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G = C20.5D12order 480 = 25·3·5

5th non-split extension by C20 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.5D12, C60.53D4, (C2×D12).5D5, (C2×C20).41D6, C4.Dic52S3, C60.7C42C2, (C10×D12).1C2, (C2×C12).42D10, C31(C20.D4), C20.5(C3⋊D4), C12.6(C5⋊D4), C153(C4.D4), C10.39(D6⋊C4), (C22×S3).Dic5, C4.12(C15⋊D4), C54(C12.46D4), C4.19(C5⋊D12), C2.3(D6⋊Dic5), (C2×C60).22C22, C6.2(C23.D5), C22.3(S3×Dic5), C30.42(C22⋊C4), (S3×C2×C10).1C4, (C2×C4).2(S3×D5), (C2×C30).81(C2×C4), (C2×C10).70(C4×S3), (C3×C4.Dic5)⋊1C2, (C2×C6).1(C2×Dic5), SmallGroup(480,35)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C20.5D12
C1C5C15C30C60C2×C60C3×C4.Dic5 — C20.5D12
C15C30C2×C30 — C20.5D12
C1C2C2×C4

Generators and relations for C20.5D12
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a49, cbc-1=a15b3 >

Subgroups: 380 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], D6 [×4], C2×C6, C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, C24, D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×2], C30, C30, C4.D4, C52C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3, C3×M4(2), C2×D12, C60 [×2], S3×C10 [×4], C2×C30, C4.Dic5, C4.Dic5, D4×C10, C12.46D4, C3×C52C8, C153C8, C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C20.D4, C3×C4.Dic5, C60.7C4, C10×D12, C20.5D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, C4.D4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, S3×D5, C23.D5, C12.46D4, S3×Dic5, C15⋊D4, C5⋊D12, C20.D4, D6⋊Dic5, C20.5D12

Smallest permutation representation of C20.5D12
On 120 points
Generators in S120
(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)
(1 84 46 99 31 114 16 69)(2 83 47 98 32 113 17 68)(3 82 48 97 33 112 18 67)(4 81 49 96 34 111 19 66)(5 80 50 95 35 110 20 65)(6 79 51 94 36 109 21 64)(7 78 52 93 37 108 22 63)(8 77 53 92 38 107 23 62)(9 76 54 91 39 106 24 61)(10 75 55 90 40 105 25 120)(11 74 56 89 41 104 26 119)(12 73 57 88 42 103 27 118)(13 72 58 87 43 102 28 117)(14 71 59 86 44 101 29 116)(15 70 60 85 45 100 30 115)
(1 114 46 99 31 84 16 69)(2 103 47 88 32 73 17 118)(3 92 48 77 33 62 18 107)(4 81 49 66 34 111 19 96)(5 70 50 115 35 100 20 85)(6 119 51 104 36 89 21 74)(7 108 52 93 37 78 22 63)(8 97 53 82 38 67 23 112)(9 86 54 71 39 116 24 101)(10 75 55 120 40 105 25 90)(11 64 56 109 41 94 26 79)(12 113 57 98 42 83 27 68)(13 102 58 87 43 72 28 117)(14 91 59 76 44 61 29 106)(15 80 60 65 45 110 30 95)

G:=sub<Sym(120)| (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), (1,84,46,99,31,114,16,69)(2,83,47,98,32,113,17,68)(3,82,48,97,33,112,18,67)(4,81,49,96,34,111,19,66)(5,80,50,95,35,110,20,65)(6,79,51,94,36,109,21,64)(7,78,52,93,37,108,22,63)(8,77,53,92,38,107,23,62)(9,76,54,91,39,106,24,61)(10,75,55,90,40,105,25,120)(11,74,56,89,41,104,26,119)(12,73,57,88,42,103,27,118)(13,72,58,87,43,102,28,117)(14,71,59,86,44,101,29,116)(15,70,60,85,45,100,30,115), (1,114,46,99,31,84,16,69)(2,103,47,88,32,73,17,118)(3,92,48,77,33,62,18,107)(4,81,49,66,34,111,19,96)(5,70,50,115,35,100,20,85)(6,119,51,104,36,89,21,74)(7,108,52,93,37,78,22,63)(8,97,53,82,38,67,23,112)(9,86,54,71,39,116,24,101)(10,75,55,120,40,105,25,90)(11,64,56,109,41,94,26,79)(12,113,57,98,42,83,27,68)(13,102,58,87,43,72,28,117)(14,91,59,76,44,61,29,106)(15,80,60,65,45,110,30,95)>;

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), (1,84,46,99,31,114,16,69)(2,83,47,98,32,113,17,68)(3,82,48,97,33,112,18,67)(4,81,49,96,34,111,19,66)(5,80,50,95,35,110,20,65)(6,79,51,94,36,109,21,64)(7,78,52,93,37,108,22,63)(8,77,53,92,38,107,23,62)(9,76,54,91,39,106,24,61)(10,75,55,90,40,105,25,120)(11,74,56,89,41,104,26,119)(12,73,57,88,42,103,27,118)(13,72,58,87,43,102,28,117)(14,71,59,86,44,101,29,116)(15,70,60,85,45,100,30,115), (1,114,46,99,31,84,16,69)(2,103,47,88,32,73,17,118)(3,92,48,77,33,62,18,107)(4,81,49,66,34,111,19,96)(5,70,50,115,35,100,20,85)(6,119,51,104,36,89,21,74)(7,108,52,93,37,78,22,63)(8,97,53,82,38,67,23,112)(9,86,54,71,39,116,24,101)(10,75,55,120,40,105,25,90)(11,64,56,109,41,94,26,79)(12,113,57,98,42,83,27,68)(13,102,58,87,43,72,28,117)(14,91,59,76,44,61,29,106)(15,80,60,65,45,110,30,95) );

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)], [(1,84,46,99,31,114,16,69),(2,83,47,98,32,113,17,68),(3,82,48,97,33,112,18,67),(4,81,49,96,34,111,19,66),(5,80,50,95,35,110,20,65),(6,79,51,94,36,109,21,64),(7,78,52,93,37,108,22,63),(8,77,53,92,38,107,23,62),(9,76,54,91,39,106,24,61),(10,75,55,90,40,105,25,120),(11,74,56,89,41,104,26,119),(12,73,57,88,42,103,27,118),(13,72,58,87,43,102,28,117),(14,71,59,86,44,101,29,116),(15,70,60,85,45,100,30,115)], [(1,114,46,99,31,84,16,69),(2,103,47,88,32,73,17,118),(3,92,48,77,33,62,18,107),(4,81,49,66,34,111,19,96),(5,70,50,115,35,100,20,85),(6,119,51,104,36,89,21,74),(7,108,52,93,37,78,22,63),(8,97,53,82,38,67,23,112),(9,86,54,71,39,116,24,101),(10,75,55,120,40,105,25,90),(11,64,56,109,41,94,26,79),(12,113,57,98,42,83,27,68),(13,102,58,87,43,72,28,117),(14,91,59,76,44,61,29,106),(15,80,60,65,45,110,30,95)])

57 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B6A6B8A8B8C8D10A···10F10G···10N12A12B12C15A15B20A20B20C20D24A24B24C24D30A···30F60A···60H
order122223445566888810···1010···101212121515202020202424242430···3060···60
size11212122222224202060602···212···12224444444202020204···44···4

57 irreducible representations

dim11111222222222244444444
type+++++++++-++++-+-
imageC1C2C2C2C4S3D4D5D6D10Dic5D12C3⋊D4C4×S3C5⋊D4C4.D4S3×D5C12.46D4C15⋊D4C5⋊D12S3×Dic5C20.D4C20.5D12
kernelC20.5D12C3×C4.Dic5C60.7C4C10×D12S3×C2×C10C4.Dic5C60C2×D12C2×C20C2×C12C22×S3C20C20C2×C10C12C15C2×C4C5C4C4C22C3C1
# reps11114122124222812222248

Matrix representation of C20.5D12 in GL8(𝔽241)

2050000000
2487000000
0022600000
000160000
00005419100
000012118700
000019273185144
0000184998756
,
1832000000
133223000000
0001770000
006400000
0000202030
0000149077240
0000234144390
00003656760
,
223209000000
10818000000
006400000
0001770000
0000202030
000016201031
0000234144390
000017156760

G:=sub<GL(8,GF(241))| [205,24,0,0,0,0,0,0,0,87,0,0,0,0,0,0,0,0,226,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,54,121,192,184,0,0,0,0,191,187,73,99,0,0,0,0,0,0,185,87,0,0,0,0,0,0,144,56],[18,133,0,0,0,0,0,0,32,223,0,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,202,149,234,36,0,0,0,0,0,0,144,56,0,0,0,0,3,77,39,76,0,0,0,0,0,240,0,0],[223,108,0,0,0,0,0,0,209,18,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,202,162,234,171,0,0,0,0,0,0,144,56,0,0,0,0,3,103,39,76,0,0,0,0,0,1,0,0] >;

C20.5D12 in GAP, Magma, Sage, TeX

C_{20}._5D_{12}
% in TeX

G:=Group("C20.5D12");
// GroupNames label

G:=SmallGroup(480,35);
// by ID

G=gap.SmallGroup(480,35);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^15*b^3>;
// generators/relations

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