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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C8.C4
 Lower central C1 — C2 — C4 — C3×C8.C4
 Upper central C1 — C12 — C2×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 >

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 2A 2B 3A 3B 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A ··· 24H 24I ··· 24P order 1 2 2 3 3 4 4 4 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 ··· 24 24 ··· 24 size 1 1 2 1 1 1 1 2 1 1 2 2 2 2 2 2 4 4 4 4 1 1 1 1 2 2 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 C3×D4 C3×Q8 C8.C4 C3×C8.C4 kernel C3×C8.C4 C2×C24 C3×M4(2) C8.C4 C24 C2×C8 M4(2) C8 C12 C2×C6 C4 C22 C3 C1 # reps 1 1 2 2 4 2 4 8 1 1 2 2 4 8

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

 8 0 0 8
,
 51 29 0 63
,
 21 40 54 52
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

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