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G = C32⋊2Q16  order 144 = 24·32

1st semidirect product of C32 and Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial

Aliases: C32⋊2Q16, C12.14D6, Dic6.2S3, C4.10S32, (C3×C6).11D4, C3⋊2(C3⋊Q16), C6.9(C3⋊D4), (C3×C12).6C22, C32⋊4C8.1C2, (C3×Dic6).1C2, C2.5(D6⋊S3), SmallGroup(144,61)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C32⋊2Q16
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×Dic6 — C32⋊2Q16
Lower central
C32 — C3×C6 — C3×C12 — C32⋊2Q16
Upper central
C1 — C2 — C4

Generators and relations for C32⋊2Q16
 G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >

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

Character table of C32⋊2Q16

 class 123A3B3C4A4B4C6A6B6C8A8B12A12B12C12D12E12F12G12H
 size 11224212122241818444412121212
ρ1111111111111111111111    trivial
ρ2111111-11111-1-11111-1-111    linear of order 2
ρ3111111-1-1111111111-1-1-1-1    linear of order 2
ρ41111111-1111-1-1111111-1-1    linear of order 2
ρ522222-20022200-2-2-2-20000    orthogonal lifted from D4
ρ622-12-12202-1-1002-1-1-1-1-100    orthogonal lifted from S3
ρ7222-1-1202-12-100-12-1-100-1-1    orthogonal lifted from S3
ρ822-12-12-202-1-1002-1-1-11100    orthogonal lifted from D6
ρ9222-1-120-2-12-100-12-1-10011    orthogonal lifted from D6
ρ102-2222000-2-2-2-√2√200000000    symplectic lifted from Q16, Schur index 2
ρ112-2222000-2-2-2√2-√200000000    symplectic lifted from Q16, Schur index 2
ρ1222-12-1-2002-1-100-2111-√-3√-300    complex lifted from C3⋊D4
ρ13222-1-1-200-12-1001-21100-√-3√-3    complex lifted from C3⋊D4
ρ14222-1-1-200-12-1001-21100√-3-√-3    complex lifted from C3⋊D4
ρ1522-12-1-2002-1-100-2111√-3-√-300    complex lifted from C3⋊D4
ρ1644-2-21400-2-2100-2-2110000    orthogonal lifted from S32
ρ174-44-2-20002-420000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ1844-2-21-400-2-210022-1-10000    symplectic lifted from D6⋊S3, Schur index 2
ρ194-4-24-2000-4220000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ204-4-2-2100022-10000-3i3i0000    complex faithful
ρ214-4-2-2100022-100003i-3i0000    complex faithful

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

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

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

C32⋊2Q16 is a maximal subgroup of
 C32⋊SD32  C32⋊Q32  C24.23D6  D12.2D6  D12.4D6  D12.30D6  D12.32D6  Dic6.19D6  Dic6.20D6  S3×C3⋊Q16  Dic6.22D6  C12.D18  He3⋊3Q16  C33⋊6Q16  C33⋊9Q16
C32⋊2Q16 is a maximal quotient of
 Dic6⋊Dic3  C12.8Dic6  C12.D18  He3⋊2Q16  C33⋊6Q16  C33⋊9Q16

Matrix representation of C32⋊2Q16 ►in GL4(𝔽5) generated by

3020
0301
1010
0201
,
3020
0104
1010
0303
,
2040
0002
1030
0100
,
0401
1030
0003
0030
G:=sub<GL(4,GF(5))| [3,0,1,0,0,3,0,2,2,0,1,0,0,1,0,1],[3,0,1,0,0,1,0,3,2,0,1,0,0,4,0,3],[2,0,1,0,0,0,0,1,4,0,3,0,0,2,0,0],[0,1,0,0,4,0,0,0,0,3,0,3,1,0,3,0] >;
 

C32⋊2Q16 in GAP, Magma, Sage, TeX 

C_3^2\rtimes_2Q_{16}
 
% in TeX
 
G:=Group("C3^2:2Q16");
 
// GroupNames label
 
G:=SmallGroup(144,61);
 
// by ID
 
G=gap.SmallGroup(144,61);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,55,218,116,50,490,3461]);
 
// 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=d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊2Q16 in TeX
Character table of C32⋊2Q16 in TeX

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