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G = C24.129D4order 128 = 27

84th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.129D4, C4.112+ 1+4, C8⋊D414C2, Q8⋊D48C2, C8.D410C2, (C2×C8).59C23, C4.Q818C22, C2.D829C22, D4.7D422C2, C4⋊C4.135C23, (C2×C4).394C24, C22⋊Q1618C2, (C2×Q16)⋊23C22, (C22×C4).491D4, C23.278(C2×D4), D4⋊C427C22, C24.4C413C2, Q8⋊C430C22, (C2×SD16)⋊20C22, (C2×D4).145C23, C23.46D49C2, C22⋊C8.40C22, (C2×Q8).133C23, C23.20D422C2, C23.48D420C2, C4⋊D4.184C22, C2.75(C233D4), (C23×C4).574C22, C22.654(C22×D4), C22⋊Q8.189C22, C2.52(D8⋊C22), C22.19C24.21C2, (C22×C4).1072C23, C22.20(C8.C22), (C2×M4(2)).81C22, (C22×Q8).316C22, C42⋊C2.152C22, (C2×C4).532(C2×D4), (C2×C22⋊Q8)⋊60C2, C2.52(C2×C8.C22), (C2×C4⋊C4).641C22, (C2×C4○D4).163C22, SmallGroup(128,1928)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.129D4
C1C2C4C2×C4C22×C4C22×Q8C2×C22⋊Q8 — C24.129D4
C1C2C2×C4 — C24.129D4
C1C22C23×C4 — C24.129D4
C1C2C2C2×C4 — C24.129D4

Generators and relations for C24.129D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, eae-1=ad=da, faf=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e3 >

Subgroups: 428 in 207 conjugacy classes, 86 normal (42 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×4], C2×C4 [×22], D4 [×8], Q8 [×8], C23 [×3], C23 [×5], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], M4(2) [×2], SD16 [×2], Q16 [×2], C22×C4 [×6], C22×C4 [×5], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C4○D4 [×2], C24, C22⋊C8 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C23×C4, C22×Q8, C2×C4○D4, C24.4C4, Q8⋊D4, C22⋊Q16, D4.7D4 [×2], C8⋊D4 [×2], C8.D4 [×2], C23.46D4, C23.48D4, C23.20D4 [×2], C2×C22⋊Q8, C22.19C24, C24.129D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C233D4, C2×C8.C22, D8⋊C22, C24.129D4

Character table of C24.129D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112244822224488888888888
ρ111111111111111111111111111    trivial
ρ21111-1-1-11-11111-1-11-1-1-1111-111-1    linear of order 2
ρ31111-1-11-1-1-11-11-11111-1-11-11-11-1    linear of order 2
ρ4111111-1-11-11-111-11-1-11-11-1-1-111    linear of order 2
ρ511111111-1111111-1-1-1-1-1-1-11111    linear of order 2
ρ61111-1-1-1111111-1-1-1111-1-1-1-111-1    linear of order 2
ρ71111-1-11-11-11-11-11-1-1-111-111-11-1    linear of order 2
ρ8111111-1-1-1-11-111-1-111-11-11-1-111    linear of order 2
ρ9111111-1-11-11-111-1-1-11-111-111-1-1    linear of order 2
ρ101111-1-11-1-1-11-11-11-11-1111-1-11-11    linear of order 2
ρ111111-1-1-11-11111-1-1-1-111-1111-1-11    linear of order 2
ρ12111111111111111-11-1-1-111-1-1-1-1    linear of order 2
ρ13111111-1-1-1-11-111-111-11-1-1111-1-1    linear of order 2
ρ141111-1-11-11-11-11-111-11-1-1-11-11-11    linear of order 2
ρ151111-1-1-1111111-1-111-1-11-1-11-1-11    linear of order 2
ρ1611111111-11111111-1111-1-1-1-1-1-1    linear of order 2
ρ17222222-2-202-22-2-2200000000000    orthogonal lifted from D4
ρ182222-2-22-202-22-22-200000000000    orthogonal lifted from D4
ρ192222-2-2-220-2-2-2-22200000000000    orthogonal lifted from D4
ρ20222222220-2-2-2-2-2-200000000000    orthogonal lifted from D4
ρ214-44-4000000-4040000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000040-40000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-444-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-4000004i0-4i00000000000000    complex lifted from D8⋊C22
ρ2644-4-400000-4i04i00000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C24.129D4
On 32 points
Generators in S32
(2 6)(4 8)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)(26 30)(28 32)
(1 5)(2 32)(3 7)(4 26)(6 28)(8 30)(9 13)(10 18)(11 15)(12 20)(14 22)(16 24)(17 21)(19 23)(25 29)(27 31)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(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 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)

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

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

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

Matrix representation of C24.129D4 in GL8(𝔽17)

10000000
1616000000
00100000
0016160000
00001000
00000100
000000160
000000016
,
160000000
016000000
00100000
00010000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
0016150000
00110000
48000000
013000000
00000004
00000040
00004000
000001300
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[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],[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],[0,0,4,0,0,0,0,0,0,0,8,13,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C24.129D4 in GAP, Magma, Sage, TeX

C_2^4._{129}D_4
% in TeX

G:=Group("C2^4.129D4");
// GroupNames label

G:=SmallGroup(128,1928);
// by ID

G=gap.SmallGroup(128,1928);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,891,352,675,4037,1027,124]);
// Polycyclic

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

Export

Character table of C24.129D4 in TeX

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