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G = (C4×C8).C4order 128 = 27

6th non-split extension by C4×C8 of C4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: (C4×C8).6C4, (C2×Q8).5D4, C41D4.4C4, C85D4.7C2, C4⋊Q8.2C22, C42.17(C2×C4), C42.3C45C2, C2.10(C42⋊C4), C22.20(C23⋊C4), (C2×C4).36(C22⋊C4), SmallGroup(128,142)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — (C4×C8).C4
C1C2C22C2×C4C2×Q8C4⋊Q8C85D4 — (C4×C8).C4
C1C2C22C2×C4C42 — (C4×C8).C4
C1C2C22C2×C4C4⋊Q8 — (C4×C8).C4
C1C2C2C22C2×C4C4⋊Q8 — (C4×C8).C4

Generators and relations for (C4×C8).C4
 G = < a,b,c | a4=b8=1, c4=b4, ab=ba, cac-1=ab2, cbc-1=ab3 >

2C2
16C2
2C4
2C4
2C4
4C4
4C4
8C22
8C22
8C22
2C2×C4
2C2×C4
2C2×C4
2C8
2C8
4Q8
4D4
4Q8
4D4
4C23
8D4
8D4
8C8
8C8
2C2×C8
2C2×D4
4C4⋊C4
4M4(2)
4C2×D4
4SD16
4SD16
4SD16
4SD16
4M4(2)
2C4.10D4
2C2×SD16
2C2×SD16
2C4.10D4

Character table of (C4×C8).C4

 class 12A2B2C4A4B4C4D4E8A8B8C8D8E8F8G8H
 size 1121644488444416161616
ρ111111111111111111    trivial
ρ2111-111111-1-1-1-11-11-1    linear of order 2
ρ3111-111111-1-1-1-1-11-11    linear of order 2
ρ41111111111111-1-1-1-1    linear of order 2
ρ5111-1111-1-11111ii-i-i    linear of order 4
ρ61111111-1-1-1-1-1-1i-i-ii    linear of order 4
ρ71111111-1-1-1-1-1-1-iii-i    linear of order 4
ρ8111-1111-1-11111-i-iii    linear of order 4
ρ92220-22-22-200000000    orthogonal lifted from D4
ρ102220-22-2-2200000000    orthogonal lifted from D4
ρ1144-400000022-2-20000    orthogonal lifted from C42⋊C4
ρ1244-4000000-2-2220000    orthogonal lifted from C42⋊C4
ρ1344400-400000000000    orthogonal lifted from C23⋊C4
ρ144-40020-200-2-22-2000000    complex faithful
ρ154-400-2020000-2-22-20000    complex faithful
ρ164-400-20200002-2-2-20000    complex faithful
ρ174-40020-2002-2-2-2000000    complex faithful

Permutation representations of (C4×C8).C4
On 16 points - transitive group 16T378
Generators in S16
(9 15 13 11)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10 2 13 5 14 6 9)(3 16 8 15 7 12 4 11)

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

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

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

G:=TransitiveGroup(16,378);

Matrix representation of (C4×C8).C4 in GL4(𝔽3) generated by

2002
0100
0010
2001
,
0002
0110
0210
2002
,
0020
1002
0001
0110
G:=sub<GL(4,GF(3))| [2,0,0,2,0,1,0,0,0,0,1,0,2,0,0,1],[0,0,0,2,0,1,2,0,0,1,1,0,2,0,0,2],[0,1,0,0,0,0,0,1,2,0,0,1,0,2,1,0] >;

(C4×C8).C4 in GAP, Magma, Sage, TeX

(C_4\times C_8).C_4
% in TeX

G:=Group("(C4xC8).C4");
// GroupNames label

G:=SmallGroup(128,142);
// by ID

G=gap.SmallGroup(128,142);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1059,520,794,745,248,1684,1411,375,172,4037]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=1,c^4=b^4,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a*b^3>;
// generators/relations

Export

Subgroup lattice of (C4×C8).C4 in TeX
Character table of (C4×C8).C4 in TeX

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