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G = C22.31C24order 64 = 26

17th central stem extension by C22 of C24

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

Aliases: C22.31C24, C23.12C23, C2.52+ 1+4, C2.42- 1+4, (C2×C4)⋊5D4, C4⋊D47C2, C4.63(C2×D4), C22⋊Q86C2, C22.2(C2×D4), C4⋊C4.27C22, C2.16(C22×D4), (C2×C4).132C23, (C2×D4).65C22, C22⋊C4.2C22, (C2×Q8).59C22, (C22×C4).62C22, (C2×C4⋊C4)⋊18C2, (C2×C4○D4)⋊5C2, SmallGroup(64,218)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.31C24
C1C2C22C2×C4C22×C4C2×C4○D4 — C22.31C24
C1C22 — C22.31C24
C1C22 — C22.31C24
C1C22 — C22.31C24

Generators and relations for C22.31C24
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f2=a, ab=ba, dcd=fcf-1=ac=ca, ede=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >

Subgroups: 233 in 147 conjugacy classes, 81 normal (9 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C2×C4 [×14], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], C22.31C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24

Character table of C22.31C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111224444222244444444
ρ11111111111111111111111    trivial
ρ21111-1-111-111-1-111-1-1-1-111-1    linear of order 2
ρ3111111-1-1-1-1111111-1-111-1-1    linear of order 2
ρ41111-1-1-1-11-11-1-111-111-11-11    linear of order 2
ρ5111111-11-11-1-1-1-11-1-111-1-11    linear of order 2
ρ61111-1-1-1111-111-1111-1-1-1-1-1    linear of order 2
ρ71111111-11-1-1-1-1-11-11-11-11-1    linear of order 2
ρ81111-1-11-1-1-1-111-111-11-1-111    linear of order 2
ρ91111-1-1-1-1-111-1-11-11111-11-1    linear of order 2
ρ10111111-1-1111111-1-1-1-1-1-111    linear of order 2
ρ111111-1-1111-11-1-11-11-1-11-1-11    linear of order 2
ρ1211111111-1-11111-1-111-1-1-1-1    linear of order 2
ρ131111-1-11-111-111-1-1-1-1111-1-1    linear of order 2
ρ141111111-1-11-1-1-1-1-111-1-11-11    linear of order 2
ρ151111-1-1-11-1-1-111-1-1-11-11111    linear of order 2
ρ16111111-111-1-1-1-1-1-11-11-111-1    linear of order 2
ρ172-22-22-20000-22-2200000000    orthogonal lifted from D4
ρ182-22-2-220000-2-22200000000    orthogonal lifted from D4
ρ192-22-2-22000022-2-200000000    orthogonal lifted from D4
ρ202-22-22-200002-22-200000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

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

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

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

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

C22.31C24 is a maximal subgroup of
(C2×C4)⋊D8  Q8⋊D4⋊C2  (C2×C4).5D8  C4⋊C4.19D4  M4(2)⋊D4  M4(2)⋊4D4  C4211D4  C4212D4  (C2×D4)⋊21D4  (C2×Q8)⋊17D4  (C2×C8)⋊13D4  (C2×C8)⋊14D4  M4(2)⋊16D4  M4(2)⋊17D4  (C2×D4).303D4  (C2×D4).304D4  C22.38C25  C22.50C25  C22.77C25  C22.78C25  C22.81C25  C22.82C25  C4⋊2+ 1+4  C4⋊2- 1+4  C22.101C25  C22.123C25  C22.148C25  C22.149C25
 C2p.2+ 1+4: C4.2+ 1+4  C4.162+ 1+4  C4.182+ 1+4  C4.192+ 1+4  C22.49C25  C22.100C25  C22.125C25  C22.128C25 ...
C22.31C24 is a maximal quotient of
C24.542C23  C24.545C23  C42.159D4  C4213D4  C24.198C23  C24.244C23  C23.316C24  C24.252C23  C24.262C23  C23.385C24  C23.398C24  C4218D4  C42.166D4  C4219D4  C42.172D4  C42.180D4  C42.181D4  C42.182D4  C4223D4  C4225D4  C24.360C23  C24.361C23  C24.587C23  C4228D4  C23.524C24  C23.525C24  C42.188D4  C23.530C24  C4230D4  C42.192D4  C24.592C23  C23.556C24  C23.559C24  C4232D4  C42.198D4  C23.571C24  C23.572C24  C24.384C23  C23.576C24  C23.581C24  C24.389C23  C23.583C24  C23.591C24  C23.593C24  C23.595C24  C23.602C24  C24.408C23  C23.605C24  C23.607C24  C23.612C24  C23.615C24  C23.622C24  C23.625C24  C23.626C24  C23.627C24  C24.459C23  C23.714C24  C4234D4  C42.199D4  C42.201D4  C4235D4
 C2p.2- 1+4: C42.293D4  C42.294D4  C42.295D4  C42.296D4  C42.297D4  C42.298D4  C42.299D4  C42.300D4 ...

Matrix representation of C22.31C24 in GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
-100000
010000
000100
001000
000001
000010
,
-100000
0-10000
00000-1
000010
000100
00-1000
,
010000
100000
000010
000001
001000
000100
,
-100000
0-10000
000100
00-1000
000001
0000-10

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,-1,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0] >;

C22.31C24 in GAP, Magma, Sage, TeX

C_2^2._{31}C_2^4
% in TeX

G:=Group("C2^2.31C2^4");
// GroupNames label

G:=SmallGroup(64,218);
// by ID

G=gap.SmallGroup(64,218);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650,188,579,69]);
// Polycyclic

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

Export

Character table of C22.31C24 in TeX

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