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G = C42.129D4order 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, C426C47C2, (C22×C4).73D4, C4.40(C4⋊D4), C4.30(C41D4), C23.578(C2×D4), C2.8(C232D4), C2.26(D4.8D4), C22.196C22≀C2, C23.37D428C2, C22.56(C4⋊D4), (C22×C4).709C23, (C2×C42).342C22, (C22×D4).59C22, (C22×Q8).48C22, C23.38C231C2, 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

C1C22×C4 — C42.129D4
C1C2C22C23C22×C4C22×Q8C2×C4.4D4 — C42.129D4
C1C2C22×C4 — C42.129D4
C1C22C22×C4 — C42.129D4
C1C2C2C22×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 [×2], C2 [×5], C4 [×4], C4 [×7], C22 [×3], C22 [×15], C8 [×3], C2×C4 [×6], C2×C4 [×15], D4 [×10], Q8 [×6], C23, C23 [×9], C42 [×2], C42 [×2], C22⋊C4 [×13], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], M4(2) [×2], M4(2) [×3], D8 [×4], SD16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×6], C4○D4 [×4], C24, C4.10D4 [×2], D4⋊C4 [×2], C4≀C2 [×2], C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×5], C4⋊Q8, C2×M4(2) [×2], C2×D8, C2×SD16, C8⋊C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C426C4, C2×C4.10D4, C23.37D4, C2×C4≀C2, C2×C4.4D4, C23.38C23, C2×C8⋊C22, C42.129D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D4.8D4 [×2], 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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