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G = C2411D4order 128 = 27

6th semidirect product of C24 and D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C2411D4, C25.63C22, C24.458C23, C23.712C24, C22.4852+ 1+4, C243C427C2, C232D445C2, C23.222(C2×D4), (C22×D4)⋊14C22, C2.59(C233D4), (C22×C4).223C23, C22.444(C22×D4), C23.10D4106C2, C23.11D4126C2, C2.C4242C22, C2.15(C24⋊C22), C2.42(C22.54C24), (C2×C4⋊C4)⋊38C22, (C2×C22≀C2)⋊16C2, (C2×C22⋊C4)⋊32C22, SmallGroup(128,1544)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2411D4
C1C2C22C23C24C22×D4C232D4 — C2411D4
C1C23 — C2411D4
C1C23 — C2411D4
C1C23 — C2411D4

Generators and relations for C2411D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=acd, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 932 in 356 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×10], C22, C22 [×6], C22 [×60], C2×C4 [×30], D4 [×28], C23, C23 [×6], C23 [×60], C22⋊C4 [×24], C4⋊C4 [×3], C22×C4, C22×C4 [×9], C2×D4 [×27], C24 [×9], C24 [×6], C2.C42 [×6], C2×C22⋊C4 [×18], C2×C4⋊C4 [×3], C22≀C2 [×8], C22×D4, C22×D4 [×6], C25, C243C4, C232D4 [×3], C23.10D4 [×6], C23.11D4 [×3], C2×C22≀C2 [×2], C2411D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×6], C233D4 [×3], C22.54C24 [×3], C24⋊C22, C2411D4

Character table of C2411D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I4J
 size 11111111444444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-11-111-11-1-1-111-1    linear of order 2
ρ31111111111-1-1-1-111-1-11-11-1-11-11    linear of order 2
ρ4111111111111111-1-1-1-1-1-1111-1-1    linear of order 2
ρ511111111-1-1-1-111111-11-1-1-11-11-1    linear of order 2
ρ611111111-1-111-1-11-11-1-1-111-1-111    linear of order 2
ρ711111111-1-111-1-111-1111-11-1-1-1-1    linear of order 2
ρ811111111-1-1-1-1111-1-11-111-11-1-11    linear of order 2
ρ911111111-1-1-1-111-1-1111-1-11-11-11    linear of order 2
ρ1011111111-1-111-1-1-1111-1-11-111-1-1    linear of order 2
ρ1111111111-1-111-1-1-1-1-1-111-1-11111    linear of order 2
ρ1211111111-1-1-1-111-11-1-1-1111-111-1    linear of order 2
ρ1311111111111111-1-11-1111-1-1-1-1-1    linear of order 2
ρ141111111111-1-1-1-1-111-1-11-111-1-11    linear of order 2
ρ151111111111-1-1-1-1-1-1-111-1111-11-1    linear of order 2
ρ1611111111111111-11-11-1-1-1-1-1-111    linear of order 2
ρ172-22-22-22-2-222-2-22000000000000    orthogonal lifted from D4
ρ182-22-22-22-22-22-22-2000000000000    orthogonal lifted from D4
ρ192-22-22-22-2-22-222-2000000000000    orthogonal lifted from D4
ρ202-22-22-22-22-2-22-22000000000000    orthogonal lifted from D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ24444-4-44-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ254-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C2411D4
On 32 points
Generators in S32
(1 31)(2 12)(3 29)(4 10)(6 18)(8 20)(9 27)(11 25)(13 24)(15 22)(26 32)(28 30)
(2 32)(4 30)(5 21)(6 18)(7 23)(8 20)(10 28)(12 26)(13 24)(14 17)(15 22)(16 19)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 31)(26 32)(27 29)(28 30)
(1 31)(2 32)(3 29)(4 30)(5 14)(6 15)(7 16)(8 13)(9 27)(10 28)(11 25)(12 26)(17 21)(18 22)(19 23)(20 24)
(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 18)(2 17)(3 20)(4 19)(5 26)(6 25)(7 28)(8 27)(9 13)(10 16)(11 15)(12 14)(21 32)(22 31)(23 30)(24 29)

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

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

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

Matrix representation of C2411D4 in GL10(ℤ)

1000000000
0100000000
00-10000000
0001000000
0000-100000
0000010000
000000-1000
0000000100
0000000010
000000000-1
,
-1000000000
0-100000000
0010000000
0001000000
0000-100000
00000-10000
0000001000
0000000-100
0000000010
000000000-1
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
000000-1000
0000000-100
00000000-10
000000000-1
,
1000000000
0100000000
00-10000000
000-1000000
0000-100000
00000-10000
000000-1000
0000000-100
00000000-10
000000000-1
,
0100000000
-1000000000
0000010000
0000100000
0001000000
0010000000
0000000001
0000000010
0000000-100
000000-1000
,
0-100000000
-1000000000
0000100000
0000010000
0010000000
0001000000
0000000010
0000000001
0000001000
0000000100

G:=sub<GL(10,Integers())| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C2411D4 in GAP, Magma, Sage, TeX

C_2^4\rtimes_{11}D_4
% in TeX

G:=Group("C2^4:11D4");
// GroupNames label

G:=SmallGroup(128,1544);
// by ID

G=gap.SmallGroup(128,1544);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,794,185]);
// Polycyclic

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

Export

Character table of C2411D4 in TeX

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