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G = C22.11C24order 64 = 26

7th central extension by C22 of C24

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

Aliases: C424C22, C22.11C24, C23.31C23, C24.11C22, C2.12+ 1+4, D47(C2×C4), (C4×D4)⋊5C2, (C2×D4)⋊11C4, C233(C2×C4), D4(C22⋊C4), C4⋊C420C22, C2.7(C23×C4), C42⋊C26C2, (C2×C4).50C23, C4.19(C22×C4), (C22×C4)⋊3C22, (C22×D4).9C2, C22⋊C418C22, (C2×D4).77C22, C22.2(C22×C4), C4⋊C4(C4⋊C4), (C2×C4)⋊4(C2×C4), (C2×C22⋊C4)⋊5C2, C22⋊C4(C22⋊C4), SmallGroup(64,199)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22.11C24
C1C2C22C23C24C22×D4 — C22.11C24
C1C2 — C22.11C24
C1C22 — C22.11C24
C1C22 — 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 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C2×C4 [×14], C2×C4 [×8], D4 [×16], C23, C23 [×12], C23 [×4], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C22.11C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×2], 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 10)(6 11)(7 12)(8 9)
(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 8)(2 10)(3 6)(4 12)(5 16)(7 14)(9 15)(11 13)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)
(1 3)(2 14)(4 16)(5 12)(6 8)(7 10)(9 11)(13 15)

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

G:=Group( (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (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), (1,8)(2,10)(3,6)(4,12)(5,16)(7,14)(9,15)(11,13), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16), (1,3)(2,14)(4,16)(5,12)(6,8)(7,10)(9,11)(13,15) );

G=PermutationGroup([(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(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)], [(1,8),(2,10),(3,6),(4,12),(5,16),(7,14),(9,15),(11,13)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16)], [(1,3),(2,14),(4,16),(5,12),(6,8),(7,10),(9,11),(13,15)])

G:=TransitiveGroup(16,68);

C22.11C24 is a maximal subgroup of
C24.C23  C24.6(C2×C4)  C42.5D4  C23.C24  C42.275C23  C42.277C23  C42.278C23  C42.14C23  C42.15C23  C42.20C23  C42.352C23  C42.356C23  C42.357C23  C22.14C25  C4×2+ 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  C429D6  C4213D6  C427D10 ...
C22.11C24 is a maximal quotient of
C23⋊C42  C24.524C23  D44C42  C23.191C24  C23.194C24  C23.195C24  C24.545C23  C24.547C23  C23.201C24  C23.203C24  C24.198C23  C23.211C24  C424Q8  C23.214C24  C23.215C24  C24.204C23  C24.205C23  D4×C22⋊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  C233M4(2)  D47M4(2)  C42.693C23  C42.297C23  C42.298C23  C42.299C23  C42.694C23  C42.300C23  C42.301C23  D10.C24
 C42⋊D2p: C4213D4  C4214D4  C429D6  C4213D6  C427D10  C4211D10  C427D14  C4211D14 ...
 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
kernelC22.11C24C2×C22⋊C4C42⋊C2C4×D4C22×D4C2×D4C2
# 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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