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G = C42⋊C8  order 128 = 27

2nd semidirect product of C42 and C8 acting via C8/C2=C4

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

Aliases: C42⋊2C8, (C2×C42).8C4, (C22×C4).5D4, C42⋊8C4.3C2, (C2×C4).29M4(2), C2.2(C42⋊3C4), C22.17(C22⋊C8), C22.39(C23⋊C4), C2.2(C42.C4), C23.151(C22⋊C4), C22.6(C4.10D4), C22.M4(2).2C2, C2.5(C22.M4(2)), (C2×C4⋊C4).6C4, (C2×C4).34(C2×C8), (C2×C4⋊C4).3C22, (C22×C4).60(C2×C4), SmallGroup(128,56)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42⋊C8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C42⋊8C4 — C42⋊C8
Lower central
C1 — C2 — C22 — C2×C4 — C42⋊C8
Upper central
C1 — C22 — C23 — C2×C4⋊C4 — C42⋊C8
Jennings
C1 — C22 — C23 — C2×C4⋊C4 — C42⋊C8

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

C1
C2
C2
C2
2C2
2C2
C22
C22
C22
2C4
2C4
2C22
2C22
4C4
4C4
4C4
4C4
8C4
C23
C2×C4
C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
4C2×C4
8C8
8C8
C42
C22×C4
C22×C4
C22×C4
C42
2C22×C4
2C22×C4
2C42
4C4⋊C4
4C2×C8
4C2×C8
4C4⋊C4
C2×C4⋊C4
C2×C42
C2×C4⋊C4
2C2.C42
2C22⋊C8
2C2.C42
2C22⋊C8
C42⋊8C4
C22.M4(2)
C22.M4(2)
C42⋊C8

Character table of C42⋊C8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111-11-11111-1-11-1-1-1-11-1111-1    linear of order 2
ρ3111111-11-11111-1-11-1-111-11-1-1-11    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ5111111-11-1-11-1-1-1-1-111-iii-i-ii-ii    linear of order 4
ρ6111111111-11-1-111-1-1-1i-iii-ii-i-i    linear of order 4
ρ7111111-11-1-11-1-1-1-1-111i-i-iii-ii-i    linear of order 4
ρ8111111111-11-1-111-1-1-1-ii-i-ii-iii    linear of order 4
ρ91-11-11-1-11-1-i-1-ii11ii-iζ87ζ85ζ83ζ83ζ85ζ87ζ8ζ8    linear of order 8
ρ101-11-11-1111i-1i-i-1-1-ii-iζ8ζ83ζ8ζ85ζ87ζ85ζ83ζ87    linear of order 8
ρ111-11-11-1-11-1i-1i-i11-i-iiζ85ζ87ζ8ζ8ζ87ζ85ζ83ζ83    linear of order 8
ρ121-11-11-1-11-1-i-1-ii11ii-iζ83ζ8ζ87ζ87ζ8ζ83ζ85ζ85    linear of order 8
ρ131-11-11-1111-i-1-ii-1-1i-iiζ87ζ85ζ87ζ83ζ8ζ83ζ85ζ8    linear of order 8
ρ141-11-11-1111i-1i-i-1-1-ii-iζ85ζ87ζ85ζ8ζ83ζ8ζ87ζ83    linear of order 8
ρ151-11-11-1-11-1i-1i-i11-i-iiζ8ζ83ζ85ζ85ζ83ζ8ζ87ζ87    linear of order 8
ρ161-11-11-1111-i-1-ii-1-1i-iiζ83ζ8ζ83ζ87ζ85ζ87ζ8ζ85    linear of order 8
ρ172222220-202-2-2-20020000000000    orthogonal lifted from D4
ρ182222220-20-2-22200-20000000000    orthogonal lifted from D4
ρ192-22-22-20-202i2-2i2i00-2i0000000000    complex lifted from M4(2)
ρ202-22-22-20-20-2i22i-2i002i0000000000    complex lifted from M4(2)
ρ214444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224-44-4-4400000000000000000000    symplectic lifted from C4.10D4, Schur index 2
ρ2344-4-4002i0-2i0000-2i2i00000000000    complex lifted from C42.C4
ρ244-4-4400-2i02i0000-2i2i00000000000    complex lifted from C42⋊3C4
ρ2544-4-400-2i02i00002i-2i00000000000    complex lifted from C42.C4
ρ264-4-44002i0-2i00002i-2i00000000000    complex lifted from C42⋊3C4

Smallest permutation representation of C42⋊C8
►On 32 points
Generators in S32
(1 5)(2 12 32 19)(3 29)(4 14 26 21)(6 16 28 23)(7 25)(8 10 30 17)(9 20)(11 15)(13 24)(18 22)(27 31)

(1 15 31 22)(2 23 32 16)(3 24 25 9)(4 10 26 17)(5 11 27 18)(6 19 28 12)(7 20 29 13)(8 14 30 21)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
 
G:=sub<Sym(32)| (1,5)(2,12,32,19)(3,29)(4,14,26,21)(6,16,28,23)(7,25)(8,10,30,17)(9,20)(11,15)(13,24)(18,22)(27,31), (1,15,31,22)(2,23,32,16)(3,24,25,9)(4,10,26,17)(5,11,27,18)(6,19,28,12)(7,20,29,13)(8,14,30,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)>;
 
G:=Group( (1,5)(2,12,32,19)(3,29)(4,14,26,21)(6,16,28,23)(7,25)(8,10,30,17)(9,20)(11,15)(13,24)(18,22)(27,31), (1,15,31,22)(2,23,32,16)(3,24,25,9)(4,10,26,17)(5,11,27,18)(6,19,28,12)(7,20,29,13)(8,14,30,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32) );
 
G=PermutationGroup([[(1,5),(2,12,32,19),(3,29),(4,14,26,21),(6,16,28,23),(7,25),(8,10,30,17),(9,20),(11,15),(13,24),(18,22),(27,31)], [(1,15,31,22),(2,23,32,16),(3,24,25,9),(4,10,26,17),(5,11,27,18),(6,19,28,12),(7,20,29,13),(8,14,30,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)]])
 

Matrix representation of C42⋊C8 ►in GL6(𝔽17)

1600000
610000
0016000
000100
008940
00151504
,
1600000
0160000
0013000
000400
00010130
007004
,
14160000
1330000
00711150
00117015
0000106
0010610

G:=sub<GL(6,GF(17))| [16,6,0,0,0,0,0,1,0,0,0,0,0,0,16,0,8,15,0,0,0,1,9,15,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,7,0,0,0,4,10,0,0,0,0,0,13,0,0,0,0,0,0,4],[14,13,0,0,0,0,16,3,0,0,0,0,0,0,7,11,0,1,0,0,11,7,0,0,0,0,15,0,10,6,0,0,0,15,6,10] >;
 

C42⋊C8 in GAP, Magma, Sage, TeX 

C_4^2\rtimes C_8
 
% in TeX
 
G:=Group("C4^2:C8");
 
// GroupNames label
 
G:=SmallGroup(128,56);
 
// by ID
 
G=gap.SmallGroup(128,56);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,352,1242,521,136,2804]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^2*b^-1>;
 
// generators/relations
 

Export

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

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