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G = C42.129D4  order 128 = 27

111st non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.129D4, M4(2).1D4, C4⋊C4.79D4, C4.2C22≀C2, (C2×D4).86D4, (C2×Q8).78D4, C42⋊6C4⋊7C2, (C22×C4).73D4, C4.40(C4⋊D4), C4.30(C4⋊1D4), C23.578(C2×D4), C2.8(C23⋊2D4), C2.26(D4.8D4), C22.196C22≀C2, C23.37D4⋊28C2, C22.56(C4⋊D4), (C22×C4).709C23, (C2×C42).342C22, (C22×D4).59C22, (C22×Q8).48C22, C23.38C23⋊1C2, C42⋊C2.47C22, (C2×M4(2)).12C22, (C2×C4≀C2)⋊24C2, (C2×C4.4D4)⋊1C2, (C2×C8⋊C22).5C2, (C2×C4).73(C4○D4), (C2×C4.10D4)⋊1C2, (C2×C4).1024(C2×D4), (C2×C4○D4).44C22, SmallGroup(128,735)

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

Derived series
C1 — C22×C4 — C42.129D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×C4.4D4 — C42.129D4
Lower central
C1 — C2 — C22×C4 — C42.129D4
Upper central
C1 — C22 — C22×C4 — C42.129D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.129D4

Generators and relations for C42.129D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b-1, dad=ab-1, bc=cb, dbd=b-1, dcd=b2c3 >

Subgroups: 448 in 191 conjugacy classes, 46 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C4.10D4, D4⋊C4, C4≀C2, C2×C42, C2×C22⋊C4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C22×Q8, C2×C4○D4, C42⋊6C4, C2×C4.10D4, C23.37D4, C2×C4≀C2, C2×C4.4D4, C23.38C23, C2×C8⋊C22, C42.129D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, D4.8D4, C42.129D4

Character table of C42.129D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112288822224444888888888
ρ111111111111111111111111111    trivial
ρ2111111-111111111111-1-11-1-1-1-1-1    linear of order 2
ρ31111111-1-111111111-11-1-1-1-1-111    linear of order 2
ρ4111111-1-1-111111111-1-11-1111-1-1    linear of order 2
ρ5111111-1-1-11111-1-1-1-11-1-11-11111    linear of order 2
ρ61111111-1-11111-1-1-1-111111-1-1-1-1    linear of order 2
ρ7111111-1111111-1-1-1-1-1-11-11-1-111    linear of order 2
ρ81111111111111-1-1-1-1-11-1-1-111-1-1    linear of order 2
ρ9222222000-2-2-2-20000-200200000    orthogonal lifted from D4
ρ102222-2-2000-22-22000000-2020000    orthogonal lifted from D4
ρ112-2-22-220002-2-22-22-22000000000    orthogonal lifted from D4
ρ122-2-22-2202-2-222-20000000000000    orthogonal lifted from D4
ρ132222-2-2000-22-2200000020-20000    orthogonal lifted from D4
ρ142-2-222-2000-2-222000000000-2200    orthogonal lifted from D4
ρ15222222000-2-2-2-20000200-200000    orthogonal lifted from D4
ρ162-2-222-2000-2-2220000000002-200    orthogonal lifted from D4
ρ172-2-22-220002-2-222-22-2000000000    orthogonal lifted from D4
ρ182222-2-2-2002-22-20000020000000    orthogonal lifted from D4
ρ192-2-22-220-22-222-20000000000000    orthogonal lifted from D4
ρ202222-2-22002-22-200000-20000000    orthogonal lifted from D4
ρ212-2-222-200022-2-200000000000-2i2i    complex lifted from C4○D4
ρ222-2-222-200022-2-2000000000002i-2i    complex lifted from C4○D4
ρ234-44-40000000002i2i-2i-2i000000000    complex lifted from D4.8D4
ρ2444-4-4000000000-2i2i2i-2i000000000    complex lifted from D4.8D4
ρ2544-4-40000000002i-2i-2i2i000000000    complex lifted from D4.8D4
ρ264-44-4000000000-2i-2i2i2i000000000    complex lifted from D4.8D4

Smallest permutation representation of C42.129D4
►On 32 points
Generators in S32
(1 16 29 22)(2 11 26 21)(3 10 31 24)(4 13 28 23)(5 12 25 18)(6 15 30 17)(7 14 27 20)(8 9 32 19)

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

(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)

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

Matrix representation of C42.129D4 ►in GL6(𝔽17)

0160000
1600000
000400
0013000
0000130
0000013
,
1600000
0160000
0001600
001000
000001
0000160
,
0130000
400000
0000016
0000160
001000
0001600
,
040000
1300000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,16,0,0,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42.129D4 in GAP, Magma, Sage, TeX 

C_4^2._{129}D_4
 
% in TeX
 
G:=Group("C4^2.129D4");
 
// GroupNames label
 
G:=SmallGroup(128,735);
 
// by ID
 
G=gap.SmallGroup(128,735);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,352,2019,1018,521,248,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,d*a*d=a*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=b^2*c^3>;
 
// generators/relations
 

Export

Character table of C42.129D4 in TeX

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