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G = C48:C4order 192 = 26·3

2nd semidirect product of C48 and C4 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C48:2C4, C16:2Dic3, C12.18C42, M5(2).3S3, C12.5M4(2), C3:C16:4C4, C8.33(C4xS3), C3:2(C16:C4), C24.82(C2xC4), (C2xC8).151D6, C6.5(C8:C4), C4.9(C8:S3), (C4xDic3).3C4, C24:C4.10C2, C4.23(C4xDic3), C8.20(C2xDic3), C2.4(C24:C4), (C2xC6).4M4(2), C12.C8.8C2, (C3xM5(2)).2C2, (C2xC24).262C22, C22.6(C8:S3), (C2xC3:C8).1C4, (C2xC12).51(C2xC4), (C2xC4).136(C4xS3), SmallGroup(192,71)

Series: Derived Chief Lower central Upper central

C1C12 — C48:C4
C1C3C6C12C2xC12C2xC24C24:C4 — C48:C4
C3C12 — C48:C4
C1C4M5(2)

Generators and relations for C48:C4
 G = < a,b | a48=b4=1, bab-1=a29 >

Subgroups: 88 in 46 conjugacy classes, 31 normal (25 characteristic)
Quotients: C1, C2, C4, C22, S3, C2xC4, Dic3, D6, C42, M4(2), C4xS3, C2xDic3, C8:C4, C8:S3, C4xDic3, C16:C4, C24:C4, C48:C4
2C2
12C4
2C6
6C8
6C2xC4
4Dic3
3C16
3C16
3C2xC8
3C42
2C3:C8
2C2xDic3
3C8:C4
3M5(2)
3C16:C4

Smallest permutation representation of C48:C4
On 48 points
Generators in S48
(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)
(2 6 26 30)(3 11)(4 16 28 40)(5 21)(7 31)(8 36 32 12)(9 41)(10 46 34 22)(14 18 38 42)(15 23)(17 33)(19 43)(20 48 44 24)(27 35)(29 45)(39 47)

G:=sub<Sym(48)| (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), (2,6,26,30)(3,11)(4,16,28,40)(5,21)(7,31)(8,36,32,12)(9,41)(10,46,34,22)(14,18,38,42)(15,23)(17,33)(19,43)(20,48,44,24)(27,35)(29,45)(39,47)>;

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), (2,6,26,30)(3,11)(4,16,28,40)(5,21)(7,31)(8,36,32,12)(9,41)(10,46,34,22)(14,18,38,42)(15,23)(17,33)(19,43)(20,48,44,24)(27,35)(29,45)(39,47) );

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)], [(2,6,26,30),(3,11),(4,16,28,40),(5,21),(7,31),(8,36,32,12),(9,41),(10,46,34,22),(14,18,38,42),(15,23),(17,33),(19,43),(20,48,44,24),(27,35),(29,45),(39,47)]])

42 conjugacy classes

class 1 2A2B 3 4A4B4C4D4E6A6B8A8B8C8D8E8F12A12B12C16A16B16C16D16E16F16G16H24A24B24C24D24E24F48A···48H
order12234444466888888121212161616161616161624242424242448···48
size1122112121224222212122244444121212122222444···4

42 irreducible representations

dim1111111122222222244
type+++++-+
imageC1C2C2C2C4C4C4C4S3Dic3D6M4(2)M4(2)C4xS3C4xS3C8:S3C8:S3C16:C4C48:C4
kernelC48:C4C12.C8C24:C4C3xM5(2)C3:C16C48C2xC3:C8C4xDic3M5(2)C16C2xC8C12C2xC6C8C2xC4C4C22C3C1
# reps1111442212122224424

Matrix representation of C48:C4 in GL4(F5) generated by

0110
2000
4003
0420
,
4000
0340
0020
0001
G:=sub<GL(4,GF(5))| [0,2,4,0,1,0,0,4,1,0,0,2,0,0,3,0],[4,0,0,0,0,3,0,0,0,4,2,0,0,0,0,1] >;

C48:C4 in GAP, Magma, Sage, TeX

C_{48}\rtimes C_4
% in TeX

G:=Group("C48:C4");
// GroupNames label

G:=SmallGroup(192,71);
// by ID

G=gap.SmallGroup(192,71);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,80,102,6278]);
// Polycyclic

G:=Group<a,b|a^48=b^4=1,b*a*b^-1=a^29>;
// generators/relations

Export

Subgroup lattice of C48:C4 in TeX

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