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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C4.10D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4.10D4
 Lower central C1 — C2 — C22 — C3×C4.10D4
 Upper central C1 — C6 — C2×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 >

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 2A 2B 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A ··· 24H order 1 2 2 3 3 4 4 4 4 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 2 1 1 2 2 4 4 1 1 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 C3×D4 C4.10D4 C3×C4.10D4 kernel C3×C4.10D4 C3×M4(2) C6×Q8 C4.10D4 C2×C12 M4(2) C2×Q8 C2×C4 C12 C4 C3 C1 # reps 1 2 1 2 4 4 2 8 2 4 1 2

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 3 1 4 6 3 5 3 2 2 1 0 0 6 3 3
,
 1 1 5 0 2 5 5 1 4 5 1 4 4 5 2 0
,
 3 0 5 1 5 2 0 4 5 5 6 5 1 0 3 3
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

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