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## G = C3×C42.C4order 192 = 26·3

### Direct product of C3 and C42.C4

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

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 146 in 64 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.10D4, C4.4D4, C4×C12, C3×C22⋊C4, C3×M4(2), C6×D4, C6×Q8, C42.C4, C3×C4.10D4, C3×C4.4D4, C3×C42.C4
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.C4, C3×C23⋊C4, C3×C42.C4

Smallest permutation representation of C3×C42.C4
On 48 points
Generators in S48
(1 14 39)(2 15 40)(3 16 33)(4 9 34)(5 10 35)(6 11 36)(7 12 37)(8 13 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)
(1 3)(2 18 6 22)(4 20 8 24)(5 7)(9 32 13 28)(10 12)(11 26 15 30)(14 16)(17 19)(21 23)(25 27)(29 31)(33 39)(34 44 38 48)(35 37)(36 46 40 42)(41 43)(45 47)
(1 23 5 19)(2 20 6 24)(3 21 7 17)(4 18 8 22)(9 30 13 26)(10 31 14 27)(11 28 15 32)(12 29 16 25)(33 45 37 41)(34 42 38 46)(35 43 39 47)(36 48 40 44)
(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)| (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (1,3)(2,18,6,22)(4,20,8,24)(5,7)(9,32,13,28)(10,12)(11,26,15,30)(14,16)(17,19)(21,23)(25,27)(29,31)(33,39)(34,44,38,48)(35,37)(36,46,40,42)(41,43)(45,47), (1,23,5,19)(2,20,6,24)(3,21,7,17)(4,18,8,22)(9,30,13,26)(10,31,14,27)(11,28,15,32)(12,29,16,25)(33,45,37,41)(34,42,38,46)(35,43,39,47)(36,48,40,44), (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( (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (1,3)(2,18,6,22)(4,20,8,24)(5,7)(9,32,13,28)(10,12)(11,26,15,30)(14,16)(17,19)(21,23)(25,27)(29,31)(33,39)(34,44,38,48)(35,37)(36,46,40,42)(41,43)(45,47), (1,23,5,19)(2,20,6,24)(3,21,7,17)(4,18,8,22)(9,30,13,26)(10,31,14,27)(11,28,15,32)(12,29,16,25)(33,45,37,41)(34,42,38,46)(35,43,39,47)(36,48,40,44), (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([[(1,14,39),(2,15,40),(3,16,33),(4,9,34),(5,10,35),(6,11,36),(7,12,37),(8,13,38),(17,29,41),(18,30,42),(19,31,43),(20,32,44),(21,25,45),(22,26,46),(23,27,47),(24,28,48)], [(1,3),(2,18,6,22),(4,20,8,24),(5,7),(9,32,13,28),(10,12),(11,26,15,30),(14,16),(17,19),(21,23),(25,27),(29,31),(33,39),(34,44,38,48),(35,37),(36,46,40,42),(41,43),(45,47)], [(1,23,5,19),(2,20,6,24),(3,21,7,17),(4,18,8,22),(9,30,13,26),(10,31,14,27),(11,28,15,32),(12,29,16,25),(33,45,37,41),(34,42,38,46),(35,43,39,47),(36,48,40,44)], [(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 2C 3A 3B 4A ··· 4E 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12A ··· 12J 24A ··· 24H order 1 2 2 2 3 3 4 ··· 4 6 6 6 6 6 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 2 8 1 1 4 ··· 4 1 1 2 2 8 8 8 8 8 8 4 ··· 4 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.C4 C3×C23⋊C4 C3×C42.C4 kernel C3×C42.C4 C3×C4.10D4 C3×C4.4D4 C42.C4 C4×C12 C6×D4 C4.10D4 C4.4D4 C42 C2×D4 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.C4 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 0 72 0 0 72 0 0 0 0 0 46 0 0 0 0 46
,
 0 46 0 0 46 0 0 0 0 0 0 46 0 0 46 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 72 0 0 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,72,0,0,0,0,0,46,0,0,0,0,46],[0,46,0,0,46,0,0,0,0,0,0,46,0,0,46,0],[0,0,0,72,0,0,1,0,1,0,0,0,0,1,0,0] >;

C3×C42.C4 in GAP, Magma, Sage, TeX

C_3\times C_4^2.C_4
% in TeX

G:=Group("C3xC4^2.C4");
// GroupNames label

G:=SmallGroup(192,161);
// by ID

G=gap.SmallGroup(192,161);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951,375,172,6053]);
// 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^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations

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