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G = C24⋊C22  order 64 = 26

4th semidirect product of C24 and C22 acting faithfully

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C24⋊4C22, C42⋊12C22, C22.55C24, C23.24C23, C2.222+ 1+4, C22≀C2⋊8C2, (C2×Q8)⋊6C22, C4.4D4⋊15C2, (C2×C4).37C23, C22⋊C4⋊10C22, (C2×D4).38C22, 2-Sylow(GL(3,4)), SmallGroup(64,242)

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

Derived series
C1 — C22 — C24⋊C22
Chief series
C1 — C2 — C22 — C23 — C24 — C22≀C2 — C24⋊C22
Lower central
C1 — C22 — C24⋊C22
Upper central
C1 — C22 — C24⋊C22
Jennings
C1 — C22 — C24⋊C22

Generators and relations for C24⋊C22
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, eae=ad=da, faf=acd, fbf=bc=cb, bd=db, ebe=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 253 in 130 conjugacy classes, 71 normal (3 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C2×D4, C2×Q8, C24, C22≀C2, C4.4D4, C24⋊C22
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C24⋊C22

Character table of C24⋊C22

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I
 size 1111444444444444444
ρ11111111111111111111    trivial
ρ21111-11-1-1-111-1-11111-1-1    linear of order 2
ρ31111-1111-11-1-11-11-1-11-1    linear of order 2
ρ4111111-1-111-11-1-11-1-1-11    linear of order 2
ρ511111-11-1111-1-1-1-1-111-1    linear of order 2
ρ61111-1-1-11-11111-1-1-11-11    linear of order 2
ρ71111-1-11-1-11-11-11-11-111    linear of order 2
ρ811111-1-1111-1-111-11-1-1-1    linear of order 2
ρ91111-11111-11-1-1-1-11-1-11    linear of order 2
ρ10111111-1-1-1-1111-1-11-11-1    linear of order 2
ρ1111111111-1-1-11-11-1-11-1-1    linear of order 2
ρ121111-11-1-11-1-1-111-1-1111    linear of order 2
ρ131111-1-11-11-111111-1-1-1-1    linear of order 2
ρ1411111-1-11-1-11-1-111-1-111    linear of order 2
ρ1511111-11-1-1-1-1-11-1111-11    linear of order 2
ρ161111-1-1-111-1-11-1-11111-1    linear of order 2
ρ174-4-44000000000000000    orthogonal lifted from 2+ 1+4
ρ184-44-4000000000000000    orthogonal lifted from 2+ 1+4
ρ1944-4-4000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C24⋊C22
►On 16 points - transitive group 16T98
Generators in S16
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)

(1 3)(2 4)(5 6)(7 8)(11 15)(12 16)

(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 12)(15 16)

(1 4)(2 3)(5 6)(7 8)(9 14)(10 13)(11 16)(12 15)

(1 6)(2 7)(3 8)(4 5)(9 12)(10 16)(11 13)(14 15)

(1 13)(2 14)(3 9)(4 10)(5 16)(6 11)(7 15)(8 12)
 
G:=sub<Sym(16)| (5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,3)(2,4)(5,6)(7,8)(11,15)(12,16), (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,12)(15,16), (1,4)(2,3)(5,6)(7,8)(9,14)(10,13)(11,16)(12,15), (1,6)(2,7)(3,8)(4,5)(9,12)(10,16)(11,13)(14,15), (1,13)(2,14)(3,9)(4,10)(5,16)(6,11)(7,15)(8,12)>;
 
G:=Group( (5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,3)(2,4)(5,6)(7,8)(11,15)(12,16), (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,12)(15,16), (1,4)(2,3)(5,6)(7,8)(9,14)(10,13)(11,16)(12,15), (1,6)(2,7)(3,8)(4,5)(9,12)(10,16)(11,13)(14,15), (1,13)(2,14)(3,9)(4,10)(5,16)(6,11)(7,15)(8,12) );
 
G=PermutationGroup([[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,3),(2,4),(5,6),(7,8),(11,15),(12,16)], [(1,3),(2,4),(5,7),(6,8),(9,13),(10,14),(11,12),(15,16)], [(1,4),(2,3),(5,6),(7,8),(9,14),(10,13),(11,16),(12,15)], [(1,6),(2,7),(3,8),(4,5),(9,12),(10,16),(11,13),(14,15)], [(1,13),(2,14),(3,9),(4,10),(5,16),(6,11),(7,15),(8,12)]])
 
G:=TransitiveGroup(16,98);
 

C24⋊C22 is a maximal subgroup of
 C42.C23  C42.5C23  C42.15D4  C42⋊C23  C22.134C25  C22.157C25  C24⋊A4  C42⋊A4  C24⋊5A4
 C42⋊D2p: C42⋊5D4  C42⋊24D6  C42⋊22D10  C42⋊22D14 ...
 C2p.2+ 1+4: C22.118C25  C22.129C25  C22.138C25  C22.147C25  C24⋊9D6  C24⋊5D10  C24⋊4D14 ...
C24⋊C22 is a maximal quotient of
 C23.257C24  C23.261C24  C23.570C24  C25⋊C22  C23.584C24  C23.612C24  C23.633C24  C23.636C24  C24.437C23  C23.663C24  C23.682C24  C23.685C24  C24.456C23  C23.711C24  C23.724C24  C23.728C24  C23.732C24  C23.735C24  C24⋊6Q8  C42⋊13Q8
 C42⋊D2p: C42⋊34D4  C42⋊24D6  C42⋊22D10  C42⋊22D14 ...
 C24⋊D2p: C24⋊11D4  C24⋊9D6  C24⋊5D10  C24⋊4D14 ...

Matrix representation of C24⋊C22 ►in GL8(ℤ)

-10000000
0-1000000
00100000
11010000
0000-1000
00000100
00000010
0000000-1
,
10000000
0-1000000
00100000
-10-1-10000
00001000
00000100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00100000
-1-1-1-20000
10000000
00010000
00000010
00000001
00001000
00000100
,
01000000
10000000
-1-1-1-20000
00010000
00000100
00001000
00000001
00000010

G:=sub<GL(8,Integers())| [-1,0,0,1,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-2,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C24⋊C22 in GAP, Magma, Sage, TeX 

C_2^4\rtimes C_2^2
 
% in TeX
 
G:=Group("C2^4:C2^2");
 
// GroupNames label
 
G:=SmallGroup(64,242);
 
// by ID
 
G=gap.SmallGroup(64,242);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650,476,1347,297]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,e*a*e=a*d=d*a,f*a*f=a*c*d,f*b*f=b*c=c*b,b*d=d*b,e*b*e=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

Export

Character table of C24⋊C22 in TeX

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