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G = C233D4order 64 = 26

2nd semidirect product of C23 and D4 acting via D4/C2=C22

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

Aliases: C233D4, C242C22, C23.10C23, C22.28C24, C2.32+ 1+4, C4⋊D45C2, C4⋊C43C22, C22≀C22C2, (C22×D4)⋊6C2, (C2×D4)⋊13C22, C22⋊C43C22, (C2×C4).16C23, (C22×C4)⋊7C22, C22.20(C2×D4), C2.13(C22×D4), C22.D42C2, (C2×C22⋊C4)⋊12C2, SmallGroup(64,215)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C233D4
C1C2C22C23C24C22×D4 — C233D4
C1C22 — C233D4
C1C22 — C233D4
C1C22 — C233D4

Generators and relations for C233D4
 G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 321 in 173 conjugacy classes, 81 normal (7 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C2×C4 [×8], C2×C4 [×6], D4 [×20], C23, C23 [×10], C23 [×10], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4 [×4], C2×D4 [×12], C2×D4 [×8], C24, C24 [×2], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C233D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], C233D4

Character table of C233D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ2111111111111-1-1-1-11-1-1-1-11    linear of order 2
ρ311111-11-1-1-1-1-11-1-1-111-1111    linear of order 2
ρ411111-11-1-1-1-1-1-11111-11-1-11    linear of order 2
ρ51111-1-1-11-111-1-1-1-11-1-11111    linear of order 2
ρ61111-1-1-11-111-1111-1-11-1-1-11    linear of order 2
ρ71111-11-1-11-1-11-111-1-1-1-1111    linear of order 2
ρ81111-11-1-11-1-111-1-11-111-1-11    linear of order 2
ρ91111111111-1-1-1-111-11-1-11-1    linear of order 2
ρ101111111111-1-111-1-1-1-111-1-1    linear of order 2
ρ1111111-11-1-1-111-11-1-1-111-11-1    linear of order 2
ρ1211111-11-1-1-1111-111-1-1-11-1-1    linear of order 2
ρ131111-1-1-11-11-1111-111-1-1-11-1    linear of order 2
ρ141111-1-1-11-11-11-1-11-11111-1-1    linear of order 2
ρ151111-11-1-11-11-11-11-11-11-11-1    linear of order 2
ρ161111-11-1-11-11-1-11-1111-11-1-1    linear of order 2
ρ172-22-2-2-2222-2000000000000    orthogonal lifted from D4
ρ182-22-22-2-2-222000000000000    orthogonal lifted from D4
ρ192-22-222-22-2-2000000000000    orthogonal lifted from D4
ρ202-22-2-222-2-22000000000000    orthogonal lifted from D4
ρ2144-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-44000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of C233D4
On 16 points - transitive group 16T87
Generators in S16
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 3)(2 16)(4 14)(5 7)(6 11)(8 9)(10 12)(13 15)
(1 13)(2 14)(3 15)(4 16)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 2)(3 4)(5 9)(6 12)(7 11)(8 10)(13 14)(15 16)

G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,3)(2,16)(4,14)(5,7)(6,11)(8,9)(10,12)(13,15), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,9)(6,12)(7,11)(8,10)(13,14)(15,16)>;

G:=Group( (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,3)(2,16)(4,14)(5,7)(6,11)(8,9)(10,12)(13,15), (1,13)(2,14)(3,15)(4,16)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,2)(3,4)(5,9)(6,12)(7,11)(8,10)(13,14)(15,16) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14)], [(1,3),(2,16),(4,14),(5,7),(6,11),(8,9),(10,12),(13,15)], [(1,13),(2,14),(3,15),(4,16),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,2),(3,4),(5,9),(6,12),(7,11),(8,10),(13,14),(15,16)])

G:=TransitiveGroup(16,87);

On 16 points - transitive group 16T119
Generators in S16
(5 10)(6 11)(7 12)(8 9)
(2 14)(4 16)(5 10)(7 12)
(1 13)(2 14)(3 15)(4 16)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 14)(10 13)(11 16)(12 15)

G:=sub<Sym(16)| (5,10)(6,11)(7,12)(8,9), (2,14)(4,16)(5,10)(7,12), (1,13)(2,14)(3,15)(4,16)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,14)(10,13)(11,16)(12,15)>;

G:=Group( (5,10)(6,11)(7,12)(8,9), (2,14)(4,16)(5,10)(7,12), (1,13)(2,14)(3,15)(4,16)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,14)(10,13)(11,16)(12,15) );

G=PermutationGroup([(5,10),(6,11),(7,12),(8,9)], [(2,14),(4,16),(5,10),(7,12)], [(1,13),(2,14),(3,15),(4,16),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5),(2,8),(3,7),(4,6),(9,14),(10,13),(11,16),(12,15)])

G:=TransitiveGroup(16,119);

C233D4 is a maximal subgroup of
C232SD16  C23.5D8  C24.16D4  C24.28D4  C24.31D4  C24.33D4  C24.36D4  C22.38C25  C22.73C25  C22.74C25  C22.79C25  C22.80C25  C4⋊2+ 1+4  C42⋊C23  C22.122C25  C22.123C25  C22.134C25  C22.149C25
 C23⋊D4p: C23⋊D8  C234D12  C233D20  C233D28 ...
 C24⋊D2p: C24⋊D4  C24⋊C23  C247D6  C2412D6  C243D10  C248D10  C242D14  C247D14 ...
 C2p.2+ 1+4: C22.48C25  C22.94C25  C22.126C25  C22.131C25  C22.132C25  C22.147C25  C6.372+ 1+4  C6.1202+ 1+4 ...
C233D4 is a maximal quotient of
C24.90D4  C24.91D4  C23.203C24  C24.195C23  C24.198C23  C24.94D4  C24.95D4  C24.96D4  C23.434C24  C23.439C24  C23.443C24  C23.461C24  C24.583C23  C23.479C24  C23.483C24  C23.491C24  C23.500C24  C23.502C24  C24.587C23  C24.97D4  C245Q8  C23.527C24  C23.530C24  C23.535C24  C24.374C23  C24.592C23  C23.556C24  C23.559C24  C24.377C23  C24.378C23  C23.568C24  C23.569C24  C23.571C24  C23.573C24  C23.578C24  C25⋊C22  C24.389C23  C23.584C24  C24.393C23  C24.395C23  C23.597C24  C24.406C23  C24.407C23  C23.608C24  C24.411C23  C23.617C24  C23.624C24  C24.420C23  C24.421C23  C23.630C24  C23.631C24  C23.632C24  C23.633C24  C23.634C24  C24.459C23  C23.714C24  C23.715C24  C23.716C24  C24.462C23
 C24⋊D2p: C247D4  C248D4  C249D4  C2410D4  C2411D4  C247D6  C2412D6  C243D10 ...
 C2p.2+ 1+4: C233D8  C234SD16  C24.121D4  C233Q16  C24.123D4  C24.124D4  C24.125D4  C24.126D4 ...

Matrix representation of C233D4 in GL6(ℤ)

100000
010000
000100
001000
00000-1
0000-10
,
100000
010000
00-1000
000-100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
010000
-100000
000010
00000-1
00-1000
000100
,
0-10000
-100000
000010
000001
001000
000100

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,-1,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C233D4 in GAP, Magma, Sage, TeX

C_2^3\rtimes_3D_4
% in TeX

G:=Group("C2^3:3D4");
// GroupNames label

G:=SmallGroup(64,215);
// by ID

G=gap.SmallGroup(64,215);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650,188,579]);
// Polycyclic

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

Export

Character table of C233D4 in TeX

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