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G = C22.45C24order 64 = 26

31st central stem extension by C22 of C24

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

Aliases: C4210C22, C24.20C22, C23.18C23, C22.45C24, C2.152+ 1+4, (C4×D4)⋊19C2, C4⋊C417C22, C22⋊Q815C2, (C2×Q8)⋊5C22, C422C24C2, C22≀C2.2C2, C4.4D412C2, C22⋊C48C22, (C2×C4).30C23, (C22×C4)⋊4C22, C42⋊C214C2, (C2×D4).34C22, C22.9(C4○D4), C22.D410C2, C2.24(C2×C4○D4), (C2×C22⋊C4)⋊15C2, SmallGroup(64,232)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.45C24
C1C2C22C23C24C2×C22⋊C4 — C22.45C24
C1C22 — C22.45C24
C1C22 — C22.45C24
C1C22 — C22.45C24

Generators and relations for C22.45C24
 G = < a,b,c,d,e,f | a2=b2=e2=f2=1, c2=a, d2=b, ab=ba, dcd-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >

Subgroups: 197 in 124 conjugacy classes, 75 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C2×C4, C2×C4 [×10], C2×C4 [×7], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C422C2 [×2], C22.45C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×4], C24, C2×C4○D4 [×2], 2+ 1+4, C22.45C24

Character table of C22.45C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O
 size 1111222244222222224444444
ρ11111111111111111111111111    trivial
ρ21111-1-1-1-111-1-11-11-1111-11-1-1-11    linear of order 2
ρ31111-1-1-1-11-1-1-1-11-1111-11-1-1111    linear of order 2
ρ4111111111-111-1-1-1-111-1-1-11-1-11    linear of order 2
ρ51111-1-1-1-11-1111-11-1-1-111-111-1-1    linear of order 2
ρ6111111111-1-1-11111-1-11-1-1-1-11-1    linear of order 2
ρ71111111111-1-1-1-1-1-1-1-1-111-11-1-1    linear of order 2
ρ81111-1-1-1-11111-11-11-1-1-1-111-11-1    linear of order 2
ρ911111-11-1-1111-11-111111-1-1-1-1-1    linear of order 2
ρ101111-11-11-11-1-1-1-1-1-1111-1-1111-1    linear of order 2
ρ111111-11-11-1-1-1-1111111-1111-1-1-1    linear of order 2
ρ1211111-11-1-1-1111-11-111-1-11-111-1    linear of order 2
ρ131111-11-11-1-111-1-1-1-1-1-1111-1-111    linear of order 2
ρ1411111-11-1-1-1-1-1-11-11-1-11-1111-11    linear of order 2
ρ1511111-11-1-11-1-11-11-1-1-1-11-11-111    linear of order 2
ρ161111-11-11-11111111-1-1-1-1-1-11-11    linear of order 2
ρ172-22-2020-20000-2i-2i2i2i000000000    complex lifted from C4○D4
ρ182-22-20-20200002i-2i-2i2i000000000    complex lifted from C4○D4
ρ192-22-20-2020000-2i2i2i-2i000000000    complex lifted from C4○D4
ρ2022-2-220-20002i-2i0000-2i2i0000000    complex lifted from C4○D4
ρ2122-2-2-2020002i-2i00002i-2i0000000    complex lifted from C4○D4
ρ2222-2-2-202000-2i2i0000-2i2i0000000    complex lifted from C4○D4
ρ2322-2-220-2000-2i2i00002i-2i0000000    complex lifted from C4○D4
ρ242-22-2020-200002i2i-2i-2i000000000    complex lifted from C4○D4
ρ254-4-44000000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C22.45C24
On 16 points - transitive group 16T81
Generators in S16
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10 15 7)(2 9 16 6)(3 12 13 5)(4 11 14 8)
(1 3)(2 14)(4 16)(5 7)(6 11)(8 9)(10 12)(13 15)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)

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

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

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

G:=TransitiveGroup(16,81);

C22.45C24 is a maximal subgroup of
C22.64C25  C22.79C25  C22.80C25  C22.83C25  C22.84C25  C22.99C25  C22.102C25  C22.110C25  C42⋊C23  C22.122C25  C22.123C25  C22.124C25  C22.134C25  C22.149C25  C22.153C25  C22.155C25  C22.157C25
 C42⋊D2p: C424D4  C4212D6  C4218D6  C4223D6  C4226D6  C4210D10  C4216D10  C4221D10 ...
 C2p.2+ 1+4: C22.70C25  C22.94C25  C22.103C25  C23.144C24  C22.127C25  C22.129C25  C22.131C25  C22.140C25 ...
C22.45C24 is a maximal quotient of
C23.224C24  C23.225C24  C23.235C24  C23.240C24  C23.241C24  C23.250C24  C24.221C23  C23.255C24  C24.223C23  C23.311C24  C23.318C24  C24.563C23  C244Q8  C24.567C23  C24.278C23  C23.372C24  C23.380C24  C23.382C24  C23.388C24  C24.577C23  C23.398C24  C23.405C24  C23.410C24  C24.309C23  C23.416C24  C23.417C24  C23.418C24  C23.420C24  C24.311C23  C23.422C24  C24.313C23  C23.426C24  C24.315C23  C23.430C24  C23.431C24  C23.432C24  C23.434C24  C24.326C23  C23.457C24  C23.461C24  C24.583C23  C24.584C23  C23.472C24  C23.473C24  C24.338C23  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C23.486C24  C23.488C24  C24.346C23  C24.347C23  C24.348C23  C23.500C24  C23.502C24  C428Q8  C24.355C23  C429Q8  C23.584C24  C23.585C24  C23.593C24  C23.597C24  C23.606C24  C23.617C24  C23.622C24  C24.420C23  C23.630C24  C23.635C24  C23.636C24  C23.637C24  C24.426C23  C24.427C23  C23.640C24  C23.641C24  C24.428C23  C23.643C24  C24.430C23  C23.645C24  C24.432C23  C23.647C24  C24.434C23  C23.649C24  C24.435C23  C23.651C24  C23.652C24  C24.437C23  C23.654C24  C23.655C24  C23.656C24  C24.438C23  C23.658C24  C23.659C24  C23.660C24  C24.440C23  C23.662C24  C23.663C24  C23.664C24  C23.678C24  C23.682C24  C23.686C24  C23.689C24  C23.696C24  C23.697C24  C23.698C24  C23.699C24
 C42⋊D2p: C4222D4  C4223D4  C4224D4  C4225D4  C4226D4  C4212D6  C4218D6  C4223D6 ...
 C24.D2p: C24.95D4  C24.96D4  C24.42D6  C24.43D6  C24.31D10  C24.32D10  C24.31D14  C24.32D14 ...
 C4⋊C4⋊D2p: C24.282C23  C24.283C23  C23.367C24  C24.290C23  C23.377C24  C23.379C24  C24.327C23  C24.332C23 ...

Matrix representation of C22.45C24 in GL4(𝔽5) generated by

4000
0400
0010
0001
,
1000
0100
0040
0004
,
3000
2200
0001
0010
,
4300
0100
0020
0002
,
1000
0100
0010
0004
,
1000
4400
0040
0004
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,2,0,0,0,2,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,3,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[1,4,0,0,0,4,0,0,0,0,4,0,0,0,0,4] >;

C22.45C24 in GAP, Magma, Sage, TeX

C_2^2._{45}C_2^4
% in TeX

G:=Group("C2^2.45C2^4");
// GroupNames label

G:=SmallGroup(64,232);
// by ID

G=gap.SmallGroup(64,232);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,297]);
// Polycyclic

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

Export

Character table of C22.45C24 in TeX

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