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G = C22.26C24order 64 = 26

12nd central stem extension by C22 of C24

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

Aliases: C23.9C23, C42.90C22, C22.26C24, (C2×C4)⋊8D4, C4(C4⋊Q8), (C4×D4)⋊9C2, C4⋊Q819C2, C41(C4○D4), C4(C41D4), C4.82(C2×D4), C43(C4⋊D4), C41D410C2, C4⋊D420C2, (C2×C42)⋊12C2, C22.1(C2×D4), C42(C4.4D4), C4.4D418C2, C4⋊C4.72C22, C2.12(C22×D4), (C2×C4).160C23, (C2×D4).63C22, (C2×Q8).56C22, C22⋊C4.14C22, (C22×C4).125C22, (C2×C4○D4)⋊3C2, (C2×C4)(C4⋊D4), (C2×C4)(C41D4), C2.13(C2×C4○D4), SmallGroup(64,213)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.26C24
C1C2C22C2×C4C42C2×C42 — C22.26C24
C1C22 — C22.26C24
C1C2×C4 — C22.26C24
C1C22 — C22.26C24

Generators and relations for C22.26C24
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e2=b, f2=a, ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, ece-1=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, df=fd, ef=fe >

Subgroups: 233 in 155 conjugacy classes, 85 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], C2×C4 [×2], C2×C4 [×12], C2×C4 [×12], D4 [×20], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C22.26C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.26C24

Character table of C22.26C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R
 size 1111224444111122222222224444
ρ11111111111111111111111111111    trivial
ρ21111-1-11-1-1111111-111-1-1-1-11-1-111-1    linear of order 2
ρ3111111-11-11-1-1-1-1-1-11-11-1-1-111-11-11    linear of order 2
ρ41111-1-1-1-111-1-1-1-1-111-1-11111-111-1-1    linear of order 2
ρ51111-1-1-111-111111-111-1-1-1-11-11-1-11    linear of order 2
ρ61111111-11-1-1-1-1-1-1-11-11-1-1-1111-11-1    linear of order 2
ρ7111111-1-1-1-111111111111111-1-1-1-1    linear of order 2
ρ81111-1-111-1-1-1-1-1-1-111-1-11111-1-1-111    linear of order 2
ρ91111111-1-11-1-1-1-11-1-11-111-1-1-11-1-11    linear of order 2
ρ101111-1-1-11-111111-1-1-1-1111-1-111-11-1    linear of order 2
ρ111111-1-11111-1-1-1-111-111-1-11-11-1-1-1-1    linear of order 2
ρ12111111-1-1111111-11-1-1-1-1-11-1-1-1-111    linear of order 2
ρ13111111-111-1-1-1-1-11-1-11-111-1-1-1-111-1    linear of order 2
ρ141111-1-11-11-11111-1-1-1-1111-1-11-11-11    linear of order 2
ρ151111-1-1-1-1-1-1-1-1-1-111-111-1-11-111111    linear of order 2
ρ1611111111-1-11111-11-1-1-1-1-11-1-111-1-1    linear of order 2
ρ172-22-2-2200002-22-20-2000002000000    orthogonal lifted from D4
ρ182-22-2-220000-22-220200000-2000000    orthogonal lifted from D4
ρ192-22-22-200002-22-20200000-2000000    orthogonal lifted from D4
ρ202-22-22-20000-22-220-2000002000000    orthogonal lifted from D4
ρ2122-2-2000000-2i2i2i-2i00002-2i2i00-20000    complex lifted from C4○D4
ρ222-2-220000002i2i-2i-2i-202i20000-2i00000    complex lifted from C4○D4
ρ2322-2-20000002i-2i-2i2i0000-2-2i2i0020000    complex lifted from C4○D4
ρ242-2-22000000-2i-2i2i2i-20-2i200002i00000    complex lifted from C4○D4
ρ2522-2-2000000-2i2i2i-2i0000-22i-2i0020000    complex lifted from C4○D4
ρ262-2-220000002i2i-2i-2i20-2i-200002i00000    complex lifted from C4○D4
ρ2722-2-20000002i-2i-2i2i000022i-2i00-20000    complex lifted from C4○D4
ρ282-2-22000000-2i-2i2i2i202i-20000-2i00000    complex lifted from C4○D4

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

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

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

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

C22.26C24 is a maximal subgroup of
C43⋊C2  C428D4  M4(2)⋊23D4  C42.694C23  C42.300C23  C42.301C23  C42.313C23  C22.33C25  C22.38C25  C22.44C25  C22.49C25  D4×C4○D4  C22.69C25  C22.70C25  C22.72C25  C22.83C25  C4⋊2+ 1+4  C4⋊2- 1+4  C22.87C25  C22.88C25  C22.89C25  C22.95C25  C22.96C25  C22.97C25  C22.100C25  C22.101C25  C22.103C25  C22.106C25  C22.108C25  C22.111C25  C22.118C25  C22.135C25  C22.136C25  C22.137C25  C22.138C25  C22.139C25  C22.140C25  C22.143C25  C22.147C25  C22.148C25  C22.149C25  C22.151C25
 C42.D2p: C42.45D4  C42.400D4  C42.315D4  C42.53D4  C42.403D4  C42.55D4  C42.59D4  C42.61D4 ...
 (C2×C4p)⋊D4: C42.681C23  C42.266C23  (C2×C12)⋊17D4  (C2×C20)⋊17D4  (C2×C28)⋊17D4 ...
 (C2×D4).D2p: M4(2)⋊19D4  C4⋊Q829C4  (C2×D4).135D4  C12⋊(C4○D4)  C20⋊(C4○D4)  C28⋊(C4○D4) ...
C22.26C24 is a maximal quotient of
C23.167C24  C439C2  C23.179C24  C4×C4⋊D4  C4×C41D4  C23.288C24  C4215D4  C23.295C24  C23.328C24  C24.262C23  C24.263C23  C23.352C24  C24.282C23  C23.364C24  C23.391C24  C23.396C24  C23.398C24  C24.308C23  C23.400C24  C23.406C24  C23.412C24  C23.419C24  C24.311C23  C23.443C24  C24.326C23  C24.327C23  C23.455C24  C23.456C24  C23.457C24  C23.458C24  C24.331C23  C24.332C23  C24.583C23  C4223D4  C23.502C24  C4224D4  C4226D4  C24.377C23  C4232D4  C24.378C23  C4246D4  C24.598C23  C4247D4  C4312C2  C4313C2  C42.385C23  C42.386C23  C42.387C23  C42.388C23  C42.389C23
 C42.D2p: C4×C4.4D4  C4×C4⋊Q8  C42.162D4  C42.168D4  C42.174D4  C42.175D4  C42.184D4  C42.185D4 ...
 C4p⋊Q8⋊C2: C42.390C23  C42.391C23  C12⋊(C4○D4)  C20⋊(C4○D4)  C28⋊(C4○D4) ...
 (C2×D4).D2p: C23.322C24  C24.264C23  C23.345C24  C4221D4  (C2×C12)⋊17D4  (C2×C20)⋊17D4  (C2×C28)⋊17D4 ...

Matrix representation of C22.26C24 in GL4(𝔽5) generated by

4000
0400
0040
0004
,
1000
0100
0040
0004
,
3400
3200
0001
0010
,
1300
0400
0002
0030
,
4000
0400
0001
0040
,
3000
0300
0030
0003
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,3,0,0,4,2,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,3,4,0,0,0,0,0,3,0,0,2,0],[4,0,0,0,0,4,0,0,0,0,0,4,0,0,1,0],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3] >;

C22.26C24 in GAP, Magma, Sage, TeX

C_2^2._{26}C_2^4
% in TeX

G:=Group("C2^2.26C2^4");
// GroupNames label

G:=SmallGroup(64,213);
// by ID

G=gap.SmallGroup(64,213);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650,158,69]);
// Polycyclic

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

Export

Character table of C22.26C24 in TeX

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