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G = (C2×C42).C4  order 128 = 27

6th non-split extension by C2×C42 of C4 acting faithfully

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

Aliases: (C2×Q8)⋊2C8, (C2×C42).6C4, (C22×C4).4D4, C2.9(C23⋊C8), (C2×C4).2M4(2), (C22×Q8).2C4, C2.1(C42⋊3C4), C22.38(C23⋊C4), C22.14(C22⋊C8), C2.1(C42.3C4), C22.9(C4.D4), C23.150(C22⋊C4), C23.67C23.2C2, C22.M4(2).1C2, (C2×C4).2(C2×C8), (C2×C4⋊C4).2C22, (C22×C4).57(C2×C4), SmallGroup(128,51)

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

Derived series
C1 — C2×C4 — (C2×C42).C4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.67C23 — (C2×C42).C4
Lower central
C1 — C2 — C22 — C2×C4 — (C2×C42).C4
Upper central
C1 — C22 — C23 — C2×C4⋊C4 — (C2×C42).C4
Jennings
C1 — C22 — C23 — C2×C4⋊C4 — (C2×C42).C4

Generators and relations for (C2×C42).C4
 G = < a,b,c,d | a2=b4=c4=1, d4=a, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab-1c-1, dcd-1=ab2c-1 >

Subgroups: 168 in 67 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C22⋊C8, C2×C42, C2×C4⋊C4, C22×Q8, C22.M4(2), C23.67C23, (C2×C42).C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C23⋊C8, C42⋊3C4, C42.3C4, (C2×C42).C4

Character table of (C2×C42).C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-11111-1-1-111-1-11-11-1-111-1    linear of order 2
ρ4111111-11111-1-1-111-1-1-11-111-1-11    linear of order 2
ρ5111111-1-1-11-1-1-1-1-1111-ii-i-i-iiii    linear of order 4
ρ6111111-1-1-11-1-1-1-1-1111i-iiii-i-i-i    linear of order 4
ρ71111111-1-11-1111-11-1-1-i-i-iiiii-i    linear of order 4
ρ81111111-1-11-1111-11-1-1iii-i-i-i-ii    linear of order 4
ρ91-11-11-1ii-i-1-ii-i-ii11-1ζ87ζ8ζ83ζ87ζ83ζ85ζ8ζ85    linear of order 8
ρ101-11-11-1ii-i-1-ii-i-ii11-1ζ83ζ85ζ87ζ83ζ87ζ8ζ85ζ8    linear of order 8
ρ111-11-11-1i-ii-1ii-i-i-i1-11ζ8ζ83ζ85ζ85ζ8ζ83ζ87ζ87    linear of order 8
ρ121-11-11-1i-ii-1ii-i-i-i1-11ζ85ζ87ζ8ζ8ζ85ζ87ζ83ζ83    linear of order 8
ρ131-11-11-1-ii-i-1-i-iiii1-11ζ87ζ85ζ83ζ83ζ87ζ85ζ8ζ8    linear of order 8
ρ141-11-11-1-i-ii-1i-iii-i11-1ζ8ζ87ζ85ζ8ζ85ζ83ζ87ζ83    linear of order 8
ρ151-11-11-1-ii-i-1-i-iiii1-11ζ83ζ8ζ87ζ87ζ83ζ8ζ85ζ85    linear of order 8
ρ161-11-11-1-i-ii-1i-iii-i11-1ζ85ζ83ζ8ζ85ζ8ζ87ζ83ζ87    linear of order 8
ρ1722222202-2-22000-2-20000000000    orthogonal lifted from D4
ρ182222220-22-2-20002-20000000000    orthogonal lifted from D4
ρ192-22-22-20-2i-2i22i0002i-20000000000    complex lifted from M4(2)
ρ202-22-22-202i2i2-2i000-2i-20000000000    complex lifted from M4(2)
ρ214-44-4-4400000000000000000000    orthogonal lifted from C4.D4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-400-200002-22000000000000    symplectic lifted from C42.3C4, Schur index 2
ρ2444-4-40020000-22-2000000000000    symplectic lifted from C42.3C4, Schur index 2
ρ254-4-44002i0000-2i-2i2i000000000000    complex lifted from C42⋊3C4
ρ264-4-4400-2i00002i2i-2i000000000000    complex lifted from C42⋊3C4

Smallest permutation representation of (C2×C42).C4
►On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)

(2 19 32 12)(3 29)(4 17 26 10)(6 23 28 16)(7 25)(8 21 30 14)(9 20)(13 24)

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

(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)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (2,19,32,12)(3,29)(4,17,26,10)(6,23,28,16)(7,25)(8,21,30,14)(9,20)(13,24), (1,15,31,22)(2,19,32,12)(3,24,25,9)(4,14,26,21)(5,11,27,18)(6,23,28,16)(7,20,29,13)(8,10,30,17), (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)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (2,19,32,12)(3,29)(4,17,26,10)(6,23,28,16)(7,25)(8,21,30,14)(9,20)(13,24), (1,15,31,22)(2,19,32,12)(3,24,25,9)(4,14,26,21)(5,11,27,18)(6,23,28,16)(7,20,29,13)(8,10,30,17), (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),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(2,19,32,12),(3,29),(4,17,26,10),(6,23,28,16),(7,25),(8,21,30,14),(9,20),(13,24)], [(1,15,31,22),(2,19,32,12),(3,24,25,9),(4,14,26,21),(5,11,27,18),(6,23,28,16),(7,20,29,13),(8,10,30,17)], [(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 (C2×C42).C4 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
400000
0130000
001000
000100
00160167
0014071
,
1600000
0160000
0081400
0016900
0026167
00111171
,
090000
800000
00130150
000011
000140
0010130

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,16,14,0,0,0,1,0,0,0,0,0,0,16,7,0,0,0,0,7,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,8,16,2,11,0,0,14,9,6,11,0,0,0,0,16,7,0,0,0,0,7,1],[0,8,0,0,0,0,9,0,0,0,0,0,0,0,13,0,0,1,0,0,0,0,1,0,0,0,15,1,4,13,0,0,0,1,0,0] >;
 

(C2×C42).C4 in GAP, Magma, Sage, TeX 

(C_2\times C_4^2).C_4
 
% in TeX
 
G:=Group("(C2xC4^2).C4");
 
// GroupNames label
 
G:=SmallGroup(128,51);
 
// by ID
 
G=gap.SmallGroup(128,51);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,723,352,1242,521,136,2804]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=a,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b^-1*c^-1,d*c*d^-1=a*b^2*c^-1>;
 
// generators/relations
 

Export

Character table of (C2×C42).C4 in TeX

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