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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

Aliases: C3×C4.4D4, C428C6, C12.39D4, (C2×Q8)⋊4C6, (C6×Q8)⋊9C2, C2.8(C6×D4), C4.4(C3×D4), (C4×C12)⋊12C2, C22⋊C45C6, (C2×D4).5C6, C6.71(C2×D4), (C6×D4).12C2, C23.2(C2×C6), C6.44(C4○D4), (C2×C6).79C23, (C2×C12).66C22, (C22×C6).2C22, C22.14(C22×C6), (C2×C4).6(C2×C6), C2.7(C3×C4○D4), (C3×C22⋊C4)⋊13C2, SmallGroup(96,171)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C4.4D4
C1C2C22C2×C6C22×C6C3×C22⋊C4 — C3×C4.4D4
C1C22 — C3×C4.4D4
C1C2×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 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C4.4D4, C4×C12, C3×C22⋊C4 [×4], C6×D4, C6×Q8, C3×C4.4D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], C2×D4, C4○D4 [×2], C3×D4 [×2], C22×C6, C4.4D4, C6×D4, C3×C4○D4 [×2], 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)])

C3×C4.4D4 is a maximal subgroup of
C42.7D6  C424Dic3  C42.Dic3  C42.61D6  C42.62D6  C42.213D6  D12.23D4  C42.64D6  C42.214D6  C42.65D6  C427D6  D12.14D4  C42.233D6  C42.137D6  C42.138D6  C42.139D6  C42.140D6  C4220D6  C42.141D6  D1210D4  Dic610D4  C4222D6  C4223D6  C42.234D6  C42.143D6  C42.144D6  C4224D6  C42.145D6

42 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4F4G4H6A···6F6G6H6I6J12A···12L12M12N12O12P
order122222334···4446···6666612···1212121212
size111144112···2441···144442···24444

42 irreducible representations

dim11111111112222
type++++++
imageC1C2C2C2C2C3C6C6C6C6D4C4○D4C3×D4C3×C4○D4
kernelC3×C4.4D4C4×C12C3×C22⋊C4C6×D4C6×Q8C4.4D4C42C22⋊C4C2×D4C2×Q8C12C6C4C2
# reps11411228222448

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

9000
0900
0030
0003
,
1300
81200
00810
0005
,
8000
0800
00111
00012
,
81100
0500
0080
0085
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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