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G = C24.33D4order 128 = 27

33rd non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.33D4, C24.181C23, C22⋊C47D4, C2.18C2≀C22, (C22×C4).84D4, C23.34(C2×D4), C233D4.6C2, C23.8Q82C2, C23.11(C4○D4), C23.23D42C2, C23.9D415C2, (C23×C4).28C22, C22.226C22≀C2, C22.68(C4⋊D4), (C22×D4).81C22, C22.38(C4.4D4), C2.18(C23.7D4), C2.17(C23.10D4), C22.34(C22.D4), (C2×C23⋊C4)⋊10C2, (C2×C22⋊C4).20C22, SmallGroup(128,776)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C24.33D4
C1C2C22C23C24C23×C4C23.23D4 — C24.33D4
C1C2C24 — C24.33D4
C1C22C24 — C24.33D4
C1C2C24 — C24.33D4

Generators and relations for C24.33D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, faf=ab=ba, ac=ca, ad=da, eae-1=abcd, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=be-1 >

Subgroups: 504 in 191 conjugacy classes, 42 normal (20 characteristic)
C1, C2 [×3], C2 [×8], C4 [×10], C22 [×3], C22 [×4], C22 [×22], C2×C4 [×30], D4 [×10], C23 [×3], C23 [×4], C23 [×12], C22⋊C4 [×2], C22⋊C4 [×15], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C22×C4 [×8], C2×D4 [×14], C24, C24 [×2], C2.C42 [×3], C23⋊C4 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C22≀C2 [×2], C4⋊D4 [×2], C22.D4 [×2], C23×C4, C22×D4 [×2], C23.9D4, C23.8Q8, C23.23D4 [×2], C2×C23⋊C4 [×2], C233D4, C24.33D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, C2≀C22, C23.7D4, C24.33D4

Character table of C24.33D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11112222228844448888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-11111-1-1-1-11111-1-1    linear of order 2
ρ311111111111-11111-111-1-1-1-1-1-11    linear of order 2
ρ41111111111-1111111-1-11-1-1-1-11-1    linear of order 2
ρ511111111111-1-1-1-1-1-1-1-11-11-1111    linear of order 2
ρ61111111111-11-1-1-1-1111-1-11-11-1-1    linear of order 2
ρ7111111111111-1-1-1-11-1-1-11-11-1-11    linear of order 2
ρ81111111111-1-1-1-1-1-1-11111-11-11-1    linear of order 2
ρ92-2-2222-2-22-20000000000020-200    orthogonal lifted from D4
ρ102-2-2222-2-22-200000000000-20200    orthogonal lifted from D4
ρ112222-22-2-2-22-2000000000000002    orthogonal lifted from D4
ρ122222-22-2-2-22200000000000000-2    orthogonal lifted from D4
ρ1322222-22-2-2-2020000-2000000000    orthogonal lifted from D4
ρ142222-2-2-222-2000000000020-2000    orthogonal lifted from D4
ρ1522222-22-2-2-20-200002000000000    orthogonal lifted from D4
ρ162222-2-2-222-20000000000-202000    orthogonal lifted from D4
ρ172-2-22-2222-2-2000000000-2i00002i0    complex lifted from C4○D4
ρ182-2-222-2-22-2200000002i-2i0000000    complex lifted from C4○D4
ρ192-2-222-2-22-220000000-2i2i0000000    complex lifted from C4○D4
ρ202-2-22-2222-2-20000000002i0000-2i0    complex lifted from C4○D4
ρ212-2-22-2-22-222002i-2i2i-2i0000000000    complex lifted from C4○D4
ρ222-2-22-2-22-22200-2i2i-2i2i0000000000    complex lifted from C4○D4
ρ2344-4-4000000002-2-220000000000    orthogonal lifted from C2≀C22
ρ2444-4-400000000-222-20000000000    orthogonal lifted from C2≀C22
ρ254-44-4000000002i2i-2i-2i0000000000    complex lifted from C23.7D4
ρ264-44-400000000-2i-2i2i2i0000000000    complex lifted from C23.7D4

Smallest permutation representation of C24.33D4
On 32 points
Generators in S32
(1 21)(2 23)(3 28)(4 26)(5 25)(6 27)(7 22)(8 24)(9 17)(10 29)(11 18)(12 32)(13 20)(14 30)(15 19)(16 31)
(1 3)(2 4)(5 7)(6 8)(9 16)(10 15)(11 12)(13 14)(17 31)(18 32)(19 29)(20 30)(21 28)(22 25)(23 26)(24 27)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 12)(11 15)(14 16)(17 20)(18 19)(21 25)(22 28)(23 27)(24 26)(29 32)(30 31)
(1 2)(3 4)(5 6)(7 8)(9 15)(10 16)(11 13)(12 14)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(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 12)(2 14)(3 11)(4 13)(5 16)(6 10)(7 9)(8 15)(17 25)(18 21)(19 27)(20 23)(22 31)(24 29)(26 30)(28 32)

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

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

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

Matrix representation of C24.33D4 in GL6(𝔽5)

010000
100000
000010
004443
001000
000001
,
400000
040000
004000
000400
000040
000004
,
400000
040000
000100
001000
004443
000001
,
100000
010000
004000
000400
000040
000004
,
300000
030000
001000
000400
004443
000111
,
100000
040000
000300
002000
003331
000222

G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,0,0,0,0,4,0,0,0,0,1,4,0,0,0,0,0,3,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,4,0,0,0,1,0,4,0,0,0,0,0,4,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,4,0,0,0,0,4,4,1,0,0,0,0,4,1,0,0,0,0,3,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,3,0,0,0,3,0,3,2,0,0,0,0,3,2,0,0,0,0,1,2] >;

C24.33D4 in GAP, Magma, Sage, TeX

C_2^4._{33}D_4
% in TeX

G:=Group("C2^4.33D4");
// GroupNames label

G:=SmallGroup(128,776);
// by ID

G=gap.SmallGroup(128,776);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,1411,4037]);
// Polycyclic

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

Export

Character table of C24.33D4 in TeX

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