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G = C60.8D4order 480 = 25·3·5

8th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.8D4, C23.2Dic15, (C6×D4).2D5, (C2×C4).3D30, (D4×C10).2S3, (C2×D4).2D15, (D4×C30).2C2, (C2×C20).73D6, (C2×C12).74D10, C32(C20.D4), C53(C12.D4), (C22×C30).2C4, C60.7C415C2, C20.20(C3⋊D4), C1510(C4.D4), C4.13(C157D4), C12.22(C5⋊D4), (C2×C60).59C22, (C22×C6).2Dic5, C6.15(C23.D5), C22.2(C2×Dic15), (C22×C10).5Dic3, C30.103(C22⋊C4), C2.4(C30.38D4), C10.26(C6.D4), (C2×C30).173(C2×C4), (C2×C6).30(C2×Dic5), (C2×C10).50(C2×Dic3), SmallGroup(480,193)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.8D4
C1C5C15C30C60C2×C60C60.7C4 — C60.8D4
C15C30C2×C30 — C60.8D4
C1C2C2×C4C2×D4

Generators and relations for C60.8D4
 G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a29, cbc-1=a15b3 >

Subgroups: 308 in 92 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C5, C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C10, C10 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, M4(2) [×2], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×3], C4.D4, C52C8 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], C4.Dic3 [×2], C6×D4, C60 [×2], C2×C30, C2×C30 [×4], C4.Dic5 [×2], D4×C10, C12.D4, C153C8 [×2], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.D4, C60.7C4 [×2], D4×C30, C60.8D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, Dic5 [×2], D10, C2×Dic3, C3⋊D4 [×2], D15, C4.D4, C2×Dic5, C5⋊D4 [×2], C6.D4, Dic15 [×2], D30, C23.D5, C12.D4, C2×Dic15, C157D4 [×2], C20.D4, C30.38D4, C60.8D4

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

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

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

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

81 conjugacy classes

class 1 2A2B2C2D 3 4A4B5A5B6A6B6C6D6E6F6G8A8B8C8D10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222344556666666888810···1010···101212151515152020202030···3030···3060···60
size11244222222224444606060602···24···444222244442···24···44···4

81 irreducible representations

dim111122222222222224444
type+++++++-+-++-+
imageC1C2C2C4S3D4D5D6Dic3D10Dic5C3⋊D4D15C5⋊D4D30Dic15C157D4C4.D4C12.D4C20.D4C60.8D4
kernelC60.8D4C60.7C4D4×C30C22×C30D4×C10C60C6×D4C2×C20C22×C10C2×C12C22×C6C20C2×D4C12C2×C4C23C4C15C5C3C1
# reps1214122122444848161248

Matrix representation of C60.8D4 in GL4(𝔽241) generated by

02400
217000
000231
00100
,
0010
000240
0100
1000
,
0010
0001
0100
240000
G:=sub<GL(4,GF(241))| [0,217,0,0,24,0,0,0,0,0,0,10,0,0,231,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,240,0,0],[0,0,0,240,0,0,1,0,1,0,0,0,0,1,0,0] >;

C60.8D4 in GAP, Magma, Sage, TeX

C_{60}._8D_4
% in TeX

G:=Group("C60.8D4");
// GroupNames label

G:=SmallGroup(480,193);
// by ID

G=gap.SmallGroup(480,193);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,675,2693,18822]);
// 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^29,c*b*c^-1=a^15*b^3>;
// generators/relations

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