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## G = C10×C4⋊C4order 160 = 25·5

### Direct product of C10 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C10×C4⋊C4
G = < a,b,c | a10=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 108 in 92 conjugacy classes, 76 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C5×C4⋊C4, C22×C20, C22×C20, C10×C4⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C4⋊C4, C2×C20, C5×D4, C5×Q8, C22×C10, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4

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

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

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

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

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 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C4 C5 C10 C10 C20 D4 Q8 C5×D4 C5×Q8 kernel C10×C4⋊C4 C5×C4⋊C4 C22×C20 C2×C20 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C2×C10 C2×C10 C22 C22 # reps 1 4 3 8 4 16 12 32 2 2 8 8

Matrix representation of C10×C4⋊C4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 23 0 0 0 0 23
,
 40 0 0 0 0 40 0 0 0 0 1 39 0 0 1 40
,
 40 0 0 0 0 32 0 0 0 0 33 23 0 0 24 8
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,23,0,0,0,0,23],[40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,32,0,0,0,0,33,24,0,0,23,8] >;

C10×C4⋊C4 in GAP, Magma, Sage, TeX

C_{10}\times C_4\rtimes C_4
% in TeX

G:=Group("C10xC4:C4");
// GroupNames label

G:=SmallGroup(160,177);
// by ID

G=gap.SmallGroup(160,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,247]);
// Polycyclic

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

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