Copied to
clipboard

G = C10.Q16order 160 = 25·5

2nd non-split extension by C10 of Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.2D4, C4.10D20, C10.4Q16, Dic106C4, C10.5SD16, C4⋊C4.3D5, C4.2(C4×D5), C20.25(C2×C4), C52(Q8⋊C4), (C2×C10).31D4, (C2×C4).36D10, C2.2(D4.D5), C2.2(C5⋊Q16), (C2×C20).11C22, (C2×Dic10).6C2, C2.6(D10⋊C4), C10.15(C22⋊C4), C22.15(C5⋊D4), (C5×C4⋊C4).3C2, (C2×C52C8).3C2, SmallGroup(160,17)

Series: Derived Chief Lower central Upper central

C1C20 — C10.Q16
C1C5C10C2×C10C2×C20C2×Dic10 — C10.Q16
C5C10C20 — C10.Q16
C1C22C2×C4C4⋊C4

Generators and relations for C10.Q16
 G = < a,b,c | a10=b8=1, c2=a5b4, bab-1=a-1, ac=ca, cbc-1=a5b-1 >

4C4
10C4
10C4
2C2×C4
5Q8
5Q8
10C8
10Q8
10C2×C4
2Dic5
2Dic5
4C20
5C2×C8
5C2×Q8
2C52C8
2C2×C20
2C2×Dic5
2Dic10
5Q8⋊C4

Smallest permutation representation of C10.Q16
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 30 50 142 11 140 34 157)(2 29 41 141 12 139 35 156)(3 28 42 150 13 138 36 155)(4 27 43 149 14 137 37 154)(5 26 44 148 15 136 38 153)(6 25 45 147 16 135 39 152)(7 24 46 146 17 134 40 151)(8 23 47 145 18 133 31 160)(9 22 48 144 19 132 32 159)(10 21 49 143 20 131 33 158)(51 116 64 99 87 101 79 124)(52 115 65 98 88 110 80 123)(53 114 66 97 89 109 71 122)(54 113 67 96 90 108 72 121)(55 112 68 95 81 107 73 130)(56 111 69 94 82 106 74 129)(57 120 70 93 83 105 75 128)(58 119 61 92 84 104 76 127)(59 118 62 91 85 103 77 126)(60 117 63 100 86 102 78 125)
(1 63 16 73)(2 64 17 74)(3 65 18 75)(4 66 19 76)(5 67 20 77)(6 68 11 78)(7 69 12 79)(8 70 13 80)(9 61 14 71)(10 62 15 72)(21 113 136 103)(22 114 137 104)(23 115 138 105)(24 116 139 106)(25 117 140 107)(26 118 131 108)(27 119 132 109)(28 120 133 110)(29 111 134 101)(30 112 135 102)(31 83 42 52)(32 84 43 53)(33 85 44 54)(34 86 45 55)(35 87 46 56)(36 88 47 57)(37 89 48 58)(38 90 49 59)(39 81 50 60)(40 82 41 51)(91 143 121 153)(92 144 122 154)(93 145 123 155)(94 146 124 156)(95 147 125 157)(96 148 126 158)(97 149 127 159)(98 150 128 160)(99 141 129 151)(100 142 130 152)

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

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

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

C10.Q16 is a maximal subgroup of
C20⋊Q8⋊C2  Dic10.D4  (C8×Dic5)⋊C2  D4⋊(C4×D5)  D42D5⋊C4  D43D20  D4.D20  D20.D4  Dic5.3Q16  Dic5⋊Q16  C408C4.C2  D5×Q8⋊C4  (Q8×D5)⋊C4  Q82D20  D104Q16  Dic5⋊SD16  Dic58SD16  Dic2015C4  Dic10⋊Q8  Dic10.Q8  D10.12SD16  C88D20  C20.(C4○D4)  C8.2D20  Dic55Q16  Dic102Q8  Dic10.2Q8  D10.8Q16  C83D20  D102Q16  C2.D87D5  C4021(C2×C4)  C4○D209C4  C4⋊C4.230D10  C4⋊C4.231D10  C4⋊C4.233D10  C4○D2010C4  C4.(C2×D20)  D4.1D20  C4×D4.D5  C42.51D10  D4.2D20  Q8.1D20  C4×C5⋊Q16  C42.59D10  C207Q16  D2017D4  Dic1017D4  C52C823D4  C4.(D4×D5)  D20.37D4  Dic10.37D4  (C2×C10)⋊Q16  C5⋊(C8.D4)  Dic10.4Q8  C42.216D10  C42.71D10  C42.82D10  Dic105Q8  C20.11Q16  Dic106Q8  C30.Q16  Dic3015C4  Dic309C4
C10.Q16 is a maximal quotient of
C4⋊Dic5⋊C4  C4.Dic20  Dic104C8  C20.2D8  C20.31C42  C30.Q16  Dic3015C4  Dic309C4

34 conjugacy classes

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

34 irreducible representations

dim1111122222222244
type+++++++-++--
imageC1C2C2C2C4D4D4D5SD16Q16D10C4×D5D20C5⋊D4D4.D5C5⋊Q16
kernelC10.Q16C2×C52C8C5×C4⋊C4C2×Dic10Dic10C20C2×C10C4⋊C4C10C10C2×C4C4C4C22C2C2
# reps1111411222244422

Matrix representation of C10.Q16 in GL5(𝔽41)

400000
01000
00100
000035
000734
,
90000
0291200
0292900
0002815
0001613
,
320000
063900
0393500
000176
0003424

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,35,34],[9,0,0,0,0,0,29,29,0,0,0,12,29,0,0,0,0,0,28,16,0,0,0,15,13],[32,0,0,0,0,0,6,39,0,0,0,39,35,0,0,0,0,0,17,34,0,0,0,6,24] >;

C10.Q16 in GAP, Magma, Sage, TeX

C_{10}.Q_{16}
% in TeX

G:=Group("C10.Q16");
// GroupNames label

G:=SmallGroup(160,17);
// by ID

G=gap.SmallGroup(160,17);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,121,31,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^10=b^8=1,c^2=a^5*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;
// generators/relations

Export

Subgroup lattice of C10.Q16 in TeX

׿
×
𝔽