Copied to
clipboard

G = C24.1C8order 192 = 26·3

1st non-split extension by C24 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C24.1C8, C24.72D4, C8.29D12, C24.18Q8, C8.14Dic6, C42.9Dic3, C8.1(C3⋊C8), C6.4(C4⋊C8), (C4×C8).11S3, C31(C8.C8), (C4×C12).14C4, (C2×C24).18C4, (C4×C24).13C2, C12.37(C2×C8), (C2×C8).321D6, C12.39(C4⋊C4), (C2×C8).9Dic3, C2.5(C12⋊C8), C12.C8.5C2, C4.19(C4⋊Dic3), (C2×C6).19M4(2), (C2×C24).416C22, C22.2(C4.Dic3), C4.8(C2×C3⋊C8), (C2×C12).301(C2×C4), (C2×C4).69(C2×Dic3), SmallGroup(192,22)

Series: Derived Chief Lower central Upper central

C1C12 — C24.1C8
C1C3C6C12C24C2×C24C12.C8 — C24.1C8
C3C6C12 — C24.1C8
C1C8C2×C8C4×C8

Generators and relations for C24.1C8
 G = < a,b | a24=1, b8=a12, bab-1=a-1 >

2C2
2C4
2C4
2C6
2C2×C4
2C12
2C12
6C16
6C16
2C2×C12
3M5(2)
3M5(2)
2C3⋊C16
2C3⋊C16
3C8.C8

Smallest permutation representation of C24.1C8
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)
(1 28 22 31 19 34 16 37 13 40 10 43 7 46 4 25)(2 27 23 30 20 33 17 36 14 39 11 42 8 45 5 48)(3 26 24 29 21 32 18 35 15 38 12 41 9 44 6 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), (1,28,22,31,19,34,16,37,13,40,10,43,7,46,4,25)(2,27,23,30,20,33,17,36,14,39,11,42,8,45,5,48)(3,26,24,29,21,32,18,35,15,38,12,41,9,44,6,47)>;

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), (1,28,22,31,19,34,16,37,13,40,10,43,7,46,4,25)(2,27,23,30,20,33,17,36,14,39,11,42,8,45,5,48)(3,26,24,29,21,32,18,35,15,38,12,41,9,44,6,47) );

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)], [(1,28,22,31,19,34,16,37,13,40,10,43,7,46,4,25),(2,27,23,30,20,33,17,36,14,39,11,42,8,45,5,48),(3,26,24,29,21,32,18,35,15,38,12,41,9,44,6,47)])

60 conjugacy classes

class 1 2A2B 3 4A4B4C···4G6A6B6C8A8B8C8D8E···8J12A···12L16A···16H24A···24P
order1223444···466688888···812···1216···1624···24
size1122112···222211112···22···212···122···2

60 irreducible representations

dim1111112222222222222
type+++++---+-+
imageC1C2C2C4C4C8S3D4Q8Dic3Dic3D6M4(2)C3⋊C8Dic6D12C4.Dic3C8.C8C24.1C8
kernelC24.1C8C12.C8C4×C24C4×C12C2×C24C24C4×C8C24C24C42C2×C8C2×C8C2×C6C8C8C8C22C3C1
# reps12122811111124224816

Matrix representation of C24.1C8 in GL2(𝔽97) generated by

40
073
,
01
470
G:=sub<GL(2,GF(97))| [4,0,0,73],[0,47,1,0] >;

C24.1C8 in GAP, Magma, Sage, TeX

C_{24}._1C_8
% in TeX

G:=Group("C24.1C8");
// GroupNames label

G:=SmallGroup(192,22);
// by ID

G=gap.SmallGroup(192,22);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,100,1123,136,102,6278]);
// Polycyclic

G:=Group<a,b|a^24=1,b^8=a^12,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C24.1C8 in TeX

׿
×
𝔽