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G = C32⋊3Q16  order 144 = 24·32

2nd semidirect product of C32 and Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial

Aliases: C32⋊3Q16, C3⋊2Dic12, C12.15D6, C6.15D12, Dic6.1S3, C3⋊C8.S3, C4.4S32, (C3×C6).12D4, C3⋊1(C3⋊Q16), C6.4(C3⋊D4), (C3×C12).7C22, (C3×Dic6).2C2, C2.7(C3⋊D12), C32⋊4Q8.2C2, (C3×C3⋊C8).1C2, SmallGroup(144,62)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C32⋊3Q16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊3Q16
Lower central
C32 — C3×C6 — C3×C12 — C32⋊3Q16
Upper central
C1 — C2 — C4

Generators and relations for C32⋊3Q16
 G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

C1
C2
C3
C3
2C3
C4
6C4
18C4
C6
C6
2C6
C32
3C8
3Q8
9Q8
C12
C12
2C12
2Dic3
6Dic3
6Dic3
6Dic3
6Dic3
6C12
C3×C6
9Q16
Dic6
C3⋊C8
3C24
3Dic6
3Dic6
3C3×Q8
6Dic6
C3×C12
2C3×Dic3
2C3⋊Dic3
3Dic12
3C3⋊Q16
C3×C3⋊C8
C3×Dic6
C32⋊4Q8
C32⋊3Q16

Character table of C32⋊3Q16

 class 123A3B3C4A4B4C6A6B6C8A8B12A12B12C12D12E12F12G24A24B24C24D
 size 1122421236224662244412126666
ρ1111111111111111111111111    trivial
ρ21111111-1111-1-11111111-1-1-1-1    linear of order 2
ρ3111111-11111-1-111111-1-1-1-1-1-1    linear of order 2
ρ4111111-1-11111111111-1-11111    linear of order 2
ρ5222-1-1200-12-1-2-2-1-12-1-1001111    orthogonal lifted from D6
ρ622-12-12202-1-10022-1-1-1-1-10000    orthogonal lifted from S3
ρ722-12-12-202-1-10022-1-1-1110000    orthogonal lifted from D6
ρ822222-20022200-2-2-2-2-2000000    orthogonal lifted from D4
ρ9222-1-1200-12-122-1-12-1-100-1-1-1-1    orthogonal lifted from S3
ρ10222-1-1-200-12-10011-21100-√3√3√3-√3    orthogonal lifted from D12
ρ11222-1-1-200-12-10011-21100√3-√3-√3√3    orthogonal lifted from D12
ρ122-2222000-2-2-2-√2√20000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ132-2222000-2-2-2√2-√20000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ142-22-1-10001-21-√2√2-√3√30-√3√300ζ87ζ32+ζ87+ζ85ζ32ζ83ζ32+ζ8ζ32+ζ8ζ83ζ3+ζ83+ζ8ζ3ζ87ζ3+ζ85ζ3+ζ85    symplectic lifted from Dic12, Schur index 2
ρ152-22-1-10001-21√2-√2√3-√30√3-√300ζ83ζ3+ζ83+ζ8ζ3ζ87ζ3+ζ85ζ3+ζ85ζ87ζ32+ζ87+ζ85ζ32ζ83ζ32+ζ8ζ32+ζ8    symplectic lifted from Dic12, Schur index 2
ρ162-22-1-10001-21√2-√2-√3√30-√3√300ζ87ζ3+ζ85ζ3+ζ85ζ83ζ3+ζ83+ζ8ζ3ζ83ζ32+ζ8ζ32+ζ8ζ87ζ32+ζ87+ζ85ζ32    symplectic lifted from Dic12, Schur index 2
ρ172-22-1-10001-21-√2√2√3-√30√3-√300ζ83ζ32+ζ8ζ32+ζ8ζ87ζ32+ζ87+ζ85ζ32ζ87ζ3+ζ85ζ3+ζ85ζ83ζ3+ζ83+ζ8ζ3    symplectic lifted from Dic12, Schur index 2
ρ1822-12-1-2002-1-100-2-2111√-3-√-30000    complex lifted from C3⋊D4
ρ1922-12-1-2002-1-100-2-2111-√-3√-30000    complex lifted from C3⋊D4
ρ2044-2-21400-2-2100-2-2-211000000    orthogonal lifted from S32
ρ2144-2-21-400-2-2100222-1-1000000    orthogonal lifted from C3⋊D12
ρ224-4-24-2000-4220000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ234-4-2-2100022-1002√3-2√30-√3√3000000    symplectic faithful, Schur index 2
ρ244-4-2-2100022-100-2√32√30√3-√3000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊3Q16
►On 48 points
Generators in S48
(1 33 25)(2 26 34)(3 35 27)(4 28 36)(5 37 29)(6 30 38)(7 39 31)(8 32 40)(9 46 20)(10 21 47)(11 48 22)(12 23 41)(13 42 24)(14 17 43)(15 44 18)(16 19 45)

(1 25 33)(2 26 34)(3 27 35)(4 28 36)(5 29 37)(6 30 38)(7 31 39)(8 32 40)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)

(1 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 46 37 42)(34 45 38 41)(35 44 39 48)(36 43 40 47)
 
G:=sub<Sym(48)| (1,33,25)(2,26,34)(3,35,27)(4,28,36)(5,37,29)(6,30,38)(7,39,31)(8,32,40)(9,46,20)(10,21,47)(11,48,22)(12,23,41)(13,42,24)(14,17,43)(15,44,18)(16,19,45), (1,25,33)(2,26,34)(3,27,35)(4,28,36)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,46,37,42)(34,45,38,41)(35,44,39,48)(36,43,40,47)>;
 
G:=Group( (1,33,25)(2,26,34)(3,35,27)(4,28,36)(5,37,29)(6,30,38)(7,39,31)(8,32,40)(9,46,20)(10,21,47)(11,48,22)(12,23,41)(13,42,24)(14,17,43)(15,44,18)(16,19,45), (1,25,33)(2,26,34)(3,27,35)(4,28,36)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,46,37,42)(34,45,38,41)(35,44,39,48)(36,43,40,47) );
 
G=PermutationGroup([(1,33,25),(2,26,34),(3,35,27),(4,28,36),(5,37,29),(6,30,38),(7,39,31),(8,32,40),(9,46,20),(10,21,47),(11,48,22),(12,23,41),(13,42,24),(14,17,43),(15,44,18),(16,19,45)], [(1,25,33),(2,26,34),(3,27,35),(4,28,36),(5,29,37),(6,30,38),(7,31,39),(8,32,40),(9,46,20),(10,47,21),(11,48,22),(12,41,23),(13,42,24),(14,43,17),(15,44,18),(16,45,19)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,9,5,13),(2,16,6,12),(3,15,7,11),(4,14,8,10),(17,32,21,28),(18,31,22,27),(19,30,23,26),(20,29,24,25),(33,46,37,42),(34,45,38,41),(35,44,39,48),(36,43,40,47)])
 

C32⋊3Q16 is a maximal subgroup of
 S3×Dic12  C24.3D6  Dic12⋊S3  D6.1D12  D12.27D6  D12.29D6  Dic6.29D6  Dic6.19D6  Dic6.D6  D12.22D6  D12.8D6  S3×C3⋊Q16  Dic6.9D6  D12.24D6  D12.15D6  C3⋊Dic36  C9⋊Dic12  He3⋊2Q16  He3⋊3Q16  C33⋊7Q16  C33⋊8Q16  C33⋊9Q16
C32⋊3Q16 is a maximal quotient of
 C6.Dic12  C12.73D12  C6.18D24  C3⋊Dic36  C9⋊Dic12  He3⋊3Q16  C33⋊7Q16  C33⋊8Q16  C33⋊9Q16

Matrix representation of C32⋊3Q16 ►in GL6(𝔽73)

100000
010000
001000
000100
0000072
0000172
,
100000
010000
00727200
001000
000010
000001
,
0410000
16410000
0072000
0007200
000001
000010
,
19130000
62540000
001000
00727200
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[19,62,0,0,0,0,13,54,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C32⋊3Q16 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_3Q_{16}
 
% in TeX
 
G:=Group("C3^2:3Q16");
 
// GroupNames label
 
G:=SmallGroup(144,62);
 
// by ID
 
G=gap.SmallGroup(144,62);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊3Q16 in TeX
Character table of C32⋊3Q16 in TeX

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