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G = C23.C8order 64 = 26

The non-split extension by C23 of C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C23.C8, C8.24D4, M5(2)⋊3C2, C4.7M4(2), (C2×C4).C8, (C2×C8).3C4, (C22×C4).4C4, C22.4(C2×C8), C2.7(C22⋊C8), (C2×C8).42C22, C4.29(C22⋊C4), (C2×M4(2)).10C2, (C2×C4).67(C2×C4), SmallGroup(64,30)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.C8
C1C2C4C8C2×C8C2×M4(2) — C23.C8
C1C2C22 — C23.C8
C1C4C2×C8 — C23.C8
C1C2C2C2C2C4C4C2×C8 — C23.C8

Generators and relations for C23.C8
 G = < a,b,c,d | a2=b2=c2=1, d8=c, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

2C2
4C2
2C4
2C22
4C22
2C2×C4
2C8
2C2×C4
2C16
2M4(2)
2M4(2)
2C16

Character table of C23.C8

 class 12A2B2C4A4B4C4D8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 1124112422224444444444
ρ11111111111111111111111    trivial
ρ2111-1111-11111-1-1-1111-11-1-1    linear of order 2
ρ3111-1111-11111-1-11-1-1-11-111    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-1-1-1i-iii-i-i-ii    linear of order 4
ρ6111-1111-1-1-1-1-111-i-iiii-ii-i    linear of order 4
ρ7111-1111-1-1-1-1-111ii-i-i-ii-ii    linear of order 4
ρ811111111-1-1-1-1-1-1-ii-i-iiii-i    linear of order 4
ρ9111-1-1-1-11i-i-ii-iiζ8ζ83ζ8ζ85ζ83ζ87ζ87ζ85    linear of order 8
ρ101111-1-1-1-1i-i-iii-iζ85ζ83ζ8ζ85ζ87ζ87ζ83ζ8    linear of order 8
ρ11111-1-1-1-11-iii-ii-iζ83ζ8ζ83ζ87ζ8ζ85ζ85ζ87    linear of order 8
ρ12111-1-1-1-11-iii-ii-iζ87ζ85ζ87ζ83ζ85ζ8ζ8ζ83    linear of order 8
ρ131111-1-1-1-1-iii-i-iiζ83ζ85ζ87ζ83ζ8ζ8ζ85ζ87    linear of order 8
ρ141111-1-1-1-1i-i-iii-iζ8ζ87ζ85ζ8ζ83ζ83ζ87ζ85    linear of order 8
ρ15111-1-1-1-11i-i-ii-iiζ85ζ87ζ85ζ8ζ87ζ83ζ83ζ8    linear of order 8
ρ161111-1-1-1-1-iii-i-iiζ87ζ8ζ83ζ87ζ85ζ85ζ8ζ83    linear of order 8
ρ1722-2022-202-22-20000000000    orthogonal lifted from D4
ρ1822-2022-20-22-220000000000    orthogonal lifted from D4
ρ1922-20-2-2202i2i-2i-2i0000000000    complex lifted from M4(2)
ρ2022-20-2-220-2i-2i2i2i0000000000    complex lifted from M4(2)
ρ214-400-4i4i0000000000000000    complex faithful
ρ224-4004i-4i0000000000000000    complex faithful

Permutation representations of C23.C8
On 16 points - transitive group 16T104
Generators in S16
(2 10)(3 11)(6 14)(7 15)
(2 10)(4 12)(6 14)(8 16)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,104);

C23.C8 is a maximal subgroup of
C24.C8  C23.1M4(2)  C42.C8  C22⋊C4.C8  C23.D8  C23.2D8  C23.SD16  C23.2SD16  M5(2).19C22  D8⋊D4  D8.D4  M5(2).C22  C23.10SD16
 C4p.M4(2): C8.5M4(2)  C8.19M4(2)  C8.25D12  C24.D4  C8.25D20  C40.D4  C20.10M4(2)  C20.29M4(2) ...
C23.C8 is a maximal quotient of
C23⋊C16  C22.M5(2)  C8.17Q16  C20.10M4(2)
 C8.D4p: C8.31D8  C8.25D12  C8.25D20  M5(2)⋊D7 ...
 (C2×C4p).C8: M5(2)⋊C4  C24.D4  C40.D4  C20.29M4(2)  C56.D4 ...

Matrix representation of C23.C8 in GL4(𝔽5) generated by

4000
0400
0010
0001
,
4000
0100
0040
0001
,
4000
0400
0040
0004
,
0003
1000
0100
0010
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,1,3,0,0,0] >;

C23.C8 in GAP, Magma, Sage, TeX

C_2^3.C_8
% in TeX

G:=Group("C2^3.C8");
// GroupNames label

G:=SmallGroup(64,30);
// by ID

G=gap.SmallGroup(64,30);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,650,489,69,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=1,d^8=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C23.C8 in TeX
Character table of C23.C8 in TeX

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