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G = C23⋊SD16  order 128 = 27

1st semidirect product of C23 and SD16 acting via SD16/C2=D4

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

Aliases: C24.8D4, C23⋊1SD16, C4⋊C4⋊2D4, (C2×Q8)⋊1D4, C23⋊C8⋊9C2, C2.5C2≀C22, C23⋊2Q8⋊1C2, (C22×C4).41D4, C22⋊SD16⋊25C2, C2.7(D4⋊4D4), C2.6(Q8⋊D4), C23.513(C2×D4), C23⋊2D4.2C2, (C22×C4).2C23, C22⋊Q8.1C22, C22.12(C2×SD16), (C22×D4).3C22, C22.123C22≀C2, C23.31D4⋊10C2, C22⋊C8.109C22, C22.27(C8.C22), C2.C42.11C22, (C2×C4).191(C2×D4), (C2×C22⋊C4).91C22, SmallGroup(128,328)

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

Derived series
C1 — C22×C4 — C23⋊SD16
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23⋊2Q8 — C23⋊SD16
Lower central
C1 — C22 — C22×C4 — C23⋊SD16
Upper central
C1 — C22 — C22×C4 — C23⋊SD16
Jennings
C1 — C2 — C22 — C22×C4 — C23⋊SD16

Generators and relations for C23⋊SD16
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, eae=ab=ba, ac=ca, dad-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 460 in 158 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C24, C2.C42, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22⋊Q8, C22⋊Q8, C2×SD16, C22×D4, C22×D4, C23⋊C8, C23.31D4, C23⋊2D4, C22⋊SD16, C23⋊2Q8, C23⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8.C22, Q8⋊D4, D4⋊4D4, C2≀C22, C23⋊SD16

Character table of C23⋊SD16

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112244884488888888888
ρ111111111111111111111111    trivial
ρ2111111-1-1-1-1111-11-1-111-11-11    linear of order 2
ρ3111111-1-11111-11-1-11-1-1-11-11    linear of order 2
ρ411111111-1-111-1-1-11-1-1-11111    linear of order 2
ρ511111111-1-111-111111-1-1-1-1-1    linear of order 2
ρ6111111-1-11111-1-11-1-11-11-11-1    linear of order 2
ρ7111111-1-1-1-11111-1-11-111-11-1    linear of order 2
ρ81111111111111-1-11-1-11-1-1-1-1    linear of order 2
ρ9222222-2-200-2-200020000000    orthogonal lifted from D4
ρ102222-2-200002-20-2002000000    orthogonal lifted from D4
ρ112222-2-20000-2200-200200000    orthogonal lifted from D4
ρ122222-2-200002-20200-2000000    orthogonal lifted from D4
ρ132222222200-2-2000-20000000    orthogonal lifted from D4
ρ142222-2-20000-2200200-200000    orthogonal lifted from D4
ρ1522-2-2-222-200000000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1622-2-2-22-2200000000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ1722-2-2-222-200000000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1822-2-2-22-2200000000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ194-4-4400000000-20000020000    orthogonal lifted from D4⋊4D4
ρ204-44-40000-220000000000000    orthogonal lifted from C2≀C22
ρ214-4-4400000000200000-20000    orthogonal lifted from D4⋊4D4
ρ224-44-400002-20000000000000    orthogonal lifted from C2≀C22
ρ2344-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2

Permutation representations of C23⋊SD16
►On 16 points - transitive group 16T405
Generators in S16
(2 10)(3 11)(6 14)(7 15)

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

(1 9)(2 10)(3 11)(4 12)(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 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)
 
G:=sub<Sym(16)| (2,10)(3,11)(6,14)(7,15), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10)>;
 
G:=Group( (2,10)(3,11)(6,14)(7,15), (2,10)(4,12)(6,14)(8,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,13)(2,16)(3,11)(4,14)(5,9)(6,12)(7,15)(8,10) );
 
G=PermutationGroup([[(2,10),(3,11),(6,14),(7,15)], [(2,10),(4,12),(6,14),(8,16)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,13),(2,16),(3,11),(4,14),(5,9),(6,12),(7,15),(8,10)]])
 
G:=TransitiveGroup(16,405);
 

Matrix representation of C23⋊SD16 ►in GL6(𝔽17)

1600000
0160000
001000
0001600
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
750000
700000
000010
000001
000100
001000
,
1600000
1510000
001000
000100
000001
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[7,7,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[16,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C23⋊SD16 in GAP, Magma, Sage, TeX 

C_2^3\rtimes {\rm SD}_{16}
 
% in TeX
 
G:=Group("C2^3:SD16");
 
// GroupNames label
 
G:=SmallGroup(128,328);
 
// by ID
 
G=gap.SmallGroup(128,328);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,232,422,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,e*a*e=a*b=b*a,a*c=c*a,d*a*d^-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=d^3>;
 
// generators/relations
 

Export

Character table of C23⋊SD16 in TeX

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