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G = C23⋊C8  order 64 = 26

The semidirect product of C23 and C8 acting via C8/C2=C4

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

Aliases: C23⋊C8, C24.1C4, C22.2M4(2), C22⋊C8⋊1C2, (C2×C4).91D4, C22.2(C2×C8), (C22×C4).1C4, C2.1(C23⋊C4), C2.3(C22⋊C8), C23.20(C2×C4), C2.1(C4.D4), (C22×C4).1C22, C22.19(C22⋊C4), (C2×C22⋊C4).1C2, SmallGroup(64,4)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C23⋊C8
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C22⋊C4 — C23⋊C8
Lower central
C1 — C2 — C22 — C23⋊C8
Upper central
C1 — C22 — C22×C4 — C23⋊C8
Jennings
C1 — C2 — C22 — C22×C4 — C23⋊C8

Generators and relations for C23⋊C8
 G = < a,b,c,d | a2=b2=c2=d8=1, ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, cd=dc >

C1
C2
C2
C2
2C2
2C2
4C2
4C2
C22
C22
C22
2C22
2C4
2C22
2C22
2C4
2C22
4C22
4C22
4C22
4C22
4C22
4C4
4C22
C23
C2×C4
C23
C23
C2×C4
2C2×C4
2C23
2C23
4C8
4C23
4C2×C4
4C2×C4
4C8
4C23
C22×C4
C22×C4
C24
2C22⋊C4
2C2×C8
2C2×C8
2C22⋊C4
C22⋊C8
C2×C22⋊C4
C22⋊C8
C23⋊C8

Character table of C23⋊C8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F8A8B8C8D8E8F8G8H
 size 1111224422224444444444
ρ11111111111111111111111    trivial
ρ211111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111-1-11111-1-1111-1-1-1-11    linear of order 2
ρ4111111-1-11111-1-1-1-1-11111-1    linear of order 2
ρ5111111-1-1-1-1-1-111ii-ii-i-ii-i    linear of order 4
ρ6111111-1-1-1-1-1-111-i-ii-iii-ii    linear of order 4
ρ711111111-1-1-1-1-1-1ii-i-iii-i-i    linear of order 4
ρ811111111-1-1-1-1-1-1-i-iii-i-iii    linear of order 4
ρ911-1-1-111-1i-ii-i-iiζ85ζ8ζ87ζ87ζ85ζ8ζ83ζ83    linear of order 8
ρ1011-1-1-111-1-ii-iii-iζ83ζ87ζ8ζ8ζ83ζ87ζ85ζ85    linear of order 8
ρ1111-1-1-111-1i-ii-i-iiζ8ζ85ζ83ζ83ζ8ζ85ζ87ζ87    linear of order 8
ρ1211-1-1-111-1-ii-iii-iζ87ζ83ζ85ζ85ζ87ζ83ζ8ζ8    linear of order 8
ρ1311-1-1-11-11i-ii-ii-iζ85ζ8ζ87ζ83ζ8ζ85ζ87ζ83    linear of order 8
ρ1411-1-1-11-11-ii-ii-iiζ87ζ83ζ85ζ8ζ83ζ87ζ85ζ8    linear of order 8
ρ1511-1-1-11-11i-ii-ii-iζ8ζ85ζ83ζ87ζ85ζ8ζ83ζ87    linear of order 8
ρ1611-1-1-11-11-ii-ii-iiζ83ζ87ζ8ζ85ζ87ζ83ζ8ζ85    linear of order 8
ρ172222-2-20022-2-20000000000    orthogonal lifted from D4
ρ182222-2-200-2-2220000000000    orthogonal lifted from D4
ρ1922-2-22-200-2i2i2i-2i0000000000    complex lifted from M4(2)
ρ2022-2-22-2002i-2i-2i2i0000000000    complex lifted from M4(2)
ρ214-44-4000000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-44000000000000000000    orthogonal lifted from C4.D4

Permutation representations of C23⋊C8
►On 16 points - transitive group 16T84
Generators in S16
(2 10)(3 15)(4 8)(6 14)(7 11)(12 16)

(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)

(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (2,10)(3,15)(4,8)(6,14)(7,11)(12,16), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (2,10)(3,15)(4,8)(6,14)(7,11)(12,16), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(2,10),(3,15),(4,8),(6,14),(7,11),(12,16)], [(1,5),(2,10),(3,7),(4,12),(6,14),(8,16),(9,13),(11,15)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,84);
 

►On 16 points - transitive group 16T85
Generators in S16
(1 5)(2 10)(3 11)(4 8)(6 14)(7 15)(9 13)(12 16)

(2 14)(4 16)(6 10)(8 12)

(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
 
G:=sub<Sym(16)| (1,5)(2,10)(3,11)(4,8)(6,14)(7,15)(9,13)(12,16), (2,14)(4,16)(6,10)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
 
G:=Group( (1,5)(2,10)(3,11)(4,8)(6,14)(7,15)(9,13)(12,16), (2,14)(4,16)(6,10)(8,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
 
G=PermutationGroup([[(1,5),(2,10),(3,11),(4,8),(6,14),(7,15),(9,13),(12,16)], [(2,14),(4,16),(6,10),(8,12)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
 
G:=TransitiveGroup(16,85);
 

C23⋊C8 is a maximal subgroup of
 C24⋊C8  C23.2M4(2)  C24.D4  C23.4D8  C23.Q16  C24.4D4  C23.8M4(2)  C25.3C4  C42.42D4  C23⋊M4(2)  C42.43D4  C23⋊C8⋊C2  C42.395D4  C24.(C2×C4)  C42.372D4  C23⋊D8  C23⋊SD16  C24.9D4  C23⋊2SD16  C23⋊Q16  C24.12D4  C23.5D8  C24.14D4  C24.15D4  C24.16D4  C24.17D4  C24.18D4
 (C22×C2p)⋊C8: C23.15M4(2)  C42.371D4  C24.3Dic3  C24.Dic5  C24.F5  C24.Dic7 ...
 C2p.(C22⋊C8): C42.393D4  (C22×S3)⋊C8  C5⋊3(C23⋊C8)  C5⋊(C23⋊C8)  (C22×D7)⋊C8 ...
C23⋊C8 is a maximal quotient of
 (C2×Q8)⋊C8  C23⋊C16  C23.M4(2)  C24⋊C8  (C2×D4)⋊C8  (C2×C42).C4  C24.C8  C23.1M4(2)  C5⋊(C23⋊C8)
 (C22×C2p)⋊C8: C23.19C42  C23.15M4(2)  C24.3Dic3  C24.Dic5  C24.F5  C24.Dic7 ...
 (C2×C4).D4p: (C2×C4).98D8  (C22×S3)⋊C8  C5⋊3(C23⋊C8)  (C22×D7)⋊C8 ...

Matrix representation of C23⋊C8 ►in GL6(𝔽17)

1600000
010000
0016000
009100
000010
0000816
,
1600000
0160000
0016000
0001600
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
010000
1300000
000010
000001
0013100
000400

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,9,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,1,4,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C23⋊C8 in GAP, Magma, Sage, TeX 

C_2^3\rtimes C_8
 
% in TeX
 
G:=Group("C2^3:C8");
 
// GroupNames label
 
G:=SmallGroup(64,4);
 
// by ID
 
G=gap.SmallGroup(64,4);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,297,117]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,d*b*d^-1=b*c=c*b,c*d=d*c>;
 
// generators/relations
 

Export

Subgroup lattice of C23⋊C8 in TeX
Character table of C23⋊C8 in TeX

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