Copied to
clipboard

G = C23.7D4order 64 = 26

7th non-split extension by C23 of D4 acting faithfully

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

Aliases: C23.7D4, C23.4C23, 2+ 1+4.2C2, (C2×C4).7D4, C23⋊C44C2, C2.19C22≀C2, C22⋊C42C22, (C22×C4)⋊2C22, C22.17(C2×D4), (C2×D4).10C22, C22.D41C2, SmallGroup(64,139)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.7D4
C1C2C22C23C2×D42+ 1+4 — C23.7D4
C1C2C23 — C23.7D4
C1C2C23 — C23.7D4
C1C2C23 — C23.7D4

Generators and relations for C23.7D4
 G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=c, ab=ba, ac=ca, dad-1=eae-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >

Subgroups: 161 in 80 conjugacy classes, 27 normal (6 characteristic)
C1, C2, C2 [×6], C4 [×7], C22 [×3], C22 [×8], C2×C4 [×3], C2×C4 [×9], D4 [×9], Q8, C23, C23 [×3], C23, C22⋊C4 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C22×C4, C2×D4 [×3], C2×D4 [×3], C4○D4 [×3], C23⋊C4 [×3], C22.D4 [×3], 2+ 1+4, C23.7D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C23.7D4

Character table of C23.7D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H
 size 1122244444444888
ρ11111111111111111    trivial
ρ211111-11-111-1-11-1-11    linear of order 2
ρ311111-1-11-1-11-1-11-11    linear of order 2
ρ4111111-1-1-1-1-11-1-111    linear of order 2
ρ511111111-1111-1-1-1-1    linear of order 2
ρ611111-11-1-11-1-1-111-1    linear of order 2
ρ711111-1-111-11-11-11-1    linear of order 2
ρ8111111-1-11-1-1111-1-1    linear of order 2
ρ922-2-2200200-200000    orthogonal lifted from D4
ρ10222-2-20-2002000000    orthogonal lifted from D4
ρ1122-22-2200000-20000    orthogonal lifted from D4
ρ1222-22-2-20000020000    orthogonal lifted from D4
ρ1322-2-2200-200200000    orthogonal lifted from D4
ρ14222-2-20200-2000000    orthogonal lifted from D4
ρ154-4000000-2i0002i000    complex faithful
ρ164-40000002i000-2i000    complex faithful

Permutation representations of C23.7D4
On 16 points - transitive group 16T146
Generators in S16
(2 6)(3 9)(4 14)(7 13)(8 10)(12 16)
(1 15)(2 6)(3 13)(4 8)(5 11)(7 9)(10 14)(12 16)
(1 11)(2 12)(3 9)(4 10)(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 10 11 4)(2 3 12 9)(5 8 15 14)(6 13 16 7)

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

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

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

G:=TransitiveGroup(16,146);

On 16 points - transitive group 16T165
Generators in S16
(1 13)(2 8)(3 5)(4 16)(6 12)(7 9)(10 14)(11 15)
(2 10)(4 12)(6 16)(8 14)
(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 13 9 7)(2 6 10 16)(3 15 11 5)(4 8 12 14)

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

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

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

G:=TransitiveGroup(16,165);

On 16 points - transitive group 16T173
Generators in S16
(1 9)(2 10)(3 13)(4 16)(5 11)(6 12)(7 14)(8 15)
(1 5)(4 7)(9 11)(14 16)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7 5 4)(2 3 6 8)(9 14 11 16)(10 15 12 13)

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

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

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

G:=TransitiveGroup(16,173);

On 16 points - transitive group 16T177
Generators in S16
(1 16)(2 10)(3 11)(4 15)(5 14)(6 12)(7 13)(8 9)
(1 3)(2 7)(4 6)(5 8)(9 14)(10 13)(11 16)(12 15)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 2 5 6)(3 7 8 4)(9 10 11 12)(13 16 15 14)

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

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

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

G:=TransitiveGroup(16,177);

C23.7D4 is a maximal subgroup of
C42.14D4  C23.7C24  C24⋊C23  2+ 1+4.C6  C23.S4
 C22⋊C4⋊D2p: C424D4  C426D4  C23.5D12  C22⋊C4⋊D6  C23.5D20  C22⋊C4⋊D10  C23.5D28  C22⋊C4⋊D14 ...
 (C2×D4).D2p: C42.13D4  C23.10C24  2+ 1+4.5S3  2+ 1+4.2D5  2+ 1+4.2D7 ...
C23.7D4 is a maximal quotient of
C24.14D4  (C2×C4).SD16  C24.17D4  C4⋊C4.20D4  C24.22D4  C24.180C23  C24.182C23
 C23.D4p: C23.5D8  C23.5D12  C23.5D20  C23.5D28 ...
 C22⋊C4⋊D2p: C24.31D4  C24.33D4  C22⋊C4⋊D6  C22⋊C4⋊D10  C22⋊C4⋊D14 ...
 (C2×D4).D2p: C4⋊C4.12D4  (C2×C4).5D8  C24.15D4  C24.16D4  C4⋊C4.18D4  C4⋊C4.19D4  C24.18D4  2+ 1+42C4 ...

Matrix representation of C23.7D4 in GL4(𝔽5) generated by

1000
0400
0040
0001
,
0001
0010
0100
1000
,
4000
0400
0040
0004
,
4441
1141
4144
4111
,
4441
4414
4144
1444
G:=sub<GL(4,GF(5))| [1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,1],[0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,1,4,4,4,1,1,1,4,4,4,1,1,1,4,1],[4,4,4,1,4,4,1,4,4,1,4,4,1,4,4,4] >;

C23.7D4 in GAP, Magma, Sage, TeX

C_2^3._7D_4
% in TeX

G:=Group("C2^3.7D4");
// GroupNames label

G:=SmallGroup(64,139);
// by ID

G=gap.SmallGroup(64,139);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,362,255,730]);
// Polycyclic

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

Export

Character table of C23.7D4 in TeX

׿
×
𝔽