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G = C22.56C24order 64 = 26

42nd central stem extension by C22 of C24

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

Aliases: C42.55C22, C23.25C23, C22.56C24, C2.232+ 1+4, C2.152- 1+4, C4⋊D418C2, C22⋊Q819C2, C4.4D416C2, C4⋊C4.39C22, (C2×C4).38C23, C42.C211C2, (C2×D4).39C22, C22⋊C4.8C22, (C2×Q8).35C22, C22.D414C2, (C22×C4).76C22, SmallGroup(64,243)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.56C24
C1C2C22C23C22×C4C22.D4 — C22.56C24
C1C22 — C22.56C24
C1C22 — C22.56C24
C1C22 — C22.56C24

Generators and relations for C22.56C24
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e2=a, ab=ba, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, fcf=abc, ede-1=abd, ef=fe >

Subgroups: 181 in 110 conjugacy classes, 71 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×11], C22, C22 [×12], C2×C4, C2×C4 [×10], C2×C4 [×4], D4 [×6], Q8 [×2], C23 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×4], C2×D4 [×6], C2×Q8 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, C22.56C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4 [×2], 2- 1+4, C22.56C24

Character table of C22.56C24

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K
 size 1111444444444444444
ρ11111111111111111111    trivial
ρ211111-1-1-1-111-1-111-11-11    linear of order 2
ρ31111-111-11-1-11-111-1-1-11    linear of order 2
ρ41111-1-1-11-1-1-1-11111-111    linear of order 2
ρ51111-11-1-11-11-1-11-1111-1    linear of order 2
ρ61111-1-111-1-11111-1-11-1-1    linear of order 2
ρ7111111-1111-1-111-1-1-1-1-1    linear of order 2
ρ811111-11-1-11-11-11-11-11-1    linear of order 2
ρ91111111-1-1-1-1-11-1-111-11    linear of order 2
ρ1011111-1-111-1-11-1-1-1-1111    linear of order 2
ρ111111-1111-111-1-1-1-1-1-111    linear of order 2
ρ121111-1-1-1-111111-1-11-1-11    linear of order 2
ρ131111-11-11-11-11-1-1111-1-1    linear of order 2
ρ141111-1-11-111-1-11-11-111-1    linear of order 2
ρ15111111-1-1-1-1111-11-1-11-1    linear of order 2
ρ1611111-1111-11-1-1-111-1-1-1    linear of order 2
ρ174-44-4000000000000000    orthogonal lifted from 2+ 1+4
ρ1844-4-4000000000000000    orthogonal lifted from 2+ 1+4
ρ194-4-44000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.56C24
On 32 points
Generators in S32
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 12)(2 9)(3 10)(4 11)(5 29)(6 30)(7 31)(8 32)(13 19)(14 20)(15 17)(16 18)(21 28)(22 25)(23 26)(24 27)
(1 25)(2 23)(3 27)(4 21)(5 17)(6 16)(7 19)(8 14)(9 26)(10 24)(11 28)(12 22)(13 31)(15 29)(18 30)(20 32)
(1 20)(2 13)(3 18)(4 15)(5 26)(6 22)(7 28)(8 24)(9 19)(10 16)(11 17)(12 14)(21 31)(23 29)(25 30)(27 32)
(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)
(5 29)(6 30)(7 31)(8 32)(13 15)(14 16)(17 19)(18 20)(21 26)(22 27)(23 28)(24 25)

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

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

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

C22.56C24 is a maximal subgroup of
C42.2C23  C42.7C23  C42.8C23  C22.120C25  C22.122C25  C22.123C25  C22.124C25  C22.134C25  C22.142C25  C22.148C25  C22.154C25  C22.155C25  C22.156C25
 C2p.2+ 1+4: C22.118C25  C22.125C25  C22.130C25  C22.131C25  C22.136C25  C22.137C25  C22.143C25  C22.147C25 ...
C22.56C24 is a maximal quotient of
C24.225C23  C23.259C24  C24.227C23  C23.261C24  C23.264C24  C23.571C24  C23.572C24  C23.576C24  C23.581C24  C24.389C23  C24.393C23  C24.394C23  C23.592C24  C24.403C23  C23.600C24  C24.407C23  C23.606C24  C23.608C24  C24.412C23  C23.630C24  C23.632C24  C24.427C23  C23.640C24  C23.641C24  C23.643C24  C24.430C23  C23.647C24  C24.435C23  C23.651C24  C23.654C24  C23.655C24  C24.440C23  C23.668C24  C23.672C24  C23.679C24  C23.681C24  C23.683C24  C23.686C24  C23.688C24  C23.698C24  C23.700C24  C23.702C24  C23.706C24  C23.708C24  C24.459C23  C23.714C24  C23.716C24  C4234D4  C23.724C24  C23.726C24  C23.727C24  C23.729C24  C23.730C24  C23.731C24  C23.732C24  C23.736C24  C23.737C24  C23.739C24  C23.741C24  C42.40Q8
 C42.D2p: C42.199D4  C42.145D6  C42.158D6  C42.145D10  C42.158D10  C42.145D14  C42.158D14 ...
 C4⋊C4.D2p: C24.401C23  C24.408C23  C23.611C24  C6.492+ 1+4  C6.592+ 1+4  C6.692+ 1+4  C10.742- 1+4  C10.262- 1+4 ...

Matrix representation of C22.56C24 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40300000
00410000
00100000
01100000
00000010
00000001
00001000
00000100
,
40000000
41000000
00400000
10010000
00004300
00000100
00000012
00000004
,
13000000
04000000
41040000
41400000
00003000
00000300
00000020
00000002
,
10000000
01000000
40400000
40040000
00001000
00004400
00000010
00000044

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,4,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,4,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4],[1,0,4,4,0,0,0,0,3,4,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2],[1,0,4,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4] >;

C22.56C24 in GAP, Magma, Sage, TeX

C_2^2._{56}C_2^4
% in TeX

G:=Group("C2^2.56C2^4");
// GroupNames label

G:=SmallGroup(64,243);
// by ID

G=gap.SmallGroup(64,243);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650,476,86,1347,297]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^2=a,a*b=b*a,d*c*d=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,e*c*e^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f=a*b*c,e*d*e^-1=a*b*d,e*f=f*e>;
// generators/relations

Export

Character table of C22.56C24 in TeX

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