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G = C2xC42.C4order 128 = 27

Direct product of C2 and C42.C4

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

Aliases: C2xC42.C4, (C2xC42).23C4, C42.25(C2xC4), (C2xQ8).117D4, (C22xC4).97D4, C4.4D4.13C4, (C22xD4).15C4, (C2xQ8).10C23, C4.10D4:17C22, C22.52(C23:C4), (C22xQ8).84C22, C23.203(C22:C4), C4.4D4.122C22, (C2xC4).7(C2xD4), (C2xD4).39(C2xC4), C2.42(C2xC23:C4), (C2xC4).99(C22xC4), (C22xC4).82(C2xC4), (C2xQ8).108(C2xC4), (C2xC4.10D4):26C2, (C2xC4).29(C22:C4), (C2xC4.4D4).14C2, C22.66(C2xC22:C4), SmallGroup(128,862)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C2xC42.C4
C1C2C22C2xC4C2xQ8C22xQ8C2xC4.4D4 — C2xC42.C4
C1C2C22C2xC4 — C2xC42.C4
C1C22C23C22xQ8 — C2xC42.C4
C1C2C22C2xQ8 — C2xC42.C4

Generators and relations for C2xC42.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, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C22:C4, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C24, C4.10D4, C4.10D4, C2xC42, C2xC22:C4, C4.4D4, C4.4D4, C2xM4(2), C22xD4, C22xQ8, C42.C4, C2xC4.10D4, C2xC4.4D4, C2xC42.C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C23:C4, C2xC22:C4, C42.C4, C2xC23:C4, C2xC42.C4

Character table of C2xC42.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 C2xC42.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 C2xC42.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] >;

C2xC42.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 C2xC42.C4 in TeX

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