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## G = C10×C4.10D4order 320 = 26·5

### Direct product of C10 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4.10D4 — C10×C4.10D4
 Lower central C1 — C2 — C22 — C10×C4.10D4
 Upper central C1 — C2×C10 — C22×C20 — C10×C4.10D4

Generators and relations for C10×C4.10D4
G = < a,b,c,d | a10=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

Subgroups: 210 in 146 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C4.10D4, C2×M4(2), C22×Q8, C40, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C2×C4.10D4, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C20, Q8×C10, Q8×C10, C5×C4.10D4, C10×M4(2), Q8×C2×C10, C10×C4.10D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C4.10D4, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C2×C4.10D4, C5×C22⋊C4, C22×C20, D4×C10, C5×C4.10D4, C10×C22⋊C4, C10×C4.10D4

Smallest permutation representation of C10×C4.10D4
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 59 41 66)(2 60 42 67)(3 51 43 68)(4 52 44 69)(5 53 45 70)(6 54 46 61)(7 55 47 62)(8 56 48 63)(9 57 49 64)(10 58 50 65)(11 26 158 33)(12 27 159 34)(13 28 160 35)(14 29 151 36)(15 30 152 37)(16 21 153 38)(17 22 154 39)(18 23 155 40)(19 24 156 31)(20 25 157 32)(71 98 83 105)(72 99 84 106)(73 100 85 107)(74 91 86 108)(75 92 87 109)(76 93 88 110)(77 94 89 101)(78 95 90 102)(79 96 81 103)(80 97 82 104)(111 138 123 145)(112 139 124 146)(113 140 125 147)(114 131 126 148)(115 132 127 149)(116 133 128 150)(117 134 129 141)(118 135 130 142)(119 136 121 143)(120 137 122 144)
(1 139 61 129 41 146 54 117)(2 140 62 130 42 147 55 118)(3 131 63 121 43 148 56 119)(4 132 64 122 44 149 57 120)(5 133 65 123 45 150 58 111)(6 134 66 124 46 141 59 112)(7 135 67 125 47 142 60 113)(8 136 68 126 48 143 51 114)(9 137 69 127 49 144 52 115)(10 138 70 128 50 145 53 116)(11 73 21 102 158 85 38 95)(12 74 22 103 159 86 39 96)(13 75 23 104 160 87 40 97)(14 76 24 105 151 88 31 98)(15 77 25 106 152 89 32 99)(16 78 26 107 153 90 33 100)(17 79 27 108 154 81 34 91)(18 80 28 109 155 82 35 92)(19 71 29 110 156 83 36 93)(20 72 30 101 157 84 37 94)
(1 37 66 152 41 30 59 15)(2 38 67 153 42 21 60 16)(3 39 68 154 43 22 51 17)(4 40 69 155 44 23 52 18)(5 31 70 156 45 24 53 19)(6 32 61 157 46 25 54 20)(7 33 62 158 47 26 55 11)(8 34 63 159 48 27 56 12)(9 35 64 160 49 28 57 13)(10 36 65 151 50 29 58 14)(71 138 105 111 83 145 98 123)(72 139 106 112 84 146 99 124)(73 140 107 113 85 147 100 125)(74 131 108 114 86 148 91 126)(75 132 109 115 87 149 92 127)(76 133 110 116 88 150 93 128)(77 134 101 117 89 141 94 129)(78 135 102 118 90 142 95 130)(79 136 103 119 81 143 96 121)(80 137 104 120 82 144 97 122)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AF 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 1 1 1 1 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + - image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 C5×D4 C4.10D4 C5×C4.10D4 kernel C10×C4.10D4 C5×C4.10D4 C10×M4(2) Q8×C2×C10 C22×C20 Q8×C10 C2×C4.10D4 C4.10D4 C2×M4(2) C22×Q8 C22×C4 C2×Q8 C2×C20 C2×C4 C10 C2 # reps 1 4 2 1 4 4 4 16 8 4 16 16 4 16 2 8

Matrix representation of C10×C4.10D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 18 31 0 0 1 0 3 13 0 0 0 0 40 40 0 0 0 0 2 1
,
 40 39 0 0 0 0 0 1 0 0 0 0 0 0 23 0 40 18 0 0 38 0 0 7 0 0 1 40 0 13 0 0 39 0 0 18
,
 32 0 0 0 0 0 9 9 0 0 0 0 0 0 28 32 38 37 0 0 32 19 20 20 0 0 23 24 0 3 0 0 35 40 0 35

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,18,3,40,2,0,0,31,13,40,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,23,38,1,39,0,0,0,0,40,0,0,0,40,0,0,0,0,0,18,7,13,18],[32,9,0,0,0,0,0,9,0,0,0,0,0,0,28,32,23,35,0,0,32,19,24,40,0,0,38,20,0,0,0,0,37,20,3,35] >;

C10×C4.10D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4._{10}D_4
% in TeX

G:=Group("C10xC4.10D4");
// GroupNames label

G:=SmallGroup(320,913);
// by ID

G=gap.SmallGroup(320,913);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,7004,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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