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G = C60.28D4order 480 = 25·3·5

28th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.28D4, C12.5D20, (C6×D20).1C2, (C2×D20).6S3, (C2×C20).40D6, C4.Dic32D5, C60.7C45C2, (C2×C12).41D10, C152(C4.D4), C52(C12.D4), C12.5(C5⋊D4), C4.19(C15⋊D4), C33(C20.46D4), C4.19(C3⋊D20), C20.77(C3⋊D4), (C2×C60).29C22, C22.3(D5×Dic3), C30.41(C22⋊C4), C6.24(D10⋊C4), (C22×D5).1Dic3, C2.3(D10⋊Dic3), C10.13(C6.D4), (D5×C2×C6).1C4, (C2×C4).1(S3×D5), (C2×C6).45(C4×D5), (C2×C30).80(C2×C4), (C5×C4.Dic3)⋊1C2, (C2×C10).21(C2×Dic3), SmallGroup(480,34)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.28D4
C1C5C15C30C60C2×C60C6×D20 — C60.28D4
C15C30C2×C30 — C60.28D4
C1C2C2×C4

Generators and relations for C60.28D4
 G = < a,b,c | a12=1, b20=a6, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a3b19 >

Subgroups: 476 in 92 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], D5 [×2], C10, C10, C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C3×D5 [×2], C30, C30, C4.D4, C52C8, C40, D20 [×2], C2×C20, C22×D5 [×2], C4.Dic3, C4.Dic3, C6×D4, C60 [×2], C6×D5 [×4], C2×C30, C4.Dic5, C5×M4(2), C2×D20, C12.D4, C5×C3⋊C8, C153C8, C3×D20 [×2], C2×C60, D5×C2×C6 [×2], C20.46D4, C5×C4.Dic3, C60.7C4, C6×D20, C60.28D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C4.D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C12.D4, D5×Dic3, C15⋊D4, C3⋊D20, C20.46D4, D10⋊Dic3, C60.28D4

Smallest permutation representation of C60.28D4
On 120 points
Generators in S120
(1 50 115 31 80 105 21 70 95 11 60 85)(2 86 61 12 96 71 22 106 41 32 116 51)(3 52 117 33 42 107 23 72 97 13 62 87)(4 88 63 14 98 73 24 108 43 34 118 53)(5 54 119 35 44 109 25 74 99 15 64 89)(6 90 65 16 100 75 26 110 45 36 120 55)(7 56 81 37 46 111 27 76 101 17 66 91)(8 92 67 18 102 77 28 112 47 38 82 57)(9 58 83 39 48 113 29 78 103 19 68 93)(10 94 69 20 104 79 30 114 49 40 84 59)
(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 30 11 20 21 10 31 40)(2 39 32 9 22 19 12 29)(3 28 13 18 23 8 33 38)(4 37 34 7 24 17 14 27)(5 26 15 16 25 6 35 36)(41 78 71 48 61 58 51 68)(42 67 52 57 62 47 72 77)(43 76 73 46 63 56 53 66)(44 65 54 55 64 45 74 75)(49 70 79 80 69 50 59 60)(81 98 91 88 101 118 111 108)(82 107 112 117 102 87 92 97)(83 96 93 86 103 116 113 106)(84 105 114 115 104 85 94 95)(89 90 99 120 109 110 119 100)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B6A6B6C6D6E6F6G8A8B8C8D10A10B10C10D12A12B15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order122223445566666668888101010101212151520202020202030···3040···4060···60
size1122020222222222020202012126060224444442222444···412···124···4

57 irreducible representations

dim11111222222222244444444
type++++++++-++++-+-+
imageC1C2C2C2C4S3D4D5D6Dic3D10C3⋊D4D20C5⋊D4C4×D5C4.D4S3×D5C12.D4C15⋊D4C3⋊D20D5×Dic3C20.46D4C60.28D4
kernelC60.28D4C5×C4.Dic3C60.7C4C6×D20D5×C2×C6C2×D20C60C4.Dic3C2×C20C22×D5C2×C12C20C12C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps11114122122444412222248

Matrix representation of C60.28D4 in GL4(𝔽241) generated by

1784500
1966300
0022248
0019319
,
0001
0024051
23819700
4416300
,
0001
0010
23819700
44300
G:=sub<GL(4,GF(241))| [178,196,0,0,45,63,0,0,0,0,222,193,0,0,48,19],[0,0,238,44,0,0,197,163,0,240,0,0,1,51,0,0],[0,0,238,44,0,0,197,3,0,1,0,0,1,0,0,0] >;

C60.28D4 in GAP, Magma, Sage, TeX

C_{60}._{28}D_4
% in TeX

G:=Group("C60.28D4");
// GroupNames label

G:=SmallGroup(480,34);
// by ID

G=gap.SmallGroup(480,34);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,100,346,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^19>;
// generators/relations

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