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## G = C3×C4.4D4order 96 = 25·3

### Direct product of C3 and C4.4D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C4.4D4
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C3×C22⋊C4 — C3×C4.4D4
 Lower central C1 — C22 — C3×C4.4D4
 Upper central C1 — C2×C6 — C3×C4.4D4

Generators and relations for C3×C4.4D4
G = < a,b,c,d | a3=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.4D4, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C3×C4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C6×D4, C3×C4○D4, C3×C4.4D4

Smallest permutation representation of C3×C4.4D4
On 48 points
Generators in S48
(1 39 47)(2 40 48)(3 37 45)(4 38 46)(5 33 41)(6 34 42)(7 35 43)(8 36 44)(9 25 15)(10 26 16)(11 27 13)(12 28 14)(17 21 31)(18 22 32)(19 23 29)(20 24 30)
(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 34 13 17)(2 35 14 18)(3 36 15 19)(4 33 16 20)(5 26 30 46)(6 27 31 47)(7 28 32 48)(8 25 29 45)(9 23 37 44)(10 24 38 41)(11 21 39 42)(12 22 40 43)
(1 18 3 20)(2 17 4 19)(5 27 7 25)(6 26 8 28)(9 41 11 43)(10 44 12 42)(13 35 15 33)(14 34 16 36)(21 38 23 40)(22 37 24 39)(29 48 31 46)(30 47 32 45)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 C4○D4 C3×D4 C3×C4○D4 kernel C3×C4.4D4 C4×C12 C3×C22⋊C4 C6×D4 C6×Q8 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C12 C6 C4 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 4 4 8

Matrix representation of C3×C4.4D4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 3 0 0 0 0 3
,
 1 3 0 0 8 12 0 0 0 0 8 10 0 0 0 5
,
 8 0 0 0 0 8 0 0 0 0 1 11 0 0 0 12
,
 8 11 0 0 0 5 0 0 0 0 8 0 0 0 8 5
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,8,0,0,3,12,0,0,0,0,8,0,0,0,10,5],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,11,12],[8,0,0,0,11,5,0,0,0,0,8,8,0,0,0,5] >;

C3×C4.4D4 in GAP, Magma, Sage, TeX

C_3\times C_4._4D_4
% in TeX

G:=Group("C3xC4.4D4");
// GroupNames label

G:=SmallGroup(96,171);
// by ID

G=gap.SmallGroup(96,171);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,295,938,122]);
// Polycyclic

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

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