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G = C22.32C24order 64 = 26

18th central stem extension by C22 of C24

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

Aliases: C427C22, C24.17C22, C23.13C23, C22.32C24, C2.62+ 1+4, (C4×D4)⋊10C2, C4⋊D48C2, C22⋊Q87C2, C22≀C24C2, C4⋊C415C22, C4.4D48C2, (C2×D4)⋊4C22, (C2×Q8)⋊3C22, C422C21C2, C22⋊C45C22, (C2×C4).19C23, (C22×C4)⋊9C22, C22.4(C4○D4), C22.D44C2, C2.15(C2×C4○D4), (C2×C22⋊C4)⋊13C2, SmallGroup(64,219)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.32C24
C1C2C22C23C24C2×C22⋊C4 — C22.32C24
C1C22 — C22.32C24
C1C22 — C22.32C24
C1C22 — C22.32C24

Generators and relations for C22.32C24
 G = < a,b,c,d,e,f | a2=b2=c2=e2=f2=1, d2=b, 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: 217 in 125 conjugacy classes, 73 normal (19 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], D4 [×9], Q8, C23, C23 [×4], C23 [×4], C42 [×2], C22⋊C4 [×14], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C24, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2 [×2], C22.32C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4 [×2], C22.32C24

Character table of C22.32C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111224444222244444444
ρ11111111111111111111111    trivial
ρ2111111-1-1-1-1-1-1-1-11-11-11111    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-1111-1-1    linear of order 2
ρ4111111-1-1-111111-11-1-111-1-1    linear of order 2
ρ511111111-11-1-1-1-1-11-1-1-1-111    linear of order 2
ρ6111111-1-11-11111-1-1-11-1-111    linear of order 2
ρ711111111-1-111111-11-1-1-1-1-1    linear of order 2
ρ8111111-1-111-1-1-1-11111-1-1-1-1    linear of order 2
ρ91111-1-1-111111-1-1-1-11-1-11-11    linear of order 2
ρ101111-1-11-1-1-1-1-111-1111-11-11    linear of order 2
ρ111111-1-1-111-1-1-11111-1-1-111-1    linear of order 2
ρ121111-1-11-1-1111-1-11-1-11-111-1    linear of order 2
ρ131111-1-1-11-11-1-1111-1-111-1-11    linear of order 2
ρ141111-1-11-11-111-1-111-1-11-1-11    linear of order 2
ρ151111-1-1-11-1-111-1-1-11111-11-1    linear of order 2
ρ161111-1-11-111-1-111-1-11-11-11-1    linear of order 2
ρ172-22-2-2200002i-2i2i-2i00000000    complex lifted from C4○D4
ρ182-22-22-200002i-2i-2i2i00000000    complex lifted from C4○D4
ρ192-22-2-220000-2i2i-2i2i00000000    complex lifted from C4○D4
ρ202-22-22-20000-2i2i2i-2i00000000    complex lifted from C4○D4
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C22.32C24
On 16 points - transitive group 16T82
Generators in S16
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 5)(2 15)(3 7)(4 13)(6 10)(8 12)(9 14)(11 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 3)(2 12)(4 10)(6 15)(8 13)(9 11)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)

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

G:=Group( (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,5)(2,15)(3,7)(4,13)(6,10)(8,12)(9,14)(11,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,3)(2,12)(4,10)(6,15)(8,13)(9,11), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,11)(10,12) );

G=PermutationGroup([(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,5),(2,15),(3,7),(4,13),(6,10),(8,12),(9,14),(11,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(2,12),(4,10),(6,15),(8,13),(9,11)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,11),(10,12)])

G:=TransitiveGroup(16,82);

C22.32C24 is a maximal subgroup of
C22.44C25  C22.79C25  C22.82C25  C22.102C25  C22.110C25  C42⋊C23  C22.122C25  C22.123C25  C22.124C25  C22.134C25  C22.149C25  C22.153C25  C22.155C25  C22.157C25
 C24.D2p: C24.9D4  C24.12D4  C24.15D4  C24.18D4  C24.41D6  C24.46D6  C24.30D10  C24.35D10 ...
 C2p.2+ 1+4: C22.48C25  C22.49C25  C22.94C25  C22.95C25  C22.103C25  C22.108C25  C22.125C25  C22.126C25 ...
C22.32C24 is a maximal quotient of
C23.194C24  C24.547C23  C23.203C24  C24.198C23  C23.211C24  C23.215C24  C24.203C23  C24.204C23  C24.205C23  C23.318C24  C23.324C24  C23.344C24  C23.350C24  C23.359C24  C23.372C24  C23.380C24  C23.405C24  C23.410C24  C23.418C24  C23.455C24  C23.457C24  C24.331C23  C23.472C24  C24.338C23  C24.340C23  C23.478C24  C23.486C24  C24.345C23  C24.346C23  C23.493C24  C24.347C23  C24.348C23  C2410D4  C245Q8  C23.530C24  C23.535C24  C24.592C23  C23.543C24  C23.546C24  C23.548C24  C24.375C23  C23.550C24  C23.551C24  C24.376C23  C23.553C24  C24.377C23  C24.379C23  C4211Q8  C23.568C24  C23.570C24  C23.578C24  C23.584C24  C23.585C24  C24.393C23  C24.395C23  C23.595C24  C23.597C24  C23.603C24  C23.608C24  C24.412C23  C23.615C24  C23.630C24  C23.633C24  C23.635C24  C23.636C24  C23.641C24  C23.643C24  C23.645C24  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C24.437C23  C23.656C24  C23.659C24  C23.660C24  C24.440C23  C23.664C24  C24.443C23  C23.668C24  C24.445C23  C23.671C24  C23.674C24  C23.675C24  C23.678C24  C23.679C24  C24.448C23  C23.681C24  C24.450C23  C23.686C24  C23.687C24  C23.688C24  C23.697C24  C23.700C24  C23.703C24  C24.456C23  C23.705C24  C23.706C24  C23.708C24  C23.724C24  C23.725C24  C23.726C24  C23.727C24  C23.728C24  C23.729C24  C23.730C24  C23.731C24  C23.732C24  C23.733C24
 C42⋊D2p: C4229D4  C4230D4  C4232D4  C4219D6  C4222D6  C4225D6  C4217D10  C4220D10 ...
 C24.D2p: C24.97D4  C24.41D6  C24.46D6  C24.30D10  C24.35D10  C24.30D14  C24.35D14 ...
 C4⋊C4⋊D2p: C23.356C24  C23.367C24  C24.327C23  C23.576C24  C24.389C23  C24.403C23  C23.602C24  C24.413C23 ...

Matrix representation of C22.32C24 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
010000
100000
000010
000001
001000
000100
,
300000
030000
000100
001000
000004
000040
,
100000
040000
001000
000400
000010
000004
,
100000
010000
001000
000100
000040
000004

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

C22.32C24 in GAP, Magma, Sage, TeX

C_2^2._{32}C_2^4
% in TeX

G:=Group("C2^2.32C2^4");
// GroupNames label

G:=SmallGroup(64,219);
// by ID

G=gap.SmallGroup(64,219);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,188,579]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=e^2=f^2=1,d^2=b,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 C22.32C24 in TeX

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