Copied to
clipboard

G = C60.7C4order 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, C1513M4(2), C20.4Dic3, C12.1Dic5, C22.Dic15, C60.57C22, C153C85C2, (C2×C20).5S3, (C2×C30).7C4, (C2×C60).7C2, (C2×C12).5D5, (C2×C4).2D15, C30.51(C2×C4), C54(C4.Dic3), C32(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

C1C30 — C60.7C4
C1C5C15C30C60C153C8 — C60.7C4
C15C30 — C60.7C4
C1C4C2×C4

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

2C2
2C6
2C10
15C8
15C8
2C30
15M4(2)
5C3⋊C8
5C3⋊C8
3C52C8
3C52C8
5C4.Dic3
3C4.Dic5

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

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

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

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

C60.7C4 is a maximal subgroup of
C60.28D4  C20.5D12  C12.6D20  C60.54D4  C60.97D4  C60.99D4  C60.105D4  C12.59D20  D607C4  C4.18D60  C60.210D4  M4(2)⋊D15  C4.D60  C60.8D4  C60.10D4  Q83Dic15  D5×C4.Dic3  D20.3Dic3  D12.2Dic5  S3×C4.Dic5  C60.36D4  D2021D6  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  C605C8  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
kernelC60.7C4C153C8C2×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

׿
×
𝔽