 107 67 78)(39 108 68 79)(40 109 69 80)(41 86 46 72)(42 87 47 73)(43 88 48 74)(44 89 49 75)(45 90 50 71)

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

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

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

C5×C2.C42 is a maximal subgroup of
(C2×D20)⋊C4  C4⋊Dic5⋊C4  (C2×C20)⋊Q8  C10.49(C4×D4)  Dic5.15C42  Dic52C42  C52(C428C4)  C52(C425C4)  C10.51(C4×D4)  C2.(C4×D20)  C4⋊Dic515C4  C10.52(C4×D4)  (C2×Dic5)⋊Q8  C2.(C20⋊Q8)  (C2×Dic5).Q8  (C2×C20).28D4  (C2×C4).Dic10  C10.(C4⋊Q8)  (C22×C4).D10  C22.58(D4×D5)  (C2×C4)⋊9D20  D102C42  D102(C4⋊C4)  D103(C4⋊C4)  C10.54(C4×D4)  C10.55(C4×D4)  (C2×C20)⋊5D4  (C2×Dic5)⋊3D4  (C2×C4).20D20  (C2×C4).21D20  C10.(C4⋊D4)  (C22×D5).Q8  (C2×C20).33D4  C22⋊C4×C20  C4⋊C4×C20

100 conjugacy classes

class 1 2A···2G4A···4L5A5B5C5D10A···10AB20A···20AV
order12···24···4555510···1020···20
size11···12···211111···12···2

100 irreducible representations

dim1111112222
type+++-
imageC1C2C4C5C10C20D4Q8C5×D4C5×Q8
kernelC5×C2.C42C22×C20C2×C20C2.C42C22×C4C2×C4C2×C10C2×C10C22C22
# reps13124124831124

Matrix representation of C5×C2.C42 in GL4(𝔽41) generated by

1000
0100
00100
00010
,
1000
0100
00400
00040
,
9000
0900
00325
001638
,
9000
0100
0001
0010
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,3,16,0,0,25,38],[9,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C5×C2.C42 in GAP, Magma, Sage, TeX

C_5\times C_2.C_4^2
% in TeX

G:=Group("C5xC2.C4^2");
// GroupNames label

G:=SmallGroup(160,45);
// by ID

G=gap.SmallGroup(160,45);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,487]);
// Polycyclic

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

׿
×
𝔽