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G = C22.29C24  order 64 = 26

15th central stem extension by C22 of C24

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

Aliases: C42⋊6C22, C22.29C24, C24.16C22, C23.11C23, C2.42+ 1+4, (C2×C4)⋊4D4, C4⋊D4⋊6C2, C4⋊1D4⋊5C2, C4.15(C2×D4), C22≀C2⋊3C2, C4⋊C4⋊14C22, C4.4D4⋊6C2, (C22×D4)⋊7C2, (C2×D4)⋊3C22, C22⋊C4⋊4C22, (C2×C4).17C23, (C22×C4)⋊8C22, (C2×Q8)⋊11C22, C22.21(C2×D4), C2.14(C22×D4), C42⋊C2⋊10C2, (C2×C4○D4)⋊4C2, SmallGroup(64,216)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C22.29C24
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C22.29C24
Lower central
C1 — C22 — C22.29C24
Upper central
C1 — C22 — C22.29C24
Jennings
C1 — C22 — C22.29C24

Generators and relations for C22.29C24
 G = < a,b,c,d,e,f | a2=b2=c2=e2=f2=1, d2=a, ab=ba, dcd-1=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >

Subgroups: 305 in 167 conjugacy classes, 81 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C22.29C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24

Character table of C22.29C24

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J
 size 1111224444442222444444
ρ11111111111111111111111    trivial
ρ21111-1-1111-11-1-1-1111-1-1-11-1    linear of order 2
ρ31111-1-11-1-111-111-1-1-11-111-1    linear of order 2
ρ41111111-1-1-111-1-1-1-1-1-11-111    linear of order 2
ρ5111111-1-1-1-1-1-1111111-1-111    linear of order 2
ρ61111-1-1-1-1-11-11-1-1111-1111-1    linear of order 2
ρ71111-1-1-111-1-1111-1-1-111-11-1    linear of order 2
ρ8111111-1111-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ91111-1-11-11-1-1111-1-11-1-11-11    linear of order 2
ρ101111111-111-1-1-1-1-1-1111-1-1-1    linear of order 2
ρ1111111111-1-1-1-11111-1-111-1-1    linear of order 2
ρ121111-1-111-11-11-1-111-11-1-1-11    linear of order 2
ρ131111-1-1-11-111-111-1-11-11-1-11    linear of order 2
ρ14111111-11-1-111-1-1-1-111-11-1-1    linear of order 2
ρ15111111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ161111-1-1-1-11-11-1-1-111-1111-11    linear of order 2
ρ172-22-22-2000000-222-2000000    orthogonal lifted from D4
ρ182-22-22-20000002-2-22000000    orthogonal lifted from D4
ρ192-22-2-220000002-22-2000000    orthogonal lifted from D4
ρ202-22-2-22000000-22-22000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C22.29C24
►On 16 points - transitive group 16T73
Generators in S16
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)

(1 12)(2 9)(3 10)(4 11)(5 14)(6 15)(7 16)(8 13)

(1 5)(2 8)(3 7)(4 6)(9 13)(10 16)(11 15)(12 14)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 9)(2 12)(3 11)(4 10)(5 8)(6 7)(13 14)(15 16)

(1 12)(2 9)(3 10)(4 11)(5 16)(6 13)(7 14)(8 15)
 
G:=sub<Sym(16)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,12)(2,9)(3,10)(4,11)(5,14)(6,15)(7,16)(8,13), (1,5)(2,8)(3,7)(4,6)(9,13)(10,16)(11,15)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9)(2,12)(3,11)(4,10)(5,8)(6,7)(13,14)(15,16), (1,12)(2,9)(3,10)(4,11)(5,16)(6,13)(7,14)(8,15)>;
 
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,12)(2,9)(3,10)(4,11)(5,14)(6,15)(7,16)(8,13), (1,5)(2,8)(3,7)(4,6)(9,13)(10,16)(11,15)(12,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9)(2,12)(3,11)(4,10)(5,8)(6,7)(13,14)(15,16), (1,12)(2,9)(3,10)(4,11)(5,16)(6,13)(7,14)(8,15) );
 
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,12),(2,9),(3,10),(4,11),(5,14),(6,15),(7,16),(8,13)], [(1,5),(2,8),(3,7),(4,6),(9,13),(10,16),(11,15),(12,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9),(2,12),(3,11),(4,10),(5,8),(6,7),(13,14),(15,16)], [(1,12),(2,9),(3,10),(4,11),(5,16),(6,13),(7,14),(8,15)]])
 
G:=TransitiveGroup(16,73);
 

C22.29C24 is a maximal subgroup of
 (C2×C8)⋊4D4  M4(2)⋊21D4  M4(2)⋊5D4  C2≀C4⋊C2  C4.4D4⋊C4  C4⋊1D4.C4  C4○D4⋊D4  (C2×Q8)⋊16D4  C42.12C23  C23.9C24  C42.14C23  C42.16C23  C42.18C23  (C2×C8)⋊11D4  (C2×D4).301D4  C42.366C23  C22.38C25  C22.73C25  C22.76C25  C22.77C25  C22.83C25  C22.87C25  C22.89C25  C22.99C25  C42⋊C23  C22.122C25
 C24.D2p: C42⋊D4  C42⋊9D4  C42⋊10D4  M4(2)⋊6D4  C24.39D4  M4(2)⋊C23  C24.45D6  C24.52D6 ...
 C2p.2+ 1+4: C4.2+ 1+4  C4.142+ 1+4  C4.152+ 1+4  C42.406C23  C42.408C23  C42.410C23  C22.49C25  C22.95C25 ...
 C8⋊pD4⋊C2: (C2×C8)⋊12D4  M4(2)⋊16D4  M4(2)⋊10D4  M4(2)⋊11D4 ...
C22.29C24 is a maximal quotient of
 C23.191C24  C24.542C23  C23.199C24  C23.203C24  C42.160D4  C23.304C24  C23.308C24  C23.311C24  C24.249C23  C24.263C23  C23.333C24  C23.335C24  C23.382C24  C24.300C23  C23.400C24  C23.402C24  C23.404C24  C42.171D4  C42.173D4  C42.175D4  C42.176D4  C42.178D4  C24⋊9D4  C24.360C23  C24⋊10D4  C24.589C23  C23.524C24  C24⋊5Q8  C42.187D4  C42.189D4  C42.190D4  C23.535C24  C42.193D4  C42.194D4  C42⋊10Q8  C24.377C23  C23.569C24  C23.570C24  C23.573C24  C23.576C24  C23.578C24  C23.585C24  C23.592C24  C24.403C23  C23.603C24  C23.606C24  C24.412C23  C23.611C24  C24.413C23  C23.618C24  C23.620C24  C24.418C23  C24.459C23  C23.715C24  C23.716C24  C42.199D4  C42.200D4
 C42⋊D2p: C42⋊13D4  C42⋊14D4  C42⋊17D4  C42⋊20D4  C42⋊21D4  C42⋊22D4  C42⋊24D4  C42⋊26D4 ...
 C2p.2+ 1+4: C42.263D4  C42.264D4  C42.265D4  C42.266D4  C42.267D4  C42.268D4  C42.269D4  C42.270D4 ...

Matrix representation of C22.29C24 ►in GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
0-10000
-100000
00-10-20
000011
000010
0001-10
,
-100000
0-10000
001200
00-1-100
00-1-101
0001-10
,
100000
0-10000
00-1-200
000100
000101
000-110
,
100000
010000
001000
000100
00-10-10
00100-1

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],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-2,1,1,-1,0,0,0,1,0,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,-1,0,0,0,2,-1,-1,1,0,0,0,0,0,-1,0,0,0,0,1,0],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1,1,-1,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,-1,1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;
 

C22.29C24 in GAP, Magma, Sage, TeX 

C_2^2._{29}C_2^4
 
% in TeX
 
G:=Group("C2^2.29C2^4");
 
// GroupNames label
 
G:=SmallGroup(64,216);
 
// by ID
 
G=gap.SmallGroup(64,216);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,188,579]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=f^2=1,d^2=a,a*b=b*a,d*c*d^-1=f*c*f=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 C22.29C24 in TeX

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