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G = C22.11C24  order 64 = 26

7th central extension by C22 of C24

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

Aliases: C42:4C22, C22.11C24, C23.31C23, C24.11C22, C2.12+ 1+4, D4:7(C2xC4), (C4xD4):5C2, (C2xD4):11C4, C23:3(C2xC4), D4o(C22:C4), C4:C4:20C22, C2.7(C23xC4), C42:C2:6C2, (C2xC4).50C23, C4.19(C22xC4), (C22xC4):3C22, (C22xD4).9C2, C22:C4:18C22, (C2xD4).77C22, C22.2(C22xC4), C4:C4o(C4:C4), (C2xC4):4(C2xC4), (C2xC22:C4):5C2, C22:C4o(C22:C4), SmallGroup(64,199)

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

Derived series
C1 — C2 — C22.11C24
Chief series
C1 — C2 — C22 — C23 — C24 — C22xD4 — C22.11C24
Lower central
C1 — C2 — C22.11C24
Upper central
C1 — C22 — C22.11C24
Jennings
C1 — C22 — C22.11C24

Generators and relations for C22.11C24
 G = < a,b,c,d,e,f | a2=b2=d2=e2=f2=1, c2=b, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ce=ec, df=fd, ef=fe >

Subgroups: 257 in 169 conjugacy classes, 121 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C22xC4, C22xC4, C2xD4, C24, C2xC22:C4, C42:C2, C4xD4, C22xD4, C22.11C24
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C24, C23xC4, 2+ 1+4, C22.11C24

Permutation representations of C22.11C24
►On 16 points - transitive group 16T68
Generators in S16
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)

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

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

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

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

(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
 
G:=sub<Sym(16)| (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15)>;
 
G:=Group( (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,10)(3,8)(4,12)(5,14)(7,16)(9,15)(11,13), (1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15) );
 
G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6),(2,10),(3,8),(4,12),(5,14),(7,16),(9,15),(11,13)], [(1,3),(2,4),(5,10),(6,11),(7,12),(8,9),(13,15),(14,16)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)]])
 
G:=TransitiveGroup(16,68);
 

C22.11C24 is a maximal subgroup of
 C24.C23  C24.6(C2xC4)  C42.5D4  C23.C24  C42.275C23  C42.277C23  C42.278C23  C42.14C23  C42.15C23  C42.20C23  C42.352C23  C42.356C23  C42.357C23  C22.14C25  C4x2+ 1+4  C22.73C25  C22.74C25  C22.79C25  C22.80C25  C22.83C25  C22.99C25  C22.102C25  C22.110C25  D10.C24
 C24.D2p: C24.21D4  C24.22D4  C24.23D4  C24.24D4  C25.C22  C24.26D4  C42:D4  C24.28D4 ...
 C2p.2+ 1+4: C22.90C25  C22.94C25  C22.97C25  C22.103C25  C22.108C25  C42:9D6  C42:13D6  C42:7D10 ...
C22.11C24 is a maximal quotient of
 C23:C42  C24.524C23  D4:4C42  C23.191C24  C23.194C24  C23.195C24  C24.545C23  C24.547C23  C23.201C24  C23.203C24  C24.198C23  C23.211C24  C42:4Q8  C23.214C24  C23.215C24  C24.204C23  C24.205C23  D4xC22:C4  C23.224C24  C23.227C24  C23.231C24  C23.235C24  C24.212C23  C23.240C24  C24.217C23  C24.218C23  C24.219C23  C23.250C24  C23.251C24  C24.221C23  C24.223C23  C23.257C24  C24.225C23  C23.259C24  C24.227C23  C23.261C24  C23.262C24  C23.263C24  C42.691C23  C23:3M4(2)  D4:7M4(2)  C42.693C23  C42.297C23  C42.298C23  C42.299C23  C42.694C23  C42.300C23  C42.301C23  D10.C24
 C42:D2p: C42:13D4  C42:14D4  C42:9D6  C42:13D6  C42:7D10  C42:11D10  C42:7D14  C42:11D14 ...
 C24.D2p: C24.90D4  C24.91D4  C24.35D6  C24.49D6  C24.24D10  C24.38D10  C24.24D14  C24.38D14 ...

34 conjugacy classes

class 1 2A2B2C2D···2M4A···4T
order12222···24···4
size11112···22···2

34 irreducible representations

dim1111114
type++++++
imageC1C2C2C2C2C42+ 1+4
kernelC22.11C24C2xC22:C4C42:C2C4xD4C22xD4C2xD4C2
# reps14281162

Matrix representation of C22.11C24 ►in GL5(F5)

10000
04000
00400
00040
00004
,
40000
01000
00100
00010
00001
,
30000
00010
00001
01000
00100
,
40000
04300
00100
00012
00004
,
40000
01000
04400
00010
00044
,
10000
01000
00100
00040
00004

G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,4,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,2,4],[4,0,0,0,0,0,1,4,0,0,0,0,4,0,0,0,0,0,1,4,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4] >;
 

C22.11C24 in GAP, Magma, Sage, TeX 

C_2^2._{11}C_2^4
 
% in TeX
 
G:=Group("C2^2.11C2^4");
 
// GroupNames label
 
G:=SmallGroup(64,199);
 
// by ID
 
G=gap.SmallGroup(64,199);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,-2,2,192,217,188,579]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=d^2=e^2=f^2=1,c^2=b,a*b=b*a,d*c*d=f*c*f=a*c=c*a,e*d*e=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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