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G = C23⋊Q16  order 128 = 27

The semidirect product of C23 and Q16 acting via Q16/C2=D4

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

Aliases: C23⋊Q16, C24.11D4, C4⋊C4.5D4, C23⋊C8.6C2, (C2×Q8).11D4, C2.11C2≀C22, C22⋊Q16⋊1C2, (C22×C4).45D4, C23.519(C2×D4), C22⋊C8.2C22, (C22×C4).8C23, C23⋊Q8.2C2, C22.11(C2×Q16), C23⋊2Q8.1C2, C22⋊Q8.5C22, C2.7(D4.9D4), C23.31D4⋊6C2, C2.6(C22⋊Q16), C22.129C22≀C2, (C22×Q8).2C22, C22.29(C8.C22), C2.C42.16C22, (C2×C4).197(C2×D4), (C2×C22⋊C4).94C22, SmallGroup(128,334)

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

Derived series
C1 — C22×C4 — C23⋊Q16
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23⋊2Q8 — C23⋊Q16
Lower central
C1 — C22 — C22×C4 — C23⋊Q16
Upper central
C1 — C22 — C22×C4 — C23⋊Q16
Jennings
C1 — C2 — C22 — C22×C4 — C23⋊Q16

Generators and relations for C23⋊Q16
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=d4, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 324 in 128 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C22⋊C4, C2×C22⋊C4, C22⋊Q8, C22⋊Q8, C2×Q16, C22×Q8, C23⋊C8, C23.31D4, C23⋊Q8, C22⋊Q16, C23⋊2Q8, C23⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C8.C22, C22⋊Q16, D4.9D4, C2≀C22, C23⋊Q16

Character table of C23⋊Q16

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ2111111-1-111-11-11-11-11-1-11-11    linear of order 2
ρ311111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-11-111-11-11-11-1    linear of order 2
ρ51111111111-11111-11-1-1-1-1-1-1    linear of order 2
ρ6111111-1-11111-11-1-1-1-111-11-1    linear of order 2
ρ71111111111-1-1-1-1-1-11-1-11111    linear of order 2
ρ8111111-1-1111-11-11-1-1-11-11-11    linear of order 2
ρ92222-2-200-2200-20200000000    orthogonal lifted from D4
ρ1022222222-2-2000000-2000000    orthogonal lifted from D4
ρ112222-2-2002-2020-2000000000    orthogonal lifted from D4
ρ122222-2-2002-20-202000000000    orthogonal lifted from D4
ρ13222222-2-2-2-20000002000000    orthogonal lifted from D4
ρ142222-2-200-220020-200000000    orthogonal lifted from D4
ρ1522-2-2-22-2200000000000√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ1622-2-2-222-200000000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ1722-2-2-22-2200000000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ1822-2-2-222-200000000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ194-44-40000000000020-200000    orthogonal lifted from C2≀C22
ρ204-44-400000000000-20200000    orthogonal lifted from C2≀C22
ρ2144-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-440000002i0000000-2i0000    complex lifted from D4.9D4
ρ234-4-44000000-2i00000002i0000    complex lifted from D4.9D4

Smallest permutation representation of C23⋊Q16
►On 32 points
Generators in S32
(1 5)(3 30)(4 27)(7 26)(8 31)(9 24)(10 21)(12 16)(13 20)(14 17)(19 23)(28 32)

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

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

(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)

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

Matrix representation of C23⋊Q16 ►in GL8(𝔽17)

160000000
016000000
00100000
00010000
000016000
00000100
000000160
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
00000010
00000001
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
00710000
001100000
167000000
71000000
00000040
000000013
00000400
00004000
,
71000000
110000000
001100000
0010160000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0],[7,1,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

C23⋊Q16 in GAP, Magma, Sage, TeX 

C_2^3\rtimes Q_{16}
 
% in TeX
 
G:=Group("C2^3:Q16");
 
// GroupNames label
 
G:=SmallGroup(128,334);
 
// by ID
 
G=gap.SmallGroup(128,334);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,232,422,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Character table of C23⋊Q16 in TeX

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