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G = C22×M4(2)  order 64 = 26

Direct product of C22 and M4(2)

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

Aliases: C22×M4(2), C8⋊4C23, C24.5C4, C4.15C24, C4○(C2×M4(2)), (C2×C4)○M4(2), (C22×C8)⋊12C2, (C2×C8)⋊15C22, (C22×C4).18C4, C23.35(C2×C4), C4.31(C22×C4), C2.10(C23×C4), (C23×C4).11C2, (C2×C4).161C23, C22.27(C22×C4), (C22×C4).127C22, (C2×C4).77(C2×C4), (C2×C4)○(C2×M4(2)), SmallGroup(64,247)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — C22×M4(2)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2)
Lower central
C1 — C2 — C22×M4(2)
Upper central
C1 — C22×C4 — C22×M4(2)
Jennings
C1 — C2 — C2 — C4 — C22×M4(2)

Generators and relations for C22×M4(2)
 G = < a,b,c,d | a2=b2=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 169 in 149 conjugacy classes, 129 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C23, C23, C23, C2×C8, M4(2), C22×C4, C22×C4, C24, C22×C8, C2×M4(2), C23×C4, C22×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C24, C2×M4(2), C23×C4, C22×M4(2)

Smallest permutation representation of C22×M4(2)
►On 32 points
Generators in S32
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)

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

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

C22×M4(2) is a maximal subgroup of
 C23.28C42  C23.29C42  C24.63D4  C24.152D4  C24.7Q8  C23.15C42  C23.17C42  C24.67D4  C24.9Q8  (C2×C8).195D4  C24.10Q8  C23⋊2M4(2)  C24.72D4  C24.75D4  C24.76D4  M4(2)⋊20D4  M4(2).45D4  M4(2)○2M4(2)  C24.73(C2×C4)  C24.98D4  C42.257C23  C24.100D4  C42.265C23  M4(2)⋊22D4  C24.110D4  M4(2)⋊14D4  M4(2)⋊15D4
C22×M4(2) is a maximal quotient of
 C42.677C23  C42.290C23  D4⋊6M4(2)  Q8⋊6M4(2)  C23⋊3M4(2)  D4⋊7M4(2)  C42.693C23  C42.302C23  Q8.4M4(2)  C42.698C23  D4⋊8M4(2)  Q8⋊7M4(2)

40 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L8A···8P
order12···222224···444448···8
size11···122221···122222···2

40 irreducible representations

dim1111112
type++++
imageC1C2C2C2C4C4M4(2)
kernelC22×M4(2)C22×C8C2×M4(2)C23×C4C22×C4C24C22
# reps121211428

Matrix representation of C22×M4(2) ►in GL4(𝔽17) generated by

16000
0100
0010
0001
,
1000
01600
0010
0001
,
13000
01600
00132
00114
,
16000
0100
0010
00416
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,16,0,0,0,0,13,11,0,0,2,4],[16,0,0,0,0,1,0,0,0,0,1,4,0,0,0,16] >;
 

C22×M4(2) in GAP, Magma, Sage, TeX 

C_2^2\times M_4(2)
 
% in TeX
 
G:=Group("C2^2xM4(2)");
 
// GroupNames label
 
G:=SmallGroup(64,247);
 
// by ID
 
G=gap.SmallGroup(64,247);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,-2,-2,96,409,88]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
 
// generators/relations
 

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