Copied to
clipboard

G = C8.2D20order 320 = 26·5

2nd non-split extension by C8 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.2D20, C40.30D4, C4.Q84D5, C4⋊C4.42D10, (C2×C8).62D10, C4.52(C2×D20), C52(C8.D4), C20.132(C2×D4), (C2×Dic20)⋊24C2, C20.33(C4○D4), C10.Q1618C2, D102Q8.7C2, C4.5(Q82D5), (C2×Dic5).51D4, (C22×D5).35D4, C22.220(D4×D5), C10.45(C4⋊D4), C2.18(C4⋊D20), (C2×C40).111C22, (C2×C20).284C23, C2.25(SD16⋊D5), C10.44(C8.C22), (C2×Dic10).89C22, (C5×C4.Q8)⋊4C2, (C2×C8⋊D5).3C2, (C2×C4×D5).40C22, (C2×C10).289(C2×D4), (C5×C4⋊C4).77C22, (C2×C52C8).61C22, (C2×C4).387(C22×D5), SmallGroup(320,495)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C8.2D20
C1C5C10C2×C10C2×C20C2×C4×D5C2×C8⋊D5 — C8.2D20
C5C10C2×C20 — C8.2D20
C1C22C2×C4C4.Q8

Generators and relations for C8.2D20
 G = < a,b,c | a8=b20=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 454 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C8, C2×C4, C2×C4 [×7], Q8 [×4], C23, D5, C10, C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], Q16 [×2], C22×C4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, Q8⋊C4 [×2], C4.Q8, C22⋊Q8 [×2], C2×M4(2), C2×Q16, C52C8, C40 [×2], Dic10 [×4], C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C8.D4, C8⋊D5 [×2], Dic20 [×2], C2×C52C8, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10 [×2], C2×C4×D5, C10.Q16 [×2], C5×C4.Q8, D102Q8 [×2], C2×C8⋊D5, C2×Dic20, C8.2D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8.C22 [×2], D20 [×2], C22×D5, C8.D4, C2×D20, D4×D5, Q82D5, C4⋊D20, SD16⋊D5 [×2], C8.2D20

Smallest permutation representation of C8.2D20
On 160 points
Generators in S160
(1 94 45 143 115 62 139 33)(2 144 140 95 116 34 46 63)(3 96 47 145 117 64 121 35)(4 146 122 97 118 36 48 65)(5 98 49 147 119 66 123 37)(6 148 124 99 120 38 50 67)(7 100 51 149 101 68 125 39)(8 150 126 81 102 40 52 69)(9 82 53 151 103 70 127 21)(10 152 128 83 104 22 54 71)(11 84 55 153 105 72 129 23)(12 154 130 85 106 24 56 73)(13 86 57 155 107 74 131 25)(14 156 132 87 108 26 58 75)(15 88 59 157 109 76 133 27)(16 158 134 89 110 28 60 77)(17 90 41 159 111 78 135 29)(18 160 136 91 112 30 42 79)(19 92 43 141 113 80 137 31)(20 142 138 93 114 32 44 61)
(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)
(1 114 115 20)(2 19 116 113)(3 112 117 18)(4 17 118 111)(5 110 119 16)(6 15 120 109)(7 108 101 14)(8 13 102 107)(9 106 103 12)(10 11 104 105)(21 24 151 154)(22 153 152 23)(25 40 155 150)(26 149 156 39)(27 38 157 148)(28 147 158 37)(29 36 159 146)(30 145 160 35)(31 34 141 144)(32 143 142 33)(41 122 135 48)(42 47 136 121)(43 140 137 46)(44 45 138 139)(49 134 123 60)(50 59 124 133)(51 132 125 58)(52 57 126 131)(53 130 127 56)(54 55 128 129)(61 94 93 62)(63 92 95 80)(64 79 96 91)(65 90 97 78)(66 77 98 89)(67 88 99 76)(68 75 100 87)(69 86 81 74)(70 73 82 85)(71 84 83 72)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444455888810···102020202020···2040···40
size1111202288204040224420202···244448···84···4

44 irreducible representations

dim111111222222224444
type+++++++++++++-++-
imageC1C2C2C2C2C2D4D4D4D5C4○D4D10D10D20C8.C22Q82D5D4×D5SD16⋊D5
kernelC8.2D20C10.Q16C5×C4.Q8D102Q8C2×C8⋊D5C2×Dic20C40C2×Dic5C22×D5C4.Q8C20C4⋊C4C2×C8C8C10C4C22C2
# reps121211211224282228

Matrix representation of C8.2D20 in GL6(𝔽41)

100000
010000
009153226
002632159
00915915
0026322632
,
010000
4000000
002251319
0016132223
0013193916
0022232528
,
010000
100000
0016392228
0028251819
002228252
0018191316

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,26,9,26,0,0,15,32,15,32,0,0,32,15,9,26,0,0,26,9,15,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,2,16,13,22,0,0,25,13,19,23,0,0,13,22,39,25,0,0,19,23,16,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,28,22,18,0,0,39,25,28,19,0,0,22,18,25,13,0,0,28,19,2,16] >;

C8.2D20 in GAP, Magma, Sage, TeX

C_8._2D_{20}
% in TeX

G:=Group("C8.2D20");
// GroupNames label

G:=SmallGroup(320,495);
// by ID

G=gap.SmallGroup(320,495);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,555,226,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations

׿
×
𝔽