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G = C23⋊D8  order 128 = 27

The semidirect product of C23 and D8 acting via D8/C2=D4

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

Aliases: C23⋊D8, C24.7D4, C4⋊C4⋊1D4, (C2×D4)⋊1D4, C23⋊C8⋊3C2, C22⋊D8⋊1C2, C2.4C2≀C22, C23⋊2D4⋊1C2, C23⋊3D4⋊1C2, C22⋊C8⋊1C22, C22.13(C2×D8), (C22×C4).40D4, C2.6(D4⋊4D4), C2.6(C22⋊D8), C23.512(C2×D4), C22.SD16⋊4C2, C4⋊D4.1C22, (C22×D4)⋊1C22, (C22×C4).1C23, C22.122C22≀C2, C22.38(C8⋊C22), C2.C42⋊3C22, (C2×C4).190(C2×D4), (C2×C22⋊C4).90C22, SmallGroup(128,327)

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

Derived series
C1 — C22×C4 — C23⋊D8
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23⋊3D4 — C23⋊D8
Lower central
C1 — C22 — C22×C4 — C23⋊D8
Upper central
C1 — C22 — C22×C4 — C23⋊D8
Jennings
C1 — C2 — C22 — C22×C4 — C23⋊D8

Generators and relations for C23⋊D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, dad-1=eae=ab=ba, ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 588 in 184 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C2×D8, C22×D4, C22×D4, C23⋊C8, C22.SD16, C23⋊2D4, C22⋊D8, C23⋊3D4, C23⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C22≀C2, C2×D8, C8⋊C22, C22⋊D8, D4⋊4D4, C2≀C22, C23⋊D8

Character table of C23⋊D8

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G8A8B8C8D
 size 11112244888844888888888
ρ111111111111111111111111    trivial
ρ2111111-1-11-1-1-1111-1-1111-11-1    linear of order 2
ρ31111111111-1-111-1111-1-1-1-1-1    linear of order 2
ρ4111111-1-11-11111-1-1-11-1-11-11    linear of order 2
ρ5111111-1-1-11-1-11111-1-11-11-11    linear of order 2
ρ611111111-1-111111-11-11-1-1-1-1    linear of order 2
ρ7111111-1-1-111111-11-1-1-11-11-1    linear of order 2
ρ811111111-1-1-1-111-1-11-1-11111    linear of order 2
ρ9222222-2-20000-2-2002000000    orthogonal lifted from D4
ρ102222-2-2000200-220-20000000    orthogonal lifted from D4
ρ11222222220000-2-200-2000000    orthogonal lifted from D4
ρ122222-2-200-20002-2000200000    orthogonal lifted from D4
ρ132222-2-2000-200-22020000000    orthogonal lifted from D4
ρ142222-2-20020002-2000-200000    orthogonal lifted from D4
ρ1522-2-2-22-2200000000000√2-√2-√2√2    orthogonal lifted from D8
ρ1622-2-2-222-200000000000√2√2-√2-√2    orthogonal lifted from D8
ρ1722-2-2-22-2200000000000-√2√2√2-√2    orthogonal lifted from D8
ρ1822-2-2-222-200000000000-√2-√2√2√2    orthogonal lifted from D8
ρ194-4-4400000000002000-20000    orthogonal lifted from D4⋊4D4
ρ204-44-40000002-200000000000    orthogonal lifted from C2≀C22
ρ214-44-4000000-2200000000000    orthogonal lifted from C2≀C22
ρ2244-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ234-4-440000000000-200020000    orthogonal lifted from D4⋊4D4

Permutation representations of C23⋊D8
►On 16 points - transitive group 16T409
Generators in S16
(1 5)(2 6)(3 12)(4 13)(7 16)(8 9)(10 14)(11 15)

(2 15)(4 9)(6 11)(8 13)

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

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

(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)
 
G:=sub<Sym(16)| (1,5)(2,6)(3,12)(4,13)(7,16)(8,9)(10,14)(11,15), (2,15)(4,9)(6,11)(8,13), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)>;
 
G:=Group( (1,5)(2,6)(3,12)(4,13)(7,16)(8,9)(10,14)(11,15), (2,15)(4,9)(6,11)(8,13), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15) );
 
G=PermutationGroup([[(1,5),(2,6),(3,12),(4,13),(7,16),(8,9),(10,14),(11,15)], [(2,15),(4,9),(6,11),(8,13)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15)]])
 
G:=TransitiveGroup(16,409);
 

Matrix representation of C23⋊D8 ►in GL6(𝔽17)

100000
010000
001000
00161600
000010
00001616
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
11110000
300000
00001615
000001
001000
00161600
,
1600000
110000
001000
00161600
00001615
000001

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[11,3,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,16,0,0,0,0,0,15,1,0,0],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1] >;
 

C23⋊D8 in GAP, Magma, Sage, TeX 

C_2^3\rtimes D_8
 
% in TeX
 
G:=Group("C2^3:D8");
 
// GroupNames label
 
G:=SmallGroup(128,327);
 
// by ID
 
G=gap.SmallGroup(128,327);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,422,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,d*a*d^-1=e*a*e=a*b=b*a,a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C23⋊D8 in TeX

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