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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×M4(2)
 Chief series C1 — C2 — C4 — C12 — C24 — C3×M4(2) — C6×M4(2)
 Lower central C1 — C2 — C6×M4(2)
 Upper central C1 — C2×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, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C2×M4(2), C2×C24, C3×M4(2), C22×C12, C6×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, M4(2), C22×C4, C2×C12, C22×C6, C2×M4(2), C3×M4(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)]])

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 8A ··· 8H 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 8 ··· 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 ··· 1 2 2 2 2 2 ··· 2 1 ··· 1 2 2 2 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 M4(2) C3×M4(2) kernel C6×M4(2) C2×C24 C3×M4(2) C22×C12 C2×M4(2) C2×C12 C22×C6 C2×C8 M4(2) C22×C4 C2×C4 C23 C6 C2 # reps 1 2 4 1 2 6 2 4 8 2 12 4 4 8

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

 9 0 0 0 1 0 0 0 1
,
 72 0 0 0 0 72 0 27 0
,
 72 0 0 0 1 0 0 0 72
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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