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G = C10.D8order 160 = 25·5

1st non-split extension by C10 of D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.6D8, C20.1Q8, C10.3Q16, C4.1Dic10, C52C81C4, C4⋊C4.1D5, C52(C2.D8), C4.11(C4×D5), C10.9(C4⋊C4), C20.22(C2×C4), C2.1(D4⋊D5), (C2×C10).28D4, (C2×C4).33D10, C4⋊Dic5.8C2, (C2×C20).8C22, C2.1(C5⋊Q16), C2.3(C10.D4), C22.12(C5⋊D4), (C5×C4⋊C4).1C2, (C2×C52C8).1C2, SmallGroup(160,14)

Series: Derived Chief Lower central Upper central

C1C20 — C10.D8
C1C5C10C2×C10C2×C20C2×C52C8 — C10.D8
C5C10C20 — C10.D8
C1C22C2×C4C4⋊C4

Generators and relations for C10.D8
 G = < a,b,c | a10=b8=1, c2=a5, bab-1=cac-1=a-1, cbc-1=b-1 >

4C4
20C4
2C2×C4
5C8
5C8
10C2×C4
4C20
4Dic5
5C2×C8
5C4⋊C4
2C2×Dic5
2C2×C20
5C2.D8

Smallest permutation representation of C10.D8
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)
(1 125 50 117 33 108 12 91)(2 124 41 116 34 107 13 100)(3 123 42 115 35 106 14 99)(4 122 43 114 36 105 15 98)(5 121 44 113 37 104 16 97)(6 130 45 112 38 103 17 96)(7 129 46 111 39 102 18 95)(8 128 47 120 40 101 19 94)(9 127 48 119 31 110 20 93)(10 126 49 118 32 109 11 92)(21 90 132 53 149 61 158 72)(22 89 133 52 150 70 159 71)(23 88 134 51 141 69 160 80)(24 87 135 60 142 68 151 79)(25 86 136 59 143 67 152 78)(26 85 137 58 144 66 153 77)(27 84 138 57 145 65 154 76)(28 83 139 56 146 64 155 75)(29 82 140 55 147 63 156 74)(30 81 131 54 148 62 157 73)
(1 136 6 131)(2 135 7 140)(3 134 8 139)(4 133 9 138)(5 132 10 137)(11 144 16 149)(12 143 17 148)(13 142 18 147)(14 141 19 146)(15 150 20 145)(21 49 26 44)(22 48 27 43)(23 47 28 42)(24 46 29 41)(25 45 30 50)(31 154 36 159)(32 153 37 158)(33 152 38 157)(34 151 39 156)(35 160 40 155)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 119 76 114)(72 118 77 113)(73 117 78 112)(74 116 79 111)(75 115 80 120)(81 125 86 130)(82 124 87 129)(83 123 88 128)(84 122 89 127)(85 121 90 126)

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

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

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

C10.D8 is a maximal subgroup of
Dic54D8  D4.D55C4  Dic5.14D8  D4.Dic10  D10.12D8  D10⋊D8  C406C4⋊C2  C52C8⋊D4  Dic54Q16  Dic5.9Q16  Q8.2Dic10  D10.7Q16  C5⋊(C8⋊D4)  D10⋊Q16  D101C8.C2  Q8⋊D56C4  C403Q8  Dic10.Q8  C8.8Dic10  (C8×D5)⋊C4  C8⋊(C4×D5)  C4.Q8⋊D5  C20.(C4○D4)  D20.Q8  C402Q8  Dic102Q8  C404Q8  D5×C2.D8  C4020(C2×C4)  D10.13D8  D10.8Q16  D202Q8  C20.47(C4⋊C4)  (C2×C10).40D8  C4⋊C4.230D10  C20.64(C4⋊C4)  C4⋊C4.233D10  C20.76(C4⋊C4)  C4⋊C4.236D10  C20.50D8  D4.3Dic10  C4×D4⋊D5  C42.51D10  C20.23Q16  Q8.3Dic10  C42.56D10  C4×C5⋊Q16  (C2×C10).D8  (C2×D4).D10  (C2×C10)⋊D8  C4.(D4×D5)  (C2×C10).Q16  C10.(C4○D8)  C22⋊Q8⋊D5  (C2×C10)⋊Q16  Dic10.4Q8  C42.215D10  C42.68D10  D20.4Q8  C20.17D8  C42.76D10  D206Q8  Dic105Q8  C60.8Q8  C30.20D8  C60.1Q8
C10.D8 is a maximal quotient of
C20.53D8  C40.2Q8  C10.SD32  C40.7Q8  C20.31C42  C60.8Q8  C30.20D8  C60.1Q8

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20L
order122244444455888810···1020···20
size11112244202022101010102···24···4

34 irreducible representations

dim1111122222222244
type++++-+++-+-+-
imageC1C2C2C2C4Q8D4D5D8Q16D10Dic10C4×D5C5⋊D4D4⋊D5C5⋊Q16
kernelC10.D8C2×C52C8C4⋊Dic5C5×C4⋊C4C52C8C20C2×C10C4⋊C4C10C10C2×C4C4C4C22C2C2
# reps1111411222244422

Matrix representation of C10.D8 in GL4(𝔽41) generated by

0600
34700
00400
00040
,
243500
211700
00030
001517
,
9000
313200
003326
00188
G:=sub<GL(4,GF(41))| [0,34,0,0,6,7,0,0,0,0,40,0,0,0,0,40],[24,21,0,0,35,17,0,0,0,0,0,15,0,0,30,17],[9,31,0,0,0,32,0,0,0,0,33,18,0,0,26,8] >;

C10.D8 in GAP, Magma, Sage, TeX

C_{10}.D_8
% in TeX

G:=Group("C10.D8");
// GroupNames label

G:=SmallGroup(160,14);
// by ID

G=gap.SmallGroup(160,14);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,121,31,297,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C10.D8 in TeX

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