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## G = C3×C42⋊C2order 96 = 25·3

### Direct product of C3 and C42⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C42⋊C2
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C22⋊C4 — C3×C42⋊C2
 Lower central C1 — C2 — C3×C42⋊C2
 Upper central C1 — C2×C12 — C3×C42⋊C2

Generators and relations for C3×C42⋊C2
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >

Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×C12, C2×C12, C22×C6, C42⋊C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C3×C42⋊C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, C2×C12, C22×C6, C42⋊C2, C22×C12, C3×C4○D4, C3×C42⋊C2

Smallest permutation representation of C3×C42⋊C2
On 48 points
Generators in S48
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 27 45)(22 28 46)(23 25 47)(24 26 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 23 39 15)(2 24 40 16)(3 21 37 13)(4 22 38 14)(5 34 28 42)(6 35 25 43)(7 36 26 44)(8 33 27 41)(9 45 17 29)(10 46 18 30)(11 47 19 31)(12 48 20 32)
(1 3)(2 38)(4 40)(5 26)(6 8)(7 28)(9 11)(10 20)(12 18)(13 15)(14 24)(16 22)(17 19)(21 23)(25 27)(29 31)(30 48)(32 46)(33 35)(34 44)(36 42)(37 39)(41 43)(45 47)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E ··· 4N 6A ··· 6F 6G 6H 6I 6J 12A ··· 12H 12I ··· 12AB order 1 2 2 2 2 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 C4○D4 C3×C4○D4 kernel C3×C42⋊C2 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C42⋊C2 C2×C12 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C6 C2 # reps 1 2 2 2 1 2 8 4 4 4 2 16 4 8

Matrix representation of C3×C42⋊C2 in GL3(𝔽13) generated by

 1 0 0 0 3 0 0 0 3
,
 5 0 0 0 8 11 0 0 5
,
 1 0 0 0 5 0 0 0 5
,
 1 0 0 0 12 0 0 5 1
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[5,0,0,0,8,0,0,11,5],[1,0,0,0,5,0,0,0,5],[1,0,0,0,12,5,0,0,1] >;

C3×C42⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes C_2
% in TeX

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

G:=SmallGroup(96,164);
// by ID

G=gap.SmallGroup(96,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,122]);
// 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,c*d=d*c>;
// generators/relations

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