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G = C23.41C23order 64 = 26

14th non-split extension by C23 of C23 acting via C23/C22=C2

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

Aliases: C23.41C23, C42.42C22, C22.38C24, C2.82- 1+4, C2.112+ 1+4, (C2×C4)⋊2Q8, C4⋊Q812C2, C4.10(C2×Q8), C42.C27C2, C22⋊Q8.9C2, C22.5(C2×Q8), C2.8(C22×Q8), C4⋊C4.32C22, (C2×C4).25C23, (C2×Q8).31C22, C42⋊C2.13C2, C22⋊C4.19C22, (C22×C4).67C22, (C2×C4⋊C4).20C2, SmallGroup(64,225)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.41C23
C1C2C22C23C22×C4C2×C4⋊C4 — C23.41C23
C1C22 — C23.41C23
C1C22 — C23.41C23
C1C22 — C23.41C23

Generators and relations for C23.41C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=f2=c, e2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, fef-1=be=eb, bf=fb, fdf-1=cd=dc, ce=ec, cf=fc >

Subgroups: 137 in 103 conjugacy classes, 81 normal (9 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C2×C4 [×18], C2×C4 [×2], Q8 [×4], C23, C42 [×4], C22⋊C4 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C23.41C23
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, C22×Q8, 2+ 1+4, 2- 1+4, C23.41C23

Character table of C23.41C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ21111-1-1-1-111-111-1-11-11-111-1    linear of order 2
ρ31111111111-1-111-1-1-1-1-1-111    linear of order 2
ρ41111-1-1-1-1111-11-11-11-11-11-1    linear of order 2
ρ51111-1-111-1-11-1-11-111-1-111-1    linear of order 2
ρ6111111-1-1-1-1-1-1-1-111-1-11111    linear of order 2
ρ71111-1-111-1-1-11-111-1-111-11-1    linear of order 2
ρ8111111-1-1-1-111-1-1-1-111-1-111    linear of order 2
ρ9111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ101111-1-1-1-111-11-11-111-11-1-11    linear of order 2
ρ111111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ121111-1-1-1-1111-1-111-1-11-11-11    linear of order 2
ρ131111-1-111-1-11-11-1-11-111-1-11    linear of order 2
ρ14111111-1-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ151111-1-111-1-1-111-11-11-1-11-11    linear of order 2
ρ16111111-1-1-1-11111-1-1-1-111-1-1    linear of order 2
ρ1722-2-2-22-222-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-22-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ1922-2-2-222-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ2022-2-22-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ214-44-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.41C23
On 32 points
Generators in S32
(2 28)(4 26)(5 20)(7 18)(10 14)(12 16)(22 30)(24 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 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 31 27 23)(2 24 28 32)(3 29 25 21)(4 22 26 30)(5 14 20 10)(6 11 17 15)(7 16 18 12)(8 9 19 13)
(1 9 3 11)(2 12 4 10)(5 32 7 30)(6 31 8 29)(13 25 15 27)(14 28 16 26)(17 23 19 21)(18 22 20 24)

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

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

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

C23.41C23 is a maximal subgroup of
(C2×C4)⋊SD16  (C2×C4)⋊Q16  (C2×C4).SD16  C4⋊C4.20D4  (C2×D4)⋊2Q8  (C2×Q8)⋊2Q8  M4(2)⋊Q8  C423Q8  C42.20C23  C42.21C23  C42.22C23  C42.23C23  C42.366C23  C42.367C23  M4(2)⋊3Q8  M4(2)⋊4Q8  C22.47C25  C22.49C25  C22.81C25  C22.82C25  C22.83C25  C22.84C25  C22.90C25  C22.93C25  C22.124C25  C22.127C25  C22.153C25
 C2p.2- 1+4: C42.423C23  C42.424C23  C42.425C23  C42.426C23  C22.50C25  C22.91C25  C22.92C25  C22.133C25 ...
C23.41C23 is a maximal quotient of
C24.545C23  C23.199C24  C23.211C24  C42.33Q8  C424Q8  C24.567C23  C42.36Q8  C42.37Q8  C428Q8  C42.38Q8  C429Q8  C23.527C24  C23.546C24  C42.39Q8  C23.559C24  C24.379C23  C4211Q8  C23.567C24  C24.421C23  C23.634C24  C24.428C23  C23.655C24  C23.663C24  C23.668C24  C23.674C24  C23.688C24  C24.454C23  C23.691C24  C23.692C24  C23.702C24  C23.705C24  C23.706C24  C23.707C24  C23.741C24  C4212Q8  C4213Q8  C42.40Q8
 C42.D2p: C42.187D4  C42.188D4  C4210Q8  C42.90D6  C42.148D6  C42.174D6  C42.90D10  C42.148D10 ...
 C4⋊C4.D2p: C24.267C23  C24.568C23  C24.268C23  C24.569C23  C426Q8  C427Q8  C42.35Q8  C24.385C23 ...

Matrix representation of C23.41C23 in GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
300000
020000
000010
000001
004000
000400
,
400000
040000
000200
002000
000003
000030
,
010000
400000
000100
004000
000004
000010

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;

C23.41C23 in GAP, Magma, Sage, TeX

C_2^3._{41}C_2^3
% in TeX

G:=Group("C2^3.41C2^3");
// GroupNames label

G:=SmallGroup(64,225);
// by ID

G=gap.SmallGroup(64,225);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,650,188,158,579]);
// Polycyclic

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

Export

Character table of C23.41C23 in TeX

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