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G = C60.7C4  order 240 = 24·3·5

1st non-split extension by C60 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C60.7C4, C4.Dic15, C4.15D30, C20.50D6, C12.51D10, C15⋊13M4(2), C20.4Dic3, C12.1Dic5, C22.Dic15, C60.57C22, C15⋊3C8⋊5C2, (C2×C20).5S3, (C2×C30).7C4, (C2×C60).7C2, (C2×C12).5D5, (C2×C4).2D15, C30.51(C2×C4), C5⋊4(C4.Dic3), C3⋊2(C4.Dic5), (C2×C6).3Dic5, C6.7(C2×Dic5), (C2×C10).5Dic3, C2.3(C2×Dic15), C10.14(C2×Dic3), SmallGroup(240,71)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C60.7C4
Chief series
C1 — C5 — C15 — C30 — C60 — C15⋊3C8 — C60.7C4
Lower central
C15 — C30 — C60.7C4
Upper central
C1 — C4 — C2×C4

Generators and relations for C60.7C4
 G = < a,b | a60=1, b4=a30, bab-1=a-1 >

C1
C2
2C2
C3
C5
C22
C4
C4
C6
2C6
C10
2C10
C15
C2×C4
15C8
15C8
C12
C12
C2×C6
C20
C20
C2×C10
C30
2C30
15M4(2)
C2×C12
5C3⋊C8
5C3⋊C8
C2×C20
3C5⋊2C8
3C5⋊2C8
C60
C60
C2×C30
5C4.Dic3
3C4.Dic5
C15⋊3C8
C15⋊3C8
C2×C60
C60.7C4

Smallest permutation representation of C60.7C4
►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)

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

C60.7C4 is a maximal subgroup of
 C60.28D4  C20.5D12  C12.6D20  C60.54D4  C60.97D4  C60.99D4  C60.105D4  C12.59D20  D60⋊7C4  C4.18D60  C60.210D4  M4(2)⋊D15  C4.D60  C60.8D4  C60.10D4  Q8⋊3Dic15  D5×C4.Dic3  D20.3Dic3  D12.2Dic5  S3×C4.Dic5  C60.36D4  D20⋊21D6  D20.37D6  D12.37D10  D60.6C4  M4(2)×D15  D4.D30  Q8.11D30  D4.Dic15  D4⋊D30  D4.9D30
C60.7C4 is a maximal quotient of
 C42.D15  C60⋊5C8  C60.212D4

66 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122344455666888810···10121212121515151520···2030···3060···60
size112211222222303030302···2222222222···22···22···2

66 irreducible representations

dim111112222222222222222
type+++++-+--+-+-+-
imageC1C2C2C4C4S3D5Dic3D6Dic3M4(2)Dic5D10Dic5D15C4.Dic3Dic15D30Dic15C4.Dic5C60.7C4
kernelC60.7C4C15⋊3C8C2×C60C60C2×C30C2×C20C2×C12C20C20C2×C10C15C12C12C2×C6C2×C4C5C4C4C22C3C1
# reps1212212111222244444816

Matrix representation of C60.7C4 ►in GL2(𝔽241) generated by

90
0134
,
01
1770
G:=sub<GL(2,GF(241))| [9,0,0,134],[0,177,1,0] >;
 

C60.7C4 in GAP, Magma, Sage, TeX 

C_{60}._7C_4
 
% in TeX
 
G:=Group("C60.7C4");
 
// GroupNames label
 
G:=SmallGroup(240,71);
 
// by ID
 
G=gap.SmallGroup(240,71);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,964,6917]);
 
// Polycyclic
 
G:=Group<a,b|a^60=1,b^4=a^30,b*a*b^-1=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C60.7C4 in TeX

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