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G = C24.4D4order 128 = 27

4th non-split extension by C24 of D4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C24.4D4, C23.2Q16, C23.6SD16, C23⋊C8.5C2, C22.18C4≀C2, (C22×C4).11D4, C2.C426C4, C2.5(C42⋊C4), C23.9D4.5C2, C23.4Q8.1C2, C22.59(C23⋊C4), C2.6(C23.D4), C23.159(C22⋊C4), C22.17(Q8⋊C4), C2.5(C23.31D4), (C2×C4⋊C4)⋊3C4, (C22×C4).6(C2×C4), (C2×C22⋊C4).85C22, SmallGroup(128,84)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.4D4
C1C2C22C23C24C2×C22⋊C4C23.4Q8 — C24.4D4
C1C2C23C22×C4 — C24.4D4
C1C22C23C2×C22⋊C4 — C24.4D4
C1C22C23C2×C22⋊C4 — C24.4D4

Generators and relations for C24.4D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=a, ab=ba, ac=ca, ad=da, eae-1=abc, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=abcde3 >

Subgroups: 236 in 77 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22 [×3], C22 [×9], C8, C2×C4 [×15], C23 [×3], C23 [×4], C22⋊C4 [×6], C4⋊C4 [×3], C2×C8, C22×C4 [×2], C22×C4 [×3], C24, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C23⋊C8, C23.9D4, C23.4Q8, C24.4D4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, C23⋊C4, Q8⋊C4, C4≀C2, C23.31D4, C23.D4, C42⋊C4, C24.4D4

Character table of C24.4D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-1-1-11-1-1-1-11111    linear of order 2
ρ31111111111-11-111-1-111-1-1-1-1    linear of order 2
ρ411111111111-11-1111-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-i-1i11i-i1-1-iii-i    linear of order 4
ρ6111111-1-1-1-1i1-i-11-ii-11-iii-i    linear of order 4
ρ7111111-1-1-1-1-i1i-11i-i-11i-i-ii    linear of order 4
ρ8111111-1-1-1-1i-1-i11-ii1-1i-i-ii    linear of order 4
ρ922222222-2-20000-200000000    orthogonal lifted from D4
ρ10222222-2-2220000-200000000    orthogonal lifted from D4
ρ112-22-22-22-200000000000-2-222    symplectic lifted from Q16, Schur index 2
ρ122-22-22-22-20000000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ132-22-2-22002i-2i1-i0-1-i001+i-1+i000000    complex lifted from C4≀C2
ρ142-22-2-2200-2i2i-1-i01-i00-1+i1+i000000    complex lifted from C4≀C2
ρ152-22-2-22002i-2i-1+i01+i00-1-i1-i000000    complex lifted from C4≀C2
ρ162-22-22-2-2200000000000-2--2-2--2    complex lifted from SD16
ρ172-22-2-2200-2i2i1+i0-1+i001-i-1-i000000    complex lifted from C4≀C2
ρ182-22-22-2-2200000000000--2-2--2-2    complex lifted from SD16
ρ194-4-440000000002000-200000    orthogonal lifted from C42⋊C4
ρ204444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ214-4-44000000000-2000200000    orthogonal lifted from C42⋊C4
ρ2244-4-40000000-2i0000002i0000    complex lifted from C23.D4
ρ2344-4-400000002i000000-2i0000    complex lifted from C23.D4

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

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

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

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

Matrix representation of C24.4D4 in GL6(𝔽17)

1600000
0160000
0016200
000100
0041301
0041310
,
1600000
0160000
0011500
0001600
000401
000410
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
11110000
300000
004111
004011
001676
000776
,
400000
13130000
00130015
001301616
0016004
0001604

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,4,4,0,0,2,1,13,13,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,15,16,4,4,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[11,3,0,0,0,0,11,0,0,0,0,0,0,0,4,4,1,0,0,0,1,0,6,7,0,0,1,1,7,7,0,0,1,1,6,6],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,13,13,16,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,15,16,4,4] >;

C24.4D4 in GAP, Magma, Sage, TeX

C_2^4._4D_4
% in TeX

G:=Group("C2^4.4D4");
// GroupNames label

G:=SmallGroup(128,84);
// by ID

G=gap.SmallGroup(128,84);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,387,520,1690,521,2804]);
// Polycyclic

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

Export

Character table of C24.4D4 in TeX

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