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G = C22.57C24order 64 = 26

43rd central stem extension by C22 of C24

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

Aliases: C42.56C22, C23.26C23, C22.57C24, C2.242+ 1+4, C2.162- 1+4, C4⋊Q818C2, C22⋊Q820C2, C422C29C2, C4⋊C4.40C22, (C2×C4).39C23, C42.C212C2, C4.4D4.9C2, (C2×D4).40C22, C22⋊C4.9C22, (C2×Q8).36C22, (C22×C4).77C22, C22.D4.3C2, SmallGroup(64,244)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.57C24
C1C2C22C23C22×C4C22.D4 — C22.57C24
C1C22 — C22.57C24
C1C22 — C22.57C24
C1C22 — C22.57C24

Generators and relations for C22.57C24
 G = < a,b,c,d,e,f | a2=b2=f2=1, c2=d2=e2=a, ab=ba, dcd-1=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: 141 in 98 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C2×C4, C2×C4 [×12], C2×C4 [×2], D4, Q8 [×3], C23 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C422C2 [×4], C4⋊Q8 [×2], C22.57C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, 2+ 1+4, 2- 1+4 [×2], C22.57C24

Character table of C22.57C24

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 1111444444444444444
ρ11111111111111111111    trivial
ρ21111111-111-1-111-1-1-1-1-1    linear of order 2
ρ31111-1-11-1-11-11-11-1111-1    linear of order 2
ρ41111-1-111-111-1-111-1-1-11    linear of order 2
ρ511111-1-1-11111-1-1-1-1-111    linear of order 2
ρ611111-1-1111-1-1-1-1111-1-1    linear of order 2
ρ71111-11-11-11-111-11-1-11-1    linear of order 2
ρ81111-11-1-1-111-11-1-111-11    linear of order 2
ρ911111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ101111111-1-1-111-1-111-1-1-1    linear of order 2
ρ111111-1-11-11-11-11-11-111-1    linear of order 2
ρ121111-1-1111-1-111-1-11-1-11    linear of order 2
ρ1311111-1-1-1-1-1-1-11111-111    linear of order 2
ρ1411111-1-11-1-11111-1-11-1-1    linear of order 2
ρ151111-11-111-11-1-11-11-11-1    linear of order 2
ρ161111-11-1-11-1-11-111-11-11    linear of order 2
ρ174-44-4000000000000000    orthogonal lifted from 2+ 1+4
ρ184-4-44000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ1944-4-4000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C22.57C24
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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(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 9 3 11)(2 12 4 10)(5 21 7 23)(6 24 8 22)(13 18 15 20)(14 17 16 19)(25 31 27 29)(26 30 28 32)
(1 21 3 23)(2 14 4 16)(5 31 7 29)(6 12 8 10)(9 20 11 18)(13 25 15 27)(17 32 19 30)(22 26 24 28)
(2 26)(4 28)(5 7)(6 17)(8 19)(9 11)(10 30)(12 32)(14 24)(16 22)(18 20)(29 31)

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

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

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

C22.57C24 is a maximal subgroup of
C42.3C23  C42.6C23  C42.10C23  C22.120C25  C22.122C25  C22.124C25  C22.133C25  C22.134C25  C22.142C25  C22.144C25  C22.146C25  C22.148C25  C22.149C25  C22.152C25  C22.153C25  C22.154C25  C22.155C25  C22.157C25
 C42.D2p: C42.14D4  C42.16D4  C42.140D6  C42.157D6  C42.165D6  C42.180D6  C42.140D10  C42.157D10 ...
 C2p.2+ 1+4: C22.118C25  C22.127C25  C22.130C25  C22.139C25  C22.140C25  C22.141C25  C22.150C25  C6.252- 1+4 ...
C22.57C24 is a maximal quotient of
C24.225C23  C24.227C23  C23.261C24  C23.263C24  C23.264C24  C24.230C23  C23.580C24  C23.583C24  C23.589C24  C23.595C24  C24.405C23  C23.602C24  C23.615C24  C23.617C24  C23.618C24  C23.622C24  C24.418C23  C23.624C24  C23.627C24  C24.420C23  C24.421C23  C23.631C24  C23.634C24  C24.426C23  C23.645C24  C23.658C24  C23.659C24  C23.662C24  C23.664C24  C24.443C23  C23.666C24  C23.669C24  C24.445C23  C23.671C24  C23.673C24  C23.674C24  C23.675C24  C23.676C24  C23.677C24  C23.687C24  C23.689C24  C23.691C24  C23.693C24  C23.694C24  C23.699C24  C23.705C24  C23.709C24  C23.714C24  C24.462C23  C4235D4  C23.727C24  C23.730C24  C23.731C24  C23.732C24  C23.733C24  C23.734C24  C23.735C24  C23.736C24  C23.737C24  C23.738C24  C23.739C24  C23.741C24  C4212Q8  C4213Q8
 C42.D2p: C42.199D4  C42.200D4  C42.201D4  C42.140D6  C42.157D6  C42.165D6  C42.180D6  C42.140D10 ...
 C4⋊C4.D2p: C23.574C24  C24.385C23  C23.590C24  C23.607C24  C23.613C24  C23.616C24  C23.620C24  C23.621C24 ...

Matrix representation of C22.57C24 in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
20000000
03000000
00300000
00020000
00000010
00000001
00004000
00000400
,
00100000
00010000
40000000
04000000
00000100
00004000
00000004
00000010
,
01000000
40000000
00010000
00400000
00002000
00000300
00000020
00000003
,
10000000
01000000
00400000
00040000
00001000
00000400
00000040
00000001

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

C22.57C24 in GAP, Magma, Sage, TeX

C_2^2._{57}C_2^4
% in TeX

G:=Group("C2^2.57C2^4");
// GroupNames label

G:=SmallGroup(64,244);
// by ID

G=gap.SmallGroup(64,244);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,103,650,476,86,1347,297]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=f^2=1,c^2=d^2=e^2=a,a*b=b*a,d*c*d^-1=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 C22.57C24 in TeX

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