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G = C32⋊5SD16  order 144 = 24·32

3rd semidirect product of C32 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial

Aliases: Dic6⋊1S3, C6.14D12, C12.13D6, C32⋊5SD16, C3⋊C8⋊3S3, C4.3S32, (C3×C6).10D4, C3⋊2(C24⋊C2), (C3×Dic6)⋊2C2, C6.3(C3⋊D4), C12⋊S3.2C2, C3⋊1(Q8⋊2S3), (C3×C12).5C22, C2.6(C3⋊D12), (C3×C3⋊C8)⋊3C2, SmallGroup(144,60)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C32⋊5SD16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊5SD16
Lower central
C32 — C3×C6 — C3×C12 — C32⋊5SD16
Upper central
C1 — C2 — C4

Generators and relations for C32⋊5SD16
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=b-1, dcd=c3 >

C1
C2
36C2
C3
C3
2C3
C4
6C4
18C22
C6
C6
2C6
12S3
12S3
12S3
12S3
C32
3C8
3Q8
9D4
C12
C12
2C12
2Dic3
6C12
6D6
6D6
6D6
6D6
C3×C6
4C3⋊S3
9SD16
C3⋊C8
Dic6
3C24
3D12
3D12
3C3×Q8
6D12
C3×C12
2C2×C3⋊S3
2C3×Dic3
3C24⋊C2
3Q8⋊2S3
C3×Dic6
C12⋊S3
C3×C3⋊C8
C32⋊5SD16

Character table of C32⋊5SD16

 class 12A2B3A3B3C4A4B6A6B6C8A8B12A12B12C12D12E12F12G24A24B24C24D
 size 1136224212224662244412126666
ρ1111111111111111111111111    trivial
ρ21111111-1111-1-111111-1-1-1-1-1-1    linear of order 2
ρ311-111111111-1-11111111-1-1-1-1    linear of order 2
ρ411-11111-11111111111-1-11111    linear of order 2
ρ5220-12-12-2-12-10022-1-1-1110000    orthogonal lifted from D6
ρ62202-1-1202-1-1-2-2-1-12-1-1001111    orthogonal lifted from D6
ρ7220-12-122-12-10022-1-1-1-1-10000    orthogonal lifted from S3
ρ82202-1-1202-1-122-1-12-1-100-1-1-1-1    orthogonal lifted from S3
ρ9220222-2022200-2-2-2-2-2000000    orthogonal lifted from D4
ρ102202-1-1-202-1-10011-21100√3-√3-√3√3    orthogonal lifted from D12
ρ112202-1-1-202-1-10011-21100-√3√3√3-√3    orthogonal lifted from D12
ρ122-2022200-2-2-2-√-2√-20000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ132-2022200-2-2-2√-2-√-20000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ14220-12-1-20-12-100-2-2111-√-3√-30000    complex lifted from C3⋊D4
ρ15220-12-1-20-12-100-2-2111√-3-√-30000    complex lifted from C3⋊D4
ρ162-202-1-100-211-√-2√-2-√3√30√3-√300-ζ87ζ32+ζ85ζ32+ζ85-ζ87ζ3+ζ85ζ3+ζ85-ζ83ζ3+ζ8ζ3+ζ8-ζ83ζ32+ζ8ζ32+ζ8    complex lifted from C24⋊C2
ρ172-202-1-100-211√-2-√-2√3-√30-√3√300-ζ83ζ3+ζ8ζ3+ζ8-ζ83ζ32+ζ8ζ32+ζ8-ζ87ζ32+ζ85ζ32+ζ85-ζ87ζ3+ζ85ζ3+ζ85    complex lifted from C24⋊C2
ρ182-202-1-100-211√-2-√-2-√3√30√3-√300-ζ83ζ32+ζ8ζ32+ζ8-ζ83ζ3+ζ8ζ3+ζ8-ζ87ζ3+ζ85ζ3+ζ85-ζ87ζ32+ζ85ζ32+ζ85    complex lifted from C24⋊C2
ρ192-202-1-100-211-√-2√-2√3-√30-√3√300-ζ87ζ3+ζ85ζ3+ζ85-ζ87ζ32+ζ85ζ32+ζ85-ζ83ζ32+ζ8ζ32+ζ8-ζ83ζ3+ζ8ζ3+ζ8    complex lifted from C24⋊C2
ρ204-40-24-2002-420000000000000    orthogonal lifted from Q8⋊2S3
ρ21440-2-21-40-2-2100222-1-1000000    orthogonal lifted from C3⋊D12
ρ22440-2-2140-2-2100-2-2-211000000    orthogonal lifted from S32
ρ234-40-2-210022-100-2√32√30-√3√3000000    orthogonal faithful
ρ244-40-2-210022-1002√3-2√30√3-√3000000    orthogonal faithful

Permutation representations of C32⋊5SD16
►On 24 points - transitive group 24T233
Generators in S24
(1 13 19)(2 20 14)(3 15 21)(4 22 16)(5 9 23)(6 24 10)(7 11 17)(8 18 12)

(1 13 19)(2 14 20)(3 15 21)(4 16 22)(5 9 23)(6 10 24)(7 11 17)(8 12 18)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

(1 3)(2 6)(5 7)(9 17)(10 20)(11 23)(12 18)(13 21)(14 24)(15 19)(16 22)
 
G:=sub<Sym(24)| (1,13,19)(2,20,14)(3,15,21)(4,22,16)(5,9,23)(6,24,10)(7,11,17)(8,18,12), (1,13,19)(2,14,20)(3,15,21)(4,16,22)(5,9,23)(6,10,24)(7,11,17)(8,12,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,3)(2,6)(5,7)(9,17)(10,20)(11,23)(12,18)(13,21)(14,24)(15,19)(16,22)>;
 
G:=Group( (1,13,19)(2,20,14)(3,15,21)(4,22,16)(5,9,23)(6,24,10)(7,11,17)(8,18,12), (1,13,19)(2,14,20)(3,15,21)(4,16,22)(5,9,23)(6,10,24)(7,11,17)(8,12,18), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,3)(2,6)(5,7)(9,17)(10,20)(11,23)(12,18)(13,21)(14,24)(15,19)(16,22) );
 
G=PermutationGroup([[(1,13,19),(2,20,14),(3,15,21),(4,22,16),(5,9,23),(6,24,10),(7,11,17),(8,18,12)], [(1,13,19),(2,14,20),(3,15,21),(4,16,22),(5,9,23),(6,10,24),(7,11,17),(8,12,18)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,3),(2,6),(5,7),(9,17),(10,20),(11,23),(12,18),(13,21),(14,24),(15,19),(16,22)]])
 
G:=TransitiveGroup(24,233);
 

C32⋊5SD16 is a maximal subgroup of
 S3×C24⋊C2  C24⋊1D6  Dic12⋊S3  D6.3D12  D12⋊18D6  D12.27D6  Dic6.29D6  Dic6⋊3D6  Dic6⋊D6  Dic6.20D6  D12⋊5D6  S3×Q8⋊2S3  Dic6.10D6  Dic6.22D6  D12.14D6  C6.D36  C18.D12  He3⋊3SD16  He3⋊5SD16  C33⋊15SD16  C33⋊17SD16  C33⋊18SD16
C32⋊5SD16 is a maximal quotient of
 C6.17D24  C6.Dic12  C12.Dic6  C6.D36  C18.D12  He3⋊4SD16  C33⋊15SD16  C33⋊17SD16  C33⋊18SD16

Matrix representation of C32⋊5SD16 ►in GL6(𝔽73)

100000
010000
001000
000100
00007272
000010
,
100000
010000
0007200
0017200
000010
000001
,
660000
6760000
0072000
0007200
000010
00007272
,
010000
100000
000100
001000
000010
00007272

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
 

C32⋊5SD16 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_5{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^2:5SD16");
 
// GroupNames label
 
G:=SmallGroup(144,60);
 
// by ID
 
G=gap.SmallGroup(144,60);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,31,218,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^3>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊5SD16 in TeX
Character table of C32⋊5SD16 in TeX

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