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G = C24.126D4order 128 = 27

81st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.126D4, C4.82+ 1+4, C8⋊D412C2, Q8⋊D47C2, C22⋊SD167C2, (C2×C8).56C23, C2.D827C22, C4⋊C4.132C23, C22⋊C814C22, (C2×C4).391C24, C23.275(C2×D4), (C22×C4).488D4, C22⋊Q870C22, D4⋊C425C22, C24.4C411C2, Q8⋊C427C22, (C2×SD16)⋊18C22, (C2×D4).143C23, C22.D819C2, (C2×Q8).130C23, (C22×Q8)⋊20C22, C4⋊D4.183C22, C23.48D419C2, C2.72(C233D4), C22.50(C8⋊C22), (C2×M4(2))⋊14C22, (C23×C4).571C22, C22.651(C22×D4), (C22×C4).1069C23, (C22×D4).382C22, C22.38(C8.C22), (C2×C4⋊C4)⋊53C22, (C2×C4).529(C2×D4), C2.51(C2×C8⋊C22), (C2×C22⋊Q8)⋊59C2, (C2×C4⋊D4).60C2, C2.50(C2×C8.C22), SmallGroup(128,1925)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.126D4
C1C2C4C2×C4C22×C4C22×D4C2×C4⋊D4 — C24.126D4
C1C2C2×C4 — C24.126D4
C1C22C23×C4 — C24.126D4
C1C2C2C2×C4 — C24.126D4

Generators and relations for C24.126D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, faf-1=ac=ca, eae-1=ad=da, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 532 in 231 conjugacy classes, 88 normal (28 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×21], C8 [×4], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×6], C23 [×3], C23 [×13], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×2], SD16 [×4], C22×C4 [×6], C22×C4 [×5], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C2×Q8 [×3], C24, C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C24.4C4, Q8⋊D4 [×2], C22⋊SD16 [×2], C8⋊D4 [×4], C22.D8 [×2], C23.48D4 [×2], C2×C4⋊D4, C2×C22⋊Q8, C24.126D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C233D4, C2×C8⋊C22, C2×C8.C22, C24.126D4

Character table of C24.126D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112222488224448888888888
ρ111111111111111111111111111    trivial
ρ21111-111-1-11111-11-1-11-11-1-1-111-1    linear of order 2
ρ3111111111-1-11111111-11-11-1-1-1-1    linear of order 2
ρ41111-111-1-1-1-111-11-1-11111-11-1-11    linear of order 2
ρ51111-1-1-1-11-1111-1-11-111-1-11-1-111    linear of order 2
ρ611111-1-11-1-11111-1-111-1-11-11-11-1    linear of order 2
ρ71111-1-1-1-111-111-1-11-11-1-11111-1-1    linear of order 2
ρ811111-1-11-11-1111-1-1111-1-1-1-11-11    linear of order 2
ρ91111-1-1-1-111-111-1-111-1-111-1-1-111    linear of order 2
ρ1011111-1-11-11-1111-1-1-1-111-111-11-1    linear of order 2
ρ111111-1-1-1-11-1111-1-111-111-1-111-1-1    linear of order 2
ρ1211111-1-11-1-11111-1-1-1-1-1111-11-11    linear of order 2
ρ13111111111-1-111111-1-1-1-1-1-11111    linear of order 2
ρ141111-111-1-1-1-111-11-11-11-111-111-1    linear of order 2
ρ151111111111111111-1-11-11-1-1-1-1-1    linear of order 2
ρ161111-111-1-11111-11-11-1-1-1-111-1-11    linear of order 2
ρ172222-222-2-200-2-22-220000000000    orthogonal lifted from D4
ρ1822222222200-2-2-2-2-20000000000    orthogonal lifted from D4
ρ192222-2-2-2-2200-2-222-20000000000    orthogonal lifted from D4
ρ2022222-2-22-200-2-2-2220000000000    orthogonal lifted from D4
ρ214-44-40000000-440000000000000    orthogonal lifted from 2+ 1+4
ρ2244-4-4-4004000000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-400000004-40000000000000    orthogonal lifted from 2+ 1+4
ρ2444-4-4400-4000000000000000000    orthogonal lifted from C8⋊C22
ρ254-4-4404-40000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4-440-440000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C24.126D4
On 32 points
Generators in S32
(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)
(2 30)(4 32)(6 26)(8 28)(10 22)(12 24)(14 18)(16 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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 5 6)(3 8 7 4)(9 22 13 18)(10 17 14 21)(11 20 15 24)(12 23 16 19)(25 26 29 30)(27 32 31 28)

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

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

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

Matrix representation of C24.126D4 in GL8(𝔽17)

160000000
016000000
00100000
00010000
00000100
00001000
000012411
000011016
,
10000000
01000000
001600000
000160000
00001000
00000100
000090160
000011016
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
00010000
00100000
160000000
01000000
00008020
000071622
000011090
00007021
,
00100000
00010000
160000000
016000000
00008020
00001011515
000011090
000000016

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,1,0,0,0,0,1,0,4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,9,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,8,7,11,7,0,0,0,0,0,16,0,0,0,0,0,0,2,2,9,2,0,0,0,0,0,2,0,1],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,10,11,0,0,0,0,0,0,1,0,0,0,0,0,0,2,15,9,0,0,0,0,0,0,15,0,16] >;

C24.126D4 in GAP, Magma, Sage, TeX

C_2^4._{126}D_4
% in TeX

G:=Group("C2^4.126D4");
// GroupNames label

G:=SmallGroup(128,1925);
// by ID

G=gap.SmallGroup(128,1925);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,891,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,f*a*f^-1=a*c=c*a,e*a*e^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

Export

Character table of C24.126D4 in TeX

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