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G = C23.38C23order 64 = 26

11st non-split extension by C23 of C23 acting via C23/C22=C2

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

Aliases: C23.38C23, C42.37C22, C22.30C24, C2.32- 1+4, C4⋊Q89C2, (C2×C4).49D4, C4.62(C2×D4), C22⋊Q85C2, C4.4D47C2, (C22×Q8)⋊5C2, C4⋊C4.26C22, (C2×C4).18C23, C22.22(C2×D4), C2.15(C22×D4), C42⋊C211C2, (C2×D4).64C22, C22.D43C2, C22⋊C4.1C22, (C2×Q8).58C22, (C22×C4).61C22, (C2×C4○D4).10C2, SmallGroup(64,217)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.38C23
C1C2C22C23C22×C4C22×Q8 — C23.38C23
C1C22 — C23.38C23
C1C22 — C23.38C23
C1C22 — C23.38C23

Generators and relations for C23.38C23
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=f2=b, dad=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: 193 in 135 conjugacy classes, 81 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×8], D4 [×6], Q8 [×10], C23, C23 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×4], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C23.38C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2- 1+4 [×2], C23.38C23

Character table of C23.38C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 1111224422224444444444
ρ11111111111111111111111    trivial
ρ21111-1-11-111-1-11-1-111-1-111-1    linear of order 2
ρ31111-1-1-1-1-1-111111-1-11-111-1    linear of order 2
ρ4111111-11-1-1-1-11-1-1-1-1-11111    linear of order 2
ρ5111111-1-11111-1-111-1-1-1-111    linear of order 2
ρ61111-1-1-1111-1-1-11-11-111-11-1    linear of order 2
ρ71111-1-111-1-111-1-11-11-11-11-1    linear of order 2
ρ81111111-1-1-1-1-1-11-1-111-1-111    linear of order 2
ρ91111111-1-1-1-1-11111-1-11-1-1-1    linear of order 2
ρ101111-1-111-1-1111-1-11-11-1-1-11    linear of order 2
ρ111111-1-1-1111-1-1111-11-1-1-1-11    linear of order 2
ρ12111111-1-111111-1-1-1111-1-1-1    linear of order 2
ρ13111111-11-1-1-1-1-1-11111-11-1-1    linear of order 2
ρ141111-1-1-1-1-1-111-11-111-111-11    linear of order 2
ρ151111-1-11-111-1-1-1-11-1-1111-11    linear of order 2
ρ16111111111111-11-1-1-1-1-11-1-1    linear of order 2
ρ172-22-2-22002-2-220000000000    orthogonal lifted from D4
ρ182-22-2-2200-222-20000000000    orthogonal lifted from D4
ρ192-22-22-200-22-220000000000    orthogonal lifted from D4
ρ202-22-22-2002-22-20000000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ224-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.38C23
On 32 points
Generators in S32
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(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)
(2 4)(5 18)(6 17)(7 20)(8 19)(9 29)(10 32)(11 31)(12 30)(13 15)(21 23)(26 28)
(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)

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

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

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

C23.38C23 is a maximal subgroup of
C4.(C4×D4)  M4(2).50D4  M4(2).D4  M4(2).7D4  C23.(C2×D4)  C4⋊Q8⋊C4  C4⋊Q8.C4  (C2×D4).137D4  D4.(C2×D4)  Q8.(C2×D4)  M4(2).C23  C42.13C23  C23.10C24  C42.15C23  C42.17C23  C42.19C23  (C2×C8)⋊11D4  C8.D4⋊C2  M4(2)⋊17D4  (C2×D4).302D4  C42.367C23  M4(2)⋊10D4  M4(2).20D4  C4.152+ 1+4  C4.162+ 1+4  C4.172+ 1+4  C42.407C23  C42.409C23  C42.411C23  C22.38C25  C22.74C25  C22.75C25  C22.78C25  C22.84C25  C22.88C25  C22.89C25  C22.99C25  C22.124C25  C22.127C25  C22.129C25  C22.130C25  C22.134C25  C22.140C25  C22.150C25  C22.151C25
 C42.D2p: C42.7D4  C42.129D4  C42.130D4  C42.92D6  C42.141D6  C42.171D6  C42.92D10  C42.141D10 ...
 C2p.2- 1+4: C22.50C25  C22.96C25  C22.98C25  C22.133C25  C22.136C25  C22.139C25  C22.141C25  C22.143C25 ...
C23.38C23 is a maximal quotient of
C23.192C24  C24.542C23  C23.199C24  C24.195C23  C24.243C23  C23.309C24  C23.313C24  C23.315C24  C24.264C23  C23.334C24  C24.565C23  C24.576C23  C24.299C23  C24.308C23  C23.401C24  C24.579C23  C23.514C24  C24.361C23  C24.587C23  C4228D4  C24.589C23  C23.525C24  C23.527C24  C24.374C23  C24.378C23  C23.572C24  C23.574C24  C24.385C23  C23.580C24  C23.581C24  C24.394C23  C23.589C24  C23.590C24  C24.401C23  C24.405C23  C23.600C24  C23.613C24  C23.616C24  C23.619C24  C23.621C24  C23.714C24  C23.716C24  C24.462C23
 C42.D2p: C42.159D4  C42.160D4  C42.161D4  C42.165D4  C42.167D4  C42.168D4  C42.169D4  C42.170D4 ...
 C2p.2- 1+4: C42.423C23  C42.424C23  C42.425C23  C42.426C23  C6.162- 1+4  C6.792- 1+4  C6.442- 1+4  C6.1052- 1+4 ...

Matrix representation of C23.38C23 in GL6(𝔽5)

100000
010000
000100
001000
001144
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
040000
001000
000400
000010
002034
,
100000
010000
000300
003000
002233
001102
,
010000
100000
000010
001144
004000
002204

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

C23.38C23 in GAP, Magma, Sage, TeX

C_2^3._{38}C_2^3
% in TeX

G:=Group("C2^3.38C2^3");
// GroupNames label

G:=SmallGroup(64,217);
// by ID

G=gap.SmallGroup(64,217);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,188,86,579]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^2=f^2=b,d*a*d=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 C23.38C23 in TeX

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