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G = C24.31D4order 128 = 27

31st non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.31D4, C24.178C23, C22⋊C46D4, (C22×C4)⋊10D4, C23.584(C2×D4), C233D4.5C2, C22.41C22≀C2, C23.9D412C2, C23.34D48C2, (C23×C4).26C22, C23.127(C4○D4), C22.12(C41D4), C22.65(C4⋊D4), C2.27(C232D4), (C22×D4).71C22, C2.16(C23.7D4), (C2×C23⋊C4)⋊9C2, (C2×C22.D4)⋊1C2, (C2×C22⋊C4).103C22, SmallGroup(128,754)

Series: Derived Chief Lower central Upper central Jennings

C1C24 — C24.31D4
C1C2C22C23C24C22×D4C2×C22.D4 — C24.31D4
C1C2C24 — C24.31D4
C1C22C24 — C24.31D4
C1C2C24 — C24.31D4

Generators and relations for C24.31D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, faf-1=ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 528 in 215 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×11], C22 [×3], C22 [×4], C22 [×22], C2×C4 [×31], D4 [×10], C23 [×3], C23 [×4], C23 [×12], C22⋊C4 [×4], C22⋊C4 [×18], C4⋊C4 [×10], C22×C4 [×6], C22×C4 [×9], C2×D4 [×14], C24, C24 [×2], C2.C42 [×2], C23⋊C4 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C22.D4 [×10], C23×C4, C22×D4 [×2], C23.9D4, C23.34D4, C2×C23⋊C4 [×2], C2×C22.D4 [×2], C233D4, C24.31D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, C23.7D4 [×2], C24.31D4

Character table of C24.31D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 11112222228844448888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-11111-1-1-11-11-11-11    linear of order 2
ρ3111111111111-1-1-1-11-1-11-1-1-1-111    linear of order 2
ρ41111111111-1-1-1-1-1-1-11111-11-1-11    linear of order 2
ρ511111111111-11111-111-1-1-1-1-11-1    linear of order 2
ρ61111111111-1111111-1-1-11-11-1-1-1    linear of order 2
ρ711111111111-1-1-1-1-1-1-1-1-111111-1    linear of order 2
ρ81111111111-11-1-1-1-1111-1-11-11-1-1    linear of order 2
ρ92222-22-22-2-2000000000200000-2    orthogonal lifted from D4
ρ102222-22-22-2-2000000000-2000002    orthogonal lifted from D4
ρ112-2-222-222-2-200000002-20000000    orthogonal lifted from D4
ρ122222-2-22-22-220000000000000-20    orthogonal lifted from D4
ρ132-2-22-222-2-2200-2-2220000000000    orthogonal lifted from D4
ρ1422222-2-2-2-22020000-2000000000    orthogonal lifted from D4
ρ152-2-22-222-2-220022-2-20000000000    orthogonal lifted from D4
ρ162-2-222-222-2-20000000-220000000    orthogonal lifted from D4
ρ172222-2-22-22-2-2000000000000020    orthogonal lifted from D4
ρ182-2-22-2-2-22220000000000-202000    orthogonal lifted from D4
ρ1922222-2-2-2-220-200002000000000    orthogonal lifted from D4
ρ202-2-22-2-2-2222000000000020-2000    orthogonal lifted from D4
ρ212-2-2222-2-22-2000000000002i0-2i00    complex lifted from C4○D4
ρ222-2-2222-2-22-200000000000-2i02i00    complex lifted from C4○D4
ρ2344-4-400000000-2i2i-2i2i0000000000    complex lifted from C23.7D4
ρ2444-4-4000000002i-2i2i-2i0000000000    complex lifted from C23.7D4
ρ254-44-4000000002i-2i-2i2i0000000000    complex lifted from C23.7D4
ρ264-44-400000000-2i2i2i-2i0000000000    complex lifted from C23.7D4

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

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

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

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

Matrix representation of C24.31D4 in GL6(𝔽5)

030000
200000
000010
000001
001000
000100
,
400000
040000
001000
003400
000010
000034
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
002200
000300
000033
000042
,
100000
040000
002000
000200
000020
000013

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

C24.31D4 in GAP, Magma, Sage, TeX

C_2^4._{31}D_4
% in TeX

G:=Group("C2^4.31D4");
// GroupNames label

G:=SmallGroup(128,754);
// by ID

G=gap.SmallGroup(128,754);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,521,4037]);
// Polycyclic

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

Export

Character table of C24.31D4 in TeX

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