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G = C2×C42.C4  order 128 = 27

Direct product of C2 and C42.C4

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

Aliases: C2×C42.C4, (C2×C42).23C4, C42.25(C2×C4), (C2×Q8).117D4, (C22×C4).97D4, C4.4D4.13C4, (C22×D4).15C4, (C2×Q8).10C23, C4.10D4⋊17C22, C22.52(C23⋊C4), (C22×Q8).84C22, C23.203(C22⋊C4), C4.4D4.122C22, (C2×C4).7(C2×D4), (C2×D4).39(C2×C4), C2.42(C2×C23⋊C4), (C2×C4).99(C22×C4), (C22×C4).82(C2×C4), (C2×Q8).108(C2×C4), (C2×C4.10D4)⋊26C2, (C2×C4).29(C22⋊C4), (C2×C4.4D4).14C2, C22.66(C2×C22⋊C4), SmallGroup(128,862)

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

Derived series
C1 — C2×C4 — C2×C42.C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C22×Q8 — C2×C4.4D4 — C2×C42.C4
Lower central
C1 — C2 — C22 — C2×C4 — C2×C42.C4
Upper central
C1 — C22 — C23 — C22×Q8 — C2×C42.C4
Jennings
C1 — C2 — C22 — C2×Q8 — C2×C42.C4

Generators and relations for C2×C42.C4
 G = < a,b,c,d | a2=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >

Subgroups: 324 in 130 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C24, C4.10D4, C4.10D4, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C2×M4(2), C22×D4, C22×Q8, C42.C4, C2×C4.10D4, C2×C4.4D4, C2×C42.C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C42.C4, C2×C23⋊C4, C2×C42.C4

Character table of C2×C42.C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11112288444444444488888888
ρ111111111111111111111111111    trivial
ρ21-11-11-11-1-1-11-1111-11-1-11-11-11-11    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41-11-11-11-1-1-11-1111-11-11-11-11-11-1    linear of order 2
ρ5111111-1-1-11-1-11111-11-1-111-1-111    linear of order 2
ρ61-11-11-1-111-1-11111-1-1-11-1-111-1-11    linear of order 2
ρ7111111-1-1-11-1-11111-1111-1-111-1-1    linear of order 2
ρ81-11-11-1-111-1-11111-1-1-1-111-1-111-1    linear of order 2
ρ9111111-1-11-111-11-1-111iiii-i-i-i-i    linear of order 4
ρ101-11-11-1-11-111-1-11-111-1-ii-iii-ii-i    linear of order 4
ρ11111111-1-11-111-11-1-111-i-i-i-iiiii    linear of order 4
ρ121-11-11-1-11-111-1-11-111-1i-ii-i-ii-ii    linear of order 4
ρ131-11-11-11-111-11-11-11-1-1i-i-ii-iii-i    linear of order 4
ρ1411111111-1-1-1-1-11-1-1-11-i-iiiii-i-i    linear of order 4
ρ151-11-11-11-111-11-11-11-1-1-iii-ii-i-ii    linear of order 4
ρ1611111111-1-1-1-1-11-1-1-11ii-i-i-i-iii    linear of order 4
ρ17222222000200-2-22-20-200000000    orthogonal lifted from D4
ρ182-22-22-2000-200-2-2220200000000    orthogonal lifted from D4
ρ192-22-22-20002002-2-2-20200000000    orthogonal lifted from D4
ρ20222222000-2002-2-220-200000000    orthogonal lifted from D4
ρ214-44-4-4400000000000000000000    orthogonal lifted from C23⋊C4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-400002i02i-2i0000-2i000000000    complex lifted from C42.C4
ρ2444-4-40000-2i0-2i2i00002i000000000    complex lifted from C42.C4
ρ254-4-4400002i0-2i-2i00002i000000000    complex lifted from C42.C4
ρ264-4-440000-2i02i2i0000-2i000000000    complex lifted from C42.C4

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

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

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

(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)| (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,11,9,3)(2,18,14,26)(4,32,16,24)(5,15,13,7)(6,22,10,30)(8,28,12,20)(17,19,29,31)(21,23,25,27), (1,23,5,19)(2,32,6,28)(3,21,7,17)(4,30,8,26)(9,27,13,31)(10,20,14,24)(11,25,15,29)(12,18,16,22), (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( (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,11,9,3)(2,18,14,26)(4,32,16,24)(5,15,13,7)(6,22,10,30)(8,28,12,20)(17,19,29,31)(21,23,25,27), (1,23,5,19)(2,32,6,28)(3,21,7,17)(4,30,8,26)(9,27,13,31)(10,20,14,24)(11,25,15,29)(12,18,16,22), (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([[(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,11,9,3),(2,18,14,26),(4,32,16,24),(5,15,13,7),(6,22,10,30),(8,28,12,20),(17,19,29,31),(21,23,25,27)], [(1,23,5,19),(2,32,6,28),(3,21,7,17),(4,30,8,26),(9,27,13,31),(10,20,14,24),(11,25,15,29),(12,18,16,22)], [(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
,
1600000
210000
0001600
0016000
0000130
0000013
,
1600000
0160000
0001300
0013000
0000013
0000130
,
440000
0130000
000010
000001
000100
0016000

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],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

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(128,862);
 
// by ID
 
G=gap.SmallGroup(128,862);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1018,248,1971,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^4=1,d^4=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

Export

Character table of C2×C42.C4 in TeX

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