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G = C6×M4(2)  order 96 = 25·3

Direct product of C6 and M4(2)

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

Aliases: C6×M4(2), C12M4(2), C2414C22, C23.4C12, C12.53C23, (C2×C8)⋊6C6, C84(C2×C6), (C2×C24)⋊14C2, (C2×C4).6C12, C4(C3×M4(2)), (C2×C12).15C4, C4.10(C2×C12), C12.47(C2×C4), C12(C3×M4(2)), (C22×C6).4C4, (C22×C4).8C6, C4.11(C22×C6), C22.6(C2×C12), C2.6(C22×C12), C6.34(C22×C4), (C22×C12).16C2, (C2×C12).127C22, (C2×C6).23(C2×C4), (C2×C4).33(C2×C6), SmallGroup(96,177)

Series: Derived Chief Lower central Upper central

C1C2 — C6×M4(2)
C1C2C4C12C24C3×M4(2) — C6×M4(2)
C1C2 — C6×M4(2)
C1C2×C12 — C6×M4(2)

Generators and relations for C6×M4(2)
 G = < a,b,c | a6=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 76 in 68 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C24 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C2×M4(2), C2×C24 [×2], C3×M4(2) [×4], C22×C12, C6×M4(2)
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], M4(2) [×2], C22×C4, C2×C12 [×6], C22×C6, C2×M4(2), C3×M4(2) [×2], C22×C12, C6×M4(2)

Smallest permutation representation of C6×M4(2)
On 48 points
Generators in S48
(1 11 34 18 45 30)(2 12 35 19 46 31)(3 13 36 20 47 32)(4 14 37 21 48 25)(5 15 38 22 41 26)(6 16 39 23 42 27)(7 9 40 24 43 28)(8 10 33 17 44 29)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)

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

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

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

C6×M4(2) is a maximal subgroup of
C24.D4  M4(2)⋊Dic3  C12.3C42  (C2×C24)⋊C4  C12.20C42  C12.4C42  M4(2)⋊4Dic3  C12.21C42  Dic34M4(2)  C12.88(C2×Q8)  C23.51D12  C23.52D12  C12.7C42  C23.8Dic6  C23.9Dic6  D66M4(2)  C24⋊D4  C2421D4  D6⋊C840C2  C23.53D12  M4(2).31D6  C23.54D12  C242D4  C243D4  C24.4D4  M4(2)⋊24D6  M4(2)⋊26D6  C24.9C23

60 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6F6G6H6I6J8A···8H12A···12H12I12J12K12L24A···24P
order122222334444446···666668···812···121212121224···24
size111122111111221···122222···21···122222···2

60 irreducible representations

dim11111111111122
type++++
imageC1C2C2C2C3C4C4C6C6C6C12C12M4(2)C3×M4(2)
kernelC6×M4(2)C2×C24C3×M4(2)C22×C12C2×M4(2)C2×C12C22×C6C2×C8M4(2)C22×C4C2×C4C23C6C2
# reps124126248212448

Matrix representation of C6×M4(2) in GL3(𝔽73) generated by

900
010
001
,
7200
0072
0270
,
7200
010
0072
G:=sub<GL(3,GF(73))| [9,0,0,0,1,0,0,0,1],[72,0,0,0,0,27,0,72,0],[72,0,0,0,1,0,0,0,72] >;

C6×M4(2) in GAP, Magma, Sage, TeX

C_6\times M_4(2)
% in TeX

G:=Group("C6xM4(2)");
// GroupNames label

G:=SmallGroup(96,177);
// by ID

G=gap.SmallGroup(96,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,601,88]);
// Polycyclic

G:=Group<a,b,c|a^6=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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