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G = C3×C8.C4order 96 = 25·3

Direct product of C3 and C8.C4

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×C8.C4, C8.1C12, C24.5C4, C12.68D4, M4(2).2C6, (C2×C8).5C6, (C2×C6).2Q8, C4.8(C2×C12), C22.(C3×Q8), C4.19(C3×D4), C6.14(C4⋊C4), C12.45(C2×C4), (C2×C24).11C2, (C3×M4(2)).4C2, (C2×C12).119C22, C2.5(C3×C4⋊C4), (C2×C4).22(C2×C6), SmallGroup(96,58)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C8.C4
C1C2C4C2×C4C2×C12C3×M4(2) — C3×C8.C4
C1C2C4 — C3×C8.C4
C1C12C2×C12 — C3×C8.C4

Generators and relations for C3×C8.C4
 G = < a,b,c | a3=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C2
2C6
2C8
2C8
2C24
2C24

Smallest permutation representation of C3×C8.C4
On 48 points
Generators in S48
(1 25 17)(2 26 18)(3 27 19)(4 28 20)(5 29 21)(6 30 22)(7 31 23)(8 32 24)(9 39 44)(10 40 45)(11 33 46)(12 34 47)(13 35 48)(14 36 41)(15 37 42)(16 38 43)
(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 41 3 47 5 45 7 43)(2 48 4 46 6 44 8 42)(9 32 15 26 13 28 11 30)(10 31 16 25 14 27 12 29)(17 36 19 34 21 40 23 38)(18 35 20 33 22 39 24 37)

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

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

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

C3×C8.C4 is a maximal subgroup of
C24.7Q8  C24.6Q8  D24.C4  C24.8D4  Dic12.C4  M4(2).25D6  D2410C4  D247C4  C24.18D4  C24.19D4  C24.42D4

42 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B6C6D8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A···24H24I···24P
order1223344466668888888812121212121224···2424···24
size112111121122222244441111222···24···4

42 irreducible representations

dim11111111222222
type++++-
imageC1C2C2C3C4C6C6C12D4Q8C3×D4C3×Q8C8.C4C3×C8.C4
kernelC3×C8.C4C2×C24C3×M4(2)C8.C4C24C2×C8M4(2)C8C12C2×C6C4C22C3C1
# reps11224248112248

Matrix representation of C3×C8.C4 in GL2(𝔽73) generated by

80
08
,
5129
063
,
2140
5452
G:=sub<GL(2,GF(73))| [8,0,0,8],[51,0,29,63],[21,54,40,52] >;

C3×C8.C4 in GAP, Magma, Sage, TeX

C_3\times C_8.C_4
% in TeX

G:=Group("C3xC8.C4");
// GroupNames label

G:=SmallGroup(96,58);
// by ID

G=gap.SmallGroup(96,58);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,79,1443,117,88]);
// Polycyclic

G:=Group<a,b,c|a^3=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×C8.C4 in TeX

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