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## G = C3×C42⋊2C2order 96 = 25·3

### Direct product of C3 and C42⋊2C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C42⋊2C2
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C3×C22⋊C4 — C3×C42⋊2C2
 Lower central C1 — C22 — C3×C42⋊2C2
 Upper central C1 — C2×C6 — C3×C42⋊2C2

Generators and relations for C3×C422C2
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 84 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C22×C6, C422C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C422C2
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C22×C6, C422C2, C3×C4○D4, C3×C422C2

Smallest permutation representation of C3×C422C2
On 48 points
Generators in S48
(1 43 19)(2 44 20)(3 41 17)(4 42 18)(5 9 29)(6 10 30)(7 11 31)(8 12 32)(13 33 37)(14 34 38)(15 35 39)(16 36 40)(21 25 45)(22 26 46)(23 27 47)(24 28 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 33 5 23)(2 34 6 24)(3 35 7 21)(4 36 8 22)(9 27 43 37)(10 28 44 38)(11 25 41 39)(12 26 42 40)(13 29 47 19)(14 30 48 20)(15 31 45 17)(16 32 46 18)
(2 6)(4 8)(10 44)(12 42)(13 45)(14 16)(15 47)(18 32)(20 30)(21 33)(22 24)(23 35)(25 37)(26 28)(27 39)(34 36)(38 40)(46 48)

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

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

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

C3×C422C2 is a maximal subgroup of
C42.159D6  C42.160D6  C4225D6  C4226D6  C42.189D6  C42.161D6  C42.162D6  C42.163D6  C42.164D6  C4227D6  C42.165D6  C42⋊C18

42 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A ··· 4F 4G 4H 4I 6A ··· 6F 6G 6H 12A ··· 12L 12M ··· 12R order 1 2 2 2 2 3 3 4 ··· 4 4 4 4 6 ··· 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 4 1 1 2 ··· 2 4 4 4 1 ··· 1 4 4 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C4○D4 C3×C4○D4 kernel C3×C42⋊2C2 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C42⋊2C2 C42 C22⋊C4 C4⋊C4 C6 C2 # reps 1 1 3 3 2 2 6 6 6 12

Matrix representation of C3×C422C2 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 12 2 0 0 12 1 0 0 0 0 0 8 0 0 5 0
,
 8 0 0 0 0 8 0 0 0 0 0 1 0 0 12 0
,
 1 0 0 0 1 12 0 0 0 0 1 0 0 0 0 12
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,2,1,0,0,0,0,0,5,0,0,8,0],[8,0,0,0,0,8,0,0,0,0,0,12,0,0,1,0],[1,1,0,0,0,12,0,0,0,0,1,0,0,0,0,12] >;

C3×C422C2 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C3xC4^2:2C2");
// GroupNames label

G:=SmallGroup(96,173);
// by ID

G=gap.SmallGroup(96,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,439,938,122]);
// Polycyclic

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

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