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G = C23⋊2SD16  order 128 = 27

2nd semidirect product of C23 and SD16 acting via SD16/C2=D4

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

Aliases: C23⋊2SD16, C24.10D4, C4⋊C4.4D4, C23⋊C8⋊10C2, (C2×D4).11D4, Q8⋊D4⋊25C2, C2.10C2≀C22, C23⋊Q8⋊1C2, (C22×C4).44D4, C22⋊C8⋊38C22, C23⋊3D4.1C2, C23.518(C2×D4), C4⋊D4.5C22, (C22×C4).7C23, (C22×Q8)⋊1C22, C2.6(D4.9D4), C2.6(C22⋊SD16), C22.SD16⋊12C2, C22.31(C2×SD16), C22.128C22≀C2, C22.40(C8⋊C22), C2.C42⋊4C22, (C2×C4).196(C2×D4), (C2×C22⋊C4).93C22, SmallGroup(128,333)

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

Derived series
C1 — C22×C4 — C23⋊2SD16
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23⋊3D4 — C23⋊2SD16
Lower central
C1 — C22 — C22×C4 — C23⋊2SD16
Upper central
C1 — C22 — C22×C4 — C23⋊2SD16
Jennings
C1 — C2 — C22 — C22×C4 — C23⋊2SD16

Generators and relations for C23⋊2SD16
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d3 >

Subgroups: 452 in 154 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×D4, C2×Q8, C24, C24, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22≀C2, C4⋊D4, C4⋊D4, C22.D4, C2×SD16, C22×D4, C22×Q8, C23⋊C8, C22.SD16, C23⋊Q8, Q8⋊D4, C23⋊3D4, C23⋊2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16, D4.9D4, C2≀C22, C23⋊2SD16

Character table of C23⋊2SD16

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112244884488888888888
ρ111111111111111111111111    trivial
ρ211111111-1-11111-11-111-1-1-1-1    linear of order 2
ρ311111111-1-111-11-1-1-1-1-11111    linear of order 2
ρ4111111111111-111-11-1-1-1-1-1-1    linear of order 2
ρ5111111-1-11-111-1-1-1111-11-11-1    linear of order 2
ρ6111111-1-1-1111-1-111-11-1-11-11    linear of order 2
ρ7111111-1-1-11111-11-1-1-111-11-1    linear of order 2
ρ8111111-1-11-1111-1-1-11-11-11-11    linear of order 2
ρ92222-2-200022-200-200000000    orthogonal lifted from D4
ρ102222-2-200-20-2200002000000    orthogonal lifted from D4
ρ11222222-2-200-2-202000000000    orthogonal lifted from D4
ρ122222-2-2000-22-200200000000    orthogonal lifted from D4
ρ132222-2-20020-220000-2000000    orthogonal lifted from D4
ρ142222222200-2-20-2000000000    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
ρ194-44-40000000000020-200000    orthogonal lifted from C2≀C22
ρ204-44-400000000000-20200000    orthogonal lifted from C2≀C22
ρ2144-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-4400000000-2i000002i0000    complex lifted from D4.9D4
ρ234-4-44000000002i00000-2i0000    complex lifted from D4.9D4

Smallest permutation representation of C23⋊2SD16
►On 32 points
Generators in S32
(2 29)(3 14)(4 20)(6 25)(7 10)(8 24)(9 22)(11 27)(13 18)(15 31)(19 30)(23 26)

(1 28)(2 18)(3 30)(4 20)(5 32)(6 22)(7 26)(8 24)(9 25)(10 23)(11 27)(12 17)(13 29)(14 19)(15 31)(16 21)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)

(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 28)(18 31)(19 26)(20 29)(21 32)(22 27)(23 30)(24 25)
 
G:=sub<Sym(32)| (2,29)(3,14)(4,20)(6,25)(7,10)(8,24)(9,22)(11,27)(13,18)(15,31)(19,30)(23,26), (1,28)(2,18)(3,30)(4,20)(5,32)(6,22)(7,26)(8,24)(9,25)(10,23)(11,27)(12,17)(13,29)(14,19)(15,31)(16,21), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25)>;
 
G:=Group( (2,29)(3,14)(4,20)(6,25)(7,10)(8,24)(9,22)(11,27)(13,18)(15,31)(19,30)(23,26), (1,28)(2,18)(3,30)(4,20)(5,32)(6,22)(7,26)(8,24)(9,25)(10,23)(11,27)(12,17)(13,29)(14,19)(15,31)(16,21), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,28)(18,31)(19,26)(20,29)(21,32)(22,27)(23,30)(24,25) );
 
G=PermutationGroup([[(2,29),(3,14),(4,20),(6,25),(7,10),(8,24),(9,22),(11,27),(13,18),(15,31),(19,30),(23,26)], [(1,28),(2,18),(3,30),(4,20),(5,32),(6,22),(7,26),(8,24),(9,25),(10,23),(11,27),(12,17),(13,29),(14,19),(15,31),(16,21)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,28),(18,31),(19,26),(20,29),(21,32),(22,27),(23,30),(24,25)]])
 

Matrix representation of C23⋊2SD16 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
100000
010000
000100
001000
000001
000010
,
100000
010000
0016000
0001600
0000160
0000016
,
5120000
550000
009998
008898
009899
009888
,
1600000
010000
001000
0001600
000010
0000016

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,9,8,9,9,0,0,9,8,8,8,0,0,9,9,9,8,0,0,8,8,9,8],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
 

C23⋊2SD16 in GAP, Magma, Sage, TeX 

C_2^3\rtimes_2{\rm SD}_{16}
 
% in TeX
 
G:=Group("C2^3:2SD16");
 
// GroupNames label
 
G:=SmallGroup(128,333);
 
// by ID
 
G=gap.SmallGroup(128,333);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

Export

Character table of C23⋊2SD16 in TeX

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