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G = C60.C4order 240 = 24·3·5

3rd non-split extension by C60 of C4 acting faithfully

metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C60.3C4, C12.3F5, C20.3Dic3, D10.2Dic3, Dic5.15D6, D5⋊(C3⋊C8), C153(C2×C8), C32(D5⋊C8), (C3×D5)⋊2C8, C15⋊C87C2, C4.3(C3⋊F5), C6.8(C2×F5), C30.8(C2×C4), (C6×D5).4C4, (C4×D5).5S3, (D5×C12).8C2, C10.1(C2×Dic3), (C3×Dic5).21C22, C51(C2×C3⋊C8), C2.1(C2×C3⋊F5), SmallGroup(240,118)

Series: Derived Chief Lower central Upper central

C1C15 — C60.C4
C1C5C15C30C3×Dic5C15⋊C8 — C60.C4
C15 — C60.C4
C1C4

Generators and relations for C60.C4
 G = < a,b | a60=1, b4=a30, bab-1=a17 >

5C2
5C2
5C4
5C22
5C6
5C6
5C2×C4
15C8
15C8
5C2×C6
5C12
15C2×C8
5C2×C12
5C3⋊C8
5C3⋊C8
3C5⋊C8
3C5⋊C8
5C2×C3⋊C8
3D5⋊C8

Smallest permutation representation of C60.C4
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 96 16 111 31 66 46 81)(2 89 5 68 32 119 35 98)(3 82 54 85 33 112 24 115)(4 75 43 102 34 105 13 72)(6 61 21 76 36 91 51 106)(7 114 10 93 37 84 40 63)(8 107 59 110 38 77 29 80)(9 100 48 67 39 70 18 97)(11 86 26 101 41 116 56 71)(12 79 15 118 42 109 45 88)(14 65 53 92 44 95 23 62)(17 104 20 83 47 74 50 113)(19 90 58 117 49 120 28 87)(22 69 25 108 52 99 55 78)(27 94 30 73 57 64 60 103)

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

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

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

C60.C4 is a maximal subgroup of
F5×C3⋊C8  C30.3C42  D60⋊C4  Dic6⋊F5  C8×C3⋊F5  C24⋊F5  D20⋊Dic3  Dic102Dic3  S3×D5⋊C8  D12.F5  Dic6.F5  C5⋊C8⋊D6  C60.59(C2×C4)  Dic10.Dic3  D20.Dic3
C60.C4 is a maximal quotient of
C24.F5  C120.C4  C4×C15⋊C8  C30.7M4(2)  Dic5.13D12

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 5 6A6B6C8A···8H 10 12A12B12C12D15A15B20A20B30A30B60A60B60C60D
order12223444456668···8101212121215152020303060606060
size11552115542101015···1542210104444444444

36 irreducible representations

dim11111122222444444
type+++++--++
imageC1C2C2C4C4C8S3D6Dic3Dic3C3⋊C8F5C2×F5C3⋊F5D5⋊C8C2×C3⋊F5C60.C4
kernelC60.C4C15⋊C8D5×C12C60C6×D5C3×D5C4×D5Dic5C20D10D5C12C6C4C3C2C1
# reps12122811114112224

Matrix representation of C60.C4 in GL6(𝔽241)

0640000
1771770000
0012120127
00114126126114
0012701212
002291152290
,
1812270000
46600000
002605959
000182208182
002153333215
001822081820

G:=sub<GL(6,GF(241))| [0,177,0,0,0,0,64,177,0,0,0,0,0,0,12,114,127,229,0,0,12,126,0,115,0,0,0,126,12,229,0,0,127,114,12,0],[181,46,0,0,0,0,227,60,0,0,0,0,0,0,26,0,215,182,0,0,0,182,33,208,0,0,59,208,33,182,0,0,59,182,215,0] >;

C60.C4 in GAP, Magma, Sage, TeX

C_{60}.C_4
% in TeX

G:=Group("C60.C4");
// GroupNames label

G:=SmallGroup(240,118);
// by ID

G=gap.SmallGroup(240,118);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,50,964,5189,1745]);
// Polycyclic

G:=Group<a,b|a^60=1,b^4=a^30,b*a*b^-1=a^17>;
// generators/relations

Export

Subgroup lattice of C60.C4 in TeX

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