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## G = C3×C42.3C4order 192 = 26·3

### Direct product of C3 and C42.3C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C3×C42.3C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C6×Q8 — C3×C4.10D4 — C3×C42.3C4
 Lower central C1 — C2 — C22 — C2×C4 — C3×C42.3C4
 Upper central C1 — C6 — C2×C6 — C6×Q8 — C3×C42.3C4

Generators and relations for C3×C42.3C4
G = < a,b,c,d | a3=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 114 in 60 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C2×C6, C42, C4⋊C4, M4(2), C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C4.10D4, C4⋊Q8, C4×C12, C3×C4⋊C4, C3×M4(2), C6×Q8, C42.3C4, C3×C4.10D4, C3×C4⋊Q8, C3×C42.3C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C23⋊C4, C3×C22⋊C4, C42.3C4, C3×C23⋊C4, C3×C42.3C4

Smallest permutation representation of C3×C42.3C4
On 48 points
Generators in S48
(1 16 39)(2 9 40)(3 10 33)(4 11 34)(5 12 35)(6 13 36)(7 14 37)(8 15 38)(17 29 41)(18 30 42)(19 31 43)(20 32 44)(21 25 45)(22 26 46)(23 27 47)(24 28 48)
(2 24 6 20)(4 22 8 18)(9 28 13 32)(11 26 15 30)(34 46 38 42)(36 44 40 48)
(1 23 5 19)(2 24 6 20)(3 21 7 17)(4 22 8 18)(9 28 13 32)(10 25 14 29)(11 26 15 30)(12 31 16 27)(33 45 37 41)(34 46 38 42)(35 43 39 47)(36 44 40 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)

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

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

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

39 conjugacy classes

 class 1 2A 2B 3A 3B 4A ··· 4E 4F 6A 6B 6C 6D 8A 8B 8C 8D 12A ··· 12J 12K 12L 24A ··· 24H order 1 2 2 3 3 4 ··· 4 4 6 6 6 6 8 8 8 8 12 ··· 12 12 12 24 ··· 24 size 1 1 2 1 1 4 ··· 4 8 1 1 2 2 8 8 8 8 4 ··· 4 8 8 8 ··· 8

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + - image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 C23⋊C4 C42.3C4 C3×C23⋊C4 C3×C42.3C4 kernel C3×C42.3C4 C3×C4.10D4 C3×C4⋊Q8 C42.3C4 C4×C12 C6×Q8 C4.10D4 C4⋊Q8 C42 C2×Q8 C2×C12 C2×C4 C6 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 1 2 2 4

Matrix representation of C3×C42.3C4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 3 0 2 3 2 3 4 5 5 4 2 0 4 2 3
,
 5 0 3 0 1 5 5 4 3 0 2 0 6 4 1 2
,
 2 6 3 0 2 1 4 5 2 3 2 3 3 4 1 2
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,3,5,0,3,2,5,4,0,3,4,2,2,4,2,3],[5,1,3,6,0,5,0,4,3,5,2,1,0,4,0,2],[2,2,2,3,6,1,3,4,3,4,2,1,0,5,3,2] >;

C3×C42.3C4 in GAP, Magma, Sage, TeX

C_3\times C_4^2._3C_4
% in TeX

G:=Group("C3xC4^2.3C4");
// GroupNames label

G:=SmallGroup(192,162);
// by ID

G=gap.SmallGroup(192,162);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1522,248,2951,375,172,6053]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^2*c>;
// generators/relations

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