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G = C12.C8order 96 = 25·3

1st non-split extension by C12 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C12.1C8, C24.4C4, C8.22D6, C32M5(2), C8.2Dic3, C24.26C22, C4.(C3⋊C8), C3⋊C165C2, (C2×C8).7S3, C6.9(C2×C8), (C2×C6).3C8, C22.(C3⋊C8), (C2×C12).8C4, (C2×C24).13C2, C12.39(C2×C4), (C2×C4).5Dic3, C4.11(C2×Dic3), C2.4(C2×C3⋊C8), SmallGroup(96,19)

Series: Derived Chief Lower central Upper central

C1C6 — C12.C8
C1C3C6C12C24C3⋊C16 — C12.C8
C3C6 — C12.C8
C1C8C2×C8

Generators and relations for C12.C8
 G = < a,b | a24=1, b4=a18, bab-1=a5 >

2C2
2C6
3C16
3C16
3M5(2)

Smallest permutation representation of C12.C8
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 46 10 43 19 40 4 37 13 34 22 31 7 28 16 25)(2 27 11 48 20 45 5 42 14 39 23 36 8 33 17 30)(3 32 12 29 21 26 6 47 15 44 24 41 9 38 18 35)

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

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

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

C12.C8 is a maximal subgroup of
C24.1C8  C12.15C42  C8.Dic6  D248C4  C24.6Q8  D24.C4  C24.8D4  D12.C8  C24.97D4  C48⋊C4  C8.25D12  C24.D4  C24.99D4  D8.Dic3  Q16.Dic3  D82Dic3  D12.4C8  S3×M5(2)  C24.78C23  D8.D6  C24.27C23  Q16⋊D6  D8.9D6  C36.C8  C24.61D6  C24.94D6  C40.51D6  C60.7C8  C120.C4  C60.C8
C12.C8 is a maximal quotient of
C24.C8  C12⋊C16  C24.98D4  C36.C8  C24.61D6  C24.94D6  C40.51D6  C60.7C8  C120.C4  C60.C8

36 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D8E8F12A12B12C12D16A···16H24A···24H
order12234446668888881212121216···1624···24
size112211222211112222226···62···2

36 irreducible representations

dim111111122222222
type++++-+-
imageC1C2C2C4C4C8C8S3Dic3D6Dic3C3⋊C8C3⋊C8M5(2)C12.C8
kernelC12.C8C3⋊C16C2×C24C24C2×C12C12C2×C6C2×C8C8C8C2×C4C4C22C3C1
# reps121224411112248

Matrix representation of C12.C8 in GL2(𝔽97) generated by

730
09
,
01
330
G:=sub<GL(2,GF(97))| [73,0,0,9],[0,33,1,0] >;

C12.C8 in GAP, Magma, Sage, TeX

C_{12}.C_8
% in TeX

G:=Group("C12.C8");
// GroupNames label

G:=SmallGroup(96,19);
// by ID

G=gap.SmallGroup(96,19);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,50,69,2309]);
// Polycyclic

G:=Group<a,b|a^24=1,b^4=a^18,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C12.C8 in TeX

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