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G = C322Q16order 144 = 24·32

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

metabelian, supersoluble, monomial

Aliases: C322Q16, C12.14D6, Dic6.2S3, C4.10S32, (C3×C6).11D4, C32(C3⋊Q16), C6.9(C3⋊D4), (C3×C12).6C22, C324C8.1C2, (C3×Dic6).1C2, C2.5(D6⋊S3), SmallGroup(144,61)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C322Q16
C1C3C32C3×C6C3×C12C3×Dic6 — C322Q16
C32C3×C6C3×C12 — C322Q16
C1C2C4

Generators and relations for C322Q16
 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 >

2C3
6C4
6C4
2C6
3Q8
3Q8
9C8
2C12
2Dic3
2Dic3
6C12
6C12
9Q16
3C3×Q8
3C3×Q8
3C3⋊C8
3C3⋊C8
6C3⋊C8
2C3×Dic3
2C3×Dic3
3C3⋊Q16
3C3⋊Q16

Character table of C322Q16

 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
ρ102-2222000-2-2-2-2200000000    symplectic lifted from Q16, Schur index 2
ρ112-2222000-2-2-22-200000000    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 C322Q16
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 19 35)(10 36 20)(11 21 37)(12 38 22)(13 23 39)(14 40 24)(15 17 33)(16 34 18)
(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 19 35)(10 36 20)(11 21 37)(12 38 22)(13 23 39)(14 40 24)(15 17 33)(16 34 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)(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 31 13 27)(10 30 14 26)(11 29 15 25)(12 28 16 32)(33 48 37 44)(34 47 38 43)(35 46 39 42)(36 45 40 41)

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

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

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

C322Q16 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  He33Q16  C336Q16  C339Q16
C322Q16 is a maximal quotient of
Dic6⋊Dic3  C12.8Dic6  C12.D18  He32Q16  C336Q16  C339Q16

Matrix representation of C322Q16 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] >;

C322Q16 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 C322Q16 in TeX
Character table of C322Q16 in TeX

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