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G = C42.C8order 128 = 27

1st non-split extension by C42 of C8 acting via C8/C2=C4

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C42.1C8, C23.16M4(2), (C2×C8).17D4, (C2×C42).10C4, C23.C8.2C2, C4.41(C23⋊C4), (C2×M4(2)).8C4, C4.13(C4.10D4), C4⋊M4(2).11C2, C22.20(C22⋊C8), (C2×M4(2)).145C22, C2.8(C22.M4(2)), (C2×C4).36(C2×C8), (C22×C4).62(C2×C4), (C2×C4).344(C22⋊C4), SmallGroup(128,59)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.C8
C1C2C4C2×C4C2×C8C2×M4(2)C4⋊M4(2) — C42.C8
C1C2C22C2×C4 — C42.C8
C1C4C2×C4C2×M4(2) — C42.C8
C1C2C2C2C2C4C2×C4C2×M4(2) — C42.C8

Generators and relations for C42.C8
 G = < a,b,c | a4=b4=1, c8=b2, ab=ba, cac-1=a-1b-1, cbc-1=a2b >

2C2
4C2
2C22
2C4
2C4
2C4
4C22
4C4
2C8
2C2×C4
2C2×C4
2C8
2C2×C4
2C2×C4
4C2×C4
4C2×C4
4C8
2C42
2C2×C8
2C22×C4
4C16
4M4(2)
4C16
4M4(2)
2C4⋊C8
2C4⋊C8
2M5(2)
2M5(2)

Character table of C42.C8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 11241124444444448888888888
ρ111111111111111111111111111    trivial
ρ21111111-1-1-1-111111-1-1-1-1-11-1111    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111-1-1-1-111111-1-1111-11-1-1-1    linear of order 2
ρ51111111-1-1-1-11-1-1-1-111-i-ii-ii-iii    linear of order 4
ρ6111111111111-1-1-1-1-1-1ii-i-i-i-iii    linear of order 4
ρ71111111-1-1-1-11-1-1-1-111ii-ii-ii-i-i    linear of order 4
ρ8111111111111-1-1-1-1-1-1-i-iiiii-i-i    linear of order 4
ρ9111-1-1-1-11-1-111-i-iii-iiζ87ζ83ζ85ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ10111-1-1-1-1-111-11ii-i-i-iiζ85ζ8ζ87ζ87ζ83ζ83ζ8ζ85    linear of order 8
ρ11111-1-1-1-11-1-111-i-iii-iiζ83ζ87ζ8ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ12111-1-1-1-11-1-111ii-i-ii-iζ8ζ85ζ83ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ13111-1-1-1-1-111-11ii-i-i-iiζ8ζ85ζ83ζ83ζ87ζ87ζ85ζ8    linear of order 8
ρ14111-1-1-1-1-111-11-i-iiii-iζ83ζ87ζ8ζ8ζ85ζ85ζ87ζ83    linear of order 8
ρ15111-1-1-1-11-1-111ii-i-ii-iζ85ζ8ζ87ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ16111-1-1-1-1-111-11-i-iiii-iζ87ζ83ζ85ζ85ζ8ζ8ζ83ζ87    linear of order 8
ρ17222-22220000-2-22-220000000000    orthogonal lifted from D4
ρ18222-22220000-22-22-20000000000    orthogonal lifted from D4
ρ192222-2-2-20000-2-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ202222-2-2-20000-22i-2i-2i2i0000000000    complex lifted from M4(2)
ρ2144-4044-40000000000000000000    orthogonal lifted from C23⋊C4
ρ2244-40-4-440000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ234-4004i-4i0-22i-2i2000000000000000    complex faithful
ρ244-400-4i4i0-2-2i2i2000000000000000    complex faithful
ρ254-400-4i4i022i-2i-2000000000000000    complex faithful
ρ264-4004i-4i02-2i2i-2000000000000000    complex faithful

Permutation representations of C42.C8
On 16 points - transitive group 16T242
Generators in S16
(1 13 9 5)(2 10)(3 7 11 15)(4 12)(6 14)(8 16)
(1 13 9 5)(2 6 10 14)(3 7 11 15)(4 16 12 8)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,242);

Matrix representation of C42.C8 in GL4(𝔽5) generated by

4000
0402
0040
0401
,
4020
0402
4010
0401
,
0002
1000
0100
0010
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,4,0,0,4,0,0,2,0,1],[4,0,4,0,0,4,0,4,2,0,1,0,0,2,0,1],[0,1,0,0,0,0,1,0,0,0,0,1,2,0,0,0] >;

C42.C8 in GAP, Magma, Sage, TeX

C_4^2.C_8
% in TeX

G:=Group("C4^2.C8");
// GroupNames label

G:=SmallGroup(128,59);
// by ID

G=gap.SmallGroup(128,59);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,352,1242,521,136,2804,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=1,c^8=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b>;
// generators/relations

Export

Subgroup lattice of C42.C8 in TeX
Character table of C42.C8 in TeX

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