Copied to
clipboard

## G = C4×M4(2)  order 64 = 26

### Direct product of C4 and M4(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2)
 Lower central C1 — C2 — C4×M4(2)
 Upper central C1 — C42 — C4×M4(2)
 Jennings C1 — C2 — C2 — C2×C4 — C4×M4(2)

Generators and relations for C4×M4(2)
G = < a,b,c | a4=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 81 in 71 conjugacy classes, 61 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C4×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C2×C42, C2×M4(2), C4×M4(2)

Smallest permutation representation of C4×M4(2)
On 32 points
Generators in S32
(1 19 26 16)(2 20 27 9)(3 21 28 10)(4 22 29 11)(5 23 30 12)(6 24 31 13)(7 17 32 14)(8 18 25 15)
(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)
(1 26)(2 31)(3 28)(4 25)(5 30)(6 27)(7 32)(8 29)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)

G:=sub<Sym(32)| (1,19,26,16)(2,20,27,9)(3,21,28,10)(4,22,29,11)(5,23,30,12)(6,24,31,13)(7,17,32,14)(8,18,25,15), (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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)>;

G:=Group( (1,19,26,16)(2,20,27,9)(3,21,28,10)(4,22,29,11)(5,23,30,12)(6,24,31,13)(7,17,32,14)(8,18,25,15), (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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19) );

G=PermutationGroup([[(1,19,26,16),(2,20,27,9),(3,21,28,10),(4,22,29,11),(5,23,30,12),(6,24,31,13),(7,17,32,14),(8,18,25,15)], [(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)], [(1,26),(2,31),(3,28),(4,25),(5,30),(6,27),(7,32),(8,29),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19)]])

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 8A ··· 8P order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 M4(2) kernel C4×M4(2) C4×C8 C8⋊C4 C2×C42 C2×M4(2) C42 M4(2) C22×C4 C4 # reps 1 2 2 1 2 4 16 4 8

Matrix representation of C4×M4(2) in GL3(𝔽17) generated by

 13 0 0 0 1 0 0 0 1
,
 4 0 0 0 4 15 0 6 13
,
 1 0 0 0 1 0 0 4 16
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[4,0,0,0,4,6,0,15,13],[1,0,0,0,1,4,0,0,16] >;

C4×M4(2) in GAP, Magma, Sage, TeX

C_4\times M_4(2)
% in TeX

G:=Group("C4xM4(2)");
// GroupNames label

G:=SmallGroup(64,85);
// by ID

G=gap.SmallGroup(64,85);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,650,117]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

׿
×
𝔽