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G = C42.9D4  order 128 = 27

9th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.9D4, (C2×D4).121D4, (C2×Q8).112D4, C4.9C42⋊14C2, C4.159(C4⋊D4), C4.C42⋊22C2, C23.13(C4○D4), (C22×C8).88C22, M4(2)⋊4C4⋊20C2, C4.21(C42⋊2C2), C4.102(C4.4D4), (C22×C4).739C23, C42⋊C22.9C2, C23.24D4.2C2, C42.6C22⋊24C2, C42⋊C2.68C22, C4.30(C22.D4), C22.18(C42⋊2C2), C2.15(C23.11D4), (C2×M4(2)).237C22, C22.35(C22.D4), M4(2).8C22.9C2, (C2×C4).1378(C2×D4), (C2×C4).778(C4○D4), (C2×C4○D4).69C22, SmallGroup(128,812)

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

Derived series
C1 — C22×C4 — C42.9D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — M4(2).8C22 — C42.9D4
Lower central
C1 — C2 — C22×C4 — C42.9D4
Upper central
C1 — C4 — C22×C4 — C42.9D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.9D4

Generators and relations for C42.9D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b, dad=ab-1, cbc-1=dbd=b-1, dcd=b-1c3 >

Subgroups: 200 in 95 conjugacy classes, 36 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, D4⋊C4, Q8⋊C4, C4≀C2, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×C4○D4, C4.9C42, C4.C42, M4(2)⋊4C4, M4(2).8C22, C23.24D4, C42⋊C22, C42.6C22, C42.9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C23.11D4, C42.9D4

Character table of C42.9D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J
 size 11222811222888884444888888
ρ111111111111111111111111111    trivial
ρ211111-11111111-111-1-1-1-11-1-1-11-1    linear of order 2
ρ311111111111-1-11-1-1-1-1-1-111-111-1    linear of order 2
ρ411111-111111-1-1-1-1-111111-11-111    linear of order 2
ρ5111111111111-11-111111-1-1-1-1-1-1    linear of order 2
ρ611111-1111111-1-1-11-1-1-1-1-1111-11    linear of order 2
ρ711111111111-1111-1-1-1-1-1-1-11-1-11    linear of order 2
ρ811111-111111-11-11-11111-11-11-1-1    linear of order 2
ρ922-22-2022-22-20-20200000000000    orthogonal lifted from D4
ρ10222-2-22-2-2-22200-2000000000000    orthogonal lifted from D4
ρ1122-22-2022-22-2020-200000000000    orthogonal lifted from D4
ρ12222-2-2-2-2-2-222002000000000000    orthogonal lifted from D4
ρ13222220-2-2-2-2-20000000000-2i02i00    complex lifted from C4○D4
ρ14222220-2-2-2-2-200000000002i0-2i00    complex lifted from C4○D4
ρ1522-22-20-2-22-22000002i-2i-2i2i000000    complex lifted from C4○D4
ρ16222-2-20222-2-20000000002i000-2i0    complex lifted from C4○D4
ρ1722-22-20-2-22-2200000-2i2i2i-2i000000    complex lifted from C4○D4
ρ1822-2-220-2-222-2000000000002i00-2i    complex lifted from C4○D4
ρ1922-2-22022-2-22-2i0002i0000000000    complex lifted from C4○D4
ρ20222-2-20222-2-2000000000-2i0002i0    complex lifted from C4○D4
ρ2122-2-220-2-222-200000000000-2i002i    complex lifted from C4○D4
ρ2222-2-22022-2-222i000-2i0000000000    complex lifted from C4○D4
ρ234-400004i-4i000000002ζ832ζ852ζ82ζ87000000    complex faithful
ρ244-40000-4i4i000000002ζ82ζ872ζ832ζ85000000    complex faithful
ρ254-400004i-4i000000002ζ872ζ82ζ852ζ83000000    complex faithful
ρ264-40000-4i4i000000002ζ852ζ832ζ872ζ8000000    complex faithful

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

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

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

Matrix representation of C42.9D4 ►in GL4(𝔽17) generated by

00156
0009
9700
0200
,
13900
0400
00139
0004
,
00210
001515
81000
9900
,
21000
151500
00210
001515
G:=sub<GL(4,GF(17))| [0,0,9,0,0,0,7,2,15,0,0,0,6,9,0,0],[13,0,0,0,9,4,0,0,0,0,13,0,0,0,9,4],[0,0,8,9,0,0,10,9,2,15,0,0,10,15,0,0],[2,15,0,0,10,15,0,0,0,0,2,15,0,0,10,15] >;
 

C42.9D4 in GAP, Magma, Sage, TeX 

C_4^2._9D_4
 
% in TeX
 
G:=Group("C4^2.9D4");
 
// GroupNames label
 
G:=SmallGroup(128,812);
 
// by ID
 
G=gap.SmallGroup(128,812);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,1018,248,1411,4037,1027]);
 
// 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,d*a*d=a*b^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b^-1*c^3>;
 
// generators/relations
 

Export

Character table of C42.9D4 in TeX

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