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G = C3×C4.10D4order 96 = 25·3

Direct product of C3 and C4.10D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×C4.10D4, C12.59D4, M4(2).1C6, (C2×C4).C12, (C2×C12).2C4, C4.10(C3×D4), (C6×Q8).6C2, (C2×Q8).3C6, C22.4(C2×C12), C6.23(C22⋊C4), (C2×C12).60C22, (C3×M4(2)).3C2, (C2×C4).2(C2×C6), (C2×C6).21(C2×C4), C2.5(C3×C22⋊C4), SmallGroup(96,51)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C4.10D4
C1C2C4C2×C4C2×C12C3×M4(2) — C3×C4.10D4
C1C2C22 — C3×C4.10D4
C1C6C2×C12 — C3×C4.10D4

Generators and relations for C3×C4.10D4
 G = < a,b,c,d | a3=b4=1, c4=b2, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >

2C2
2C4
2C4
2C6
2C8
2Q8
2C8
2Q8
2C12
2C12
2C24
2C24
2C3×Q8
2C3×Q8

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

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

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

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

C3×C4.10D4 is a maximal subgroup of   (C2×C4).D12  (C2×C12).D4  M4(2).21D6  D12.4D4  D12.5D4  D12.6D4  D12.7D4

33 conjugacy classes

class 1 2A2B3A3B4A4B4C4D6A6B6C6D8A8B8C8D12A12B12C12D12E12F12G12H24A···24H
order12233444466668888121212121212121224···24
size11211224411224444222244444···4

33 irreducible representations

dim111111112244
type++++-
imageC1C2C2C3C4C6C6C12D4C3×D4C4.10D4C3×C4.10D4
kernelC3×C4.10D4C3×M4(2)C6×Q8C4.10D4C2×C12M4(2)C2×Q8C2×C4C12C4C3C1
# reps121244282412

Matrix representation of C3×C4.10D4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
0314
6353
2210
0633
,
1150
2551
4514
4520
,
3051
5204
5565
1033
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,6,2,0,3,3,2,6,1,5,1,3,4,3,0,3],[1,2,4,4,1,5,5,5,5,5,1,2,0,1,4,0],[3,5,5,1,0,2,5,0,5,0,6,3,1,4,5,3] >;

C3×C4.10D4 in GAP, Magma, Sage, TeX

C_3\times C_4._{10}D_4
% in TeX

G:=Group("C3xC4.10D4");
// GroupNames label

G:=SmallGroup(96,51);
// by ID

G=gap.SmallGroup(96,51);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,295,1443,1090,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C4.10D4 in TeX

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