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G = C23.9D4order 64 = 26

2nd non-split extension by C23 of D4 acting via D4/C2=C22

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

Aliases: C23.9D4, C23.2Q8, C22.1C42, C24.1C22, C22⋊C43C4, (C22×C4)⋊1C4, C23.5(C2×C4), C2.3(C23⋊C4), C22.4(C4⋊C4), C22.7(C22⋊C4), C2.7(C2.C42), (C2×C22⋊C4).2C2, SmallGroup(64,23)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.9D4
C1C2C22C23C24C2×C22⋊C4 — C23.9D4
C1C2C22 — C23.9D4
C1C22C24 — C23.9D4
C1C2C24 — C23.9D4

Generators and relations for C23.9D4
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=abc, ab=ba, eae-1=ac=ca, ad=da, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd-1 >

Subgroups: 149 in 71 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C23 [×3], C23 [×4], C23 [×2], C22⋊C4 [×4], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C24, C2×C22⋊C4, C2×C22⋊C4 [×2], C23.9D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4

Character table of C23.9D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1111-1-1-11-1    linear of order 2
ρ41111111111-1-1-1-1-1-1-1111-11    linear of order 2
ρ51-11-11-111-1-1-iii-iii-i11-1-i-1    linear of order 4
ρ61-11-11-111-1-1i-i-iiii-i-1-11-i1    linear of order 4
ρ71111-1-1-111-111-1-1i-i-ii-i-iii    linear of order 4
ρ81-11-11-111-1-1i-i-ii-i-ii11-1i-1    linear of order 4
ρ91-11-11-111-1-1-iii-i-i-ii-1-11i1    linear of order 4
ρ101111-1-1-111-111-1-1-iii-iii-i-i    linear of order 4
ρ111-11-1-11-11-11i-ii-i-11-1-ii-i1i    linear of order 4
ρ121-11-1-11-11-11-ii-ii-11-1i-ii1-i    linear of order 4
ρ131111-1-1-111-1-1-111i-i-i-iiii-i    linear of order 4
ρ141-11-1-11-11-11-ii-ii1-11-ii-i-1i    linear of order 4
ρ151-11-1-11-11-11i-ii-i1-11i-ii-1-i    linear of order 4
ρ161111-1-1-111-1-1-111-iiii-i-i-ii    linear of order 4
ρ172222-222-2-2-2000000000000    orthogonal lifted from D4
ρ182-22-222-2-22-2000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-22000000000000    orthogonal lifted from D4
ρ202-22-2-2-22-222000000000000    symplectic lifted from Q8, Schur index 2
ρ2144-4-4000000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-44000000000000000000    orthogonal lifted from C23⋊C4

Permutation representations of C23.9D4
On 16 points - transitive group 16T77
Generators in S16
(5 15)(6 16)(7 13)(8 14)
(1 3)(2 12)(4 10)(5 13)(6 8)(7 15)(9 11)(14 16)
(1 9)(2 10)(3 11)(4 12)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 11 13)(2 14 4 16)(3 7 9 15)(6 10 8 12)

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

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

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

G:=TransitiveGroup(16,77);

On 16 points - transitive group 16T88
Generators in S16
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)
(1 3)(2 10)(4 12)(5 15)(6 8)(7 13)(9 11)(14 16)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5 11 13)(2 14)(3 7 9 15)(4 16)(6 12)(8 10)

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

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

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

G:=TransitiveGroup(16,88);

On 16 points - transitive group 16T91
Generators in S16
(5 15)(6 16)(7 13)(8 14)
(2 10)(4 12)(5 15)(7 13)
(1 9)(2 10)(3 11)(4 12)(5 15)(6 16)(7 13)(8 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 7 9 13)(2 16)(3 5 11 15)(4 14)(6 10)(8 12)

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

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

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

G:=TransitiveGroup(16,91);

On 16 points - transitive group 16T96
Generators in S16
(1 7)(2 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(1 7)(2 11)(3 5)(4 9)(6 14)(8 16)(10 15)(12 13)
(1 15)(2 16)(3 13)(4 14)(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 5 15 12)(3 7 13 10)(6 9)(8 11)

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

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

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

G:=TransitiveGroup(16,96);

C23.9D4 is a maximal subgroup of
C24.162C23  C4×C23⋊C4  C24.167C23  C24.169C23  C24.C23  C24.174C23  C24.176C23  C24⋊D4  C242Q8  C24.180C23  C24⋊Q8  C24.182C23  C24.4C23  (C22×F5)⋊C4  C22⋊F5⋊C4
 C24.D2p: C24.D4  C23.4D8  C23.Q16  C24.4D4  C24.68D4  C24.22D4  C25.C22  C24.26D4 ...
C23.9D4 is a maximal quotient of
C23.19C42  C23.21C42  C24.2Q8  C24.3Q8  C23.C42  C23.8C42  C23.2C42  C23.3C42  (C2×Q8).Q8  (C22×C8)⋊C4  (C22×F5)⋊C4  C22⋊F5⋊C4
 C24.D2p: C24.46D4  C23.8D8  C23.30D8  C24.48D4  C24.4Q8  C24.5D4  C24.6D4  C24.12D6 ...

Matrix representation of C23.9D4 in GL6(𝔽5)

100000
010000
000400
004000
001442
000001
,
400000
040000
001000
000100
000040
004104
,
100000
010000
004000
000400
000040
000004
,
440000
210000
000010
004113
004000
004104
,
330000
020000
000040
004113
000400
004014

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

C23.9D4 in GAP, Magma, Sage, TeX

C_2^3._9D_4
% in TeX

G:=Group("C2^3.9D4");
// GroupNames label

G:=SmallGroup(64,23);
// by ID

G=gap.SmallGroup(64,23);
# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,489]);
// Polycyclic

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

Export

Character table of C23.9D4 in TeX

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