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G = C32⋊Q32  order 288 = 25·32

The semidirect product of C32 and Q32 acting via Q32/C4=D4

non-abelian, soluble, monomial

Aliases: C32⋊Q32, C4.3S3≀C2, (C3×C6).3D8, (C3×C12).7D4, C2.5(C32⋊D8), C32⋊2Q16.C2, C32⋊2C16.2C2, C32⋊4C8.3C22, SmallGroup(288,384)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C32⋊4C8 — C32⋊Q32
Chief series
C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2Q16 — C32⋊Q32
Lower central
C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊Q32
Upper central
C1 — C2 — C4

Generators and relations for C32⋊Q32
 G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=dad-1=b, cbc-1=a-1, dbd-1=a, dcd-1=c-1 >

C1
C2
2C3
2C3
C4
12C4
12C4
2C6
2C6
C32
6Q8
6Q8
9C8
2C12
2C12
4Dic3
4Dic3
12C12
12C12
C3×C6
9Q16
9C16
9Q16
2Dic6
2Dic6
6C3×Q8
6C3⋊C8
6C3⋊C8
6C3×Q8
C3×C12
4C3×Dic3
4C3×Dic3
9Q32
6C3⋊Q16
6C3⋊Q16
C32⋊4C8
2C3×Dic6
2C3×Dic6
C32⋊2Q16
C32⋊2Q16
C32⋊2C16
C32⋊Q32

Character table of C32⋊Q32

 class 123A3B4A4B4C6A6B8A8B12A12B12C12D12E12F16A16B16C16D
 size 114422424441818882424242418181818
ρ1111111111111111111111    trivial
ρ211111-1-1111111-1-1-1-11111    linear of order 2
ρ311111-11111111-11-11-1-1-1-1    linear of order 2
ρ4111111-11111111-11-1-1-1-1-1    linear of order 2
ρ5222220022-2-22200000000    orthogonal lifted from D4
ρ62222-2002200-2-20000√2√2-√2-√2    orthogonal lifted from D8
ρ72222-2002200-2-20000-√2-√2√2√2    orthogonal lifted from D8
ρ82-222000-2-2-√2√2000000-ζ167+ζ16ζ167-ζ16-ζ165+ζ163ζ165-ζ163    symplectic lifted from Q32, Schur index 2
ρ92-222000-2-2√2-√2000000ζ165-ζ163-ζ165+ζ163-ζ167+ζ16ζ167-ζ16    symplectic lifted from Q32, Schur index 2
ρ102-222000-2-2√2-√2000000-ζ165+ζ163ζ165-ζ163ζ167-ζ16-ζ167+ζ16    symplectic lifted from Q32, Schur index 2
ρ112-222000-2-2-√2√2000000ζ167-ζ16-ζ167+ζ16ζ165-ζ163-ζ165+ζ163    symplectic lifted from Q32, Schur index 2
ρ12441-240-21-200-2101010000    orthogonal lifted from S3≀C2
ρ13441-24021-200-210-10-10000    orthogonal lifted from S3≀C2
ρ1444-214-20-21001-210100000    orthogonal lifted from S3≀C2
ρ1544-21420-21001-2-10-100000    orthogonal lifted from S3≀C2
ρ16441-2-4001-2002-10-√-30√-30000    complex lifted from C32⋊D8
ρ1744-21-400-2100-12-√-30√-300000    complex lifted from C32⋊D8
ρ1844-21-400-2100-12√-30-√-300000    complex lifted from C32⋊D8
ρ19441-2-4001-2002-10√-30-√-30000    complex lifted from C32⋊D8
ρ208-8-420004-2000000000000    symplectic faithful, Schur index 2
ρ218-82-4000-24000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C32⋊Q32
►On 96 points
Generators in S96
(2 25 61)(4 63 27)(6 29 49)(8 51 31)(10 17 53)(12 55 19)(14 21 57)(16 59 23)(34 90 78)(36 80 92)(38 94 66)(40 68 96)(42 82 70)(44 72 84)(46 86 74)(48 76 88)

(1 24 60)(3 62 26)(5 28 64)(7 50 30)(9 32 52)(11 54 18)(13 20 56)(15 58 22)(33 89 77)(35 79 91)(37 93 65)(39 67 95)(41 81 69)(43 71 83)(45 85 73)(47 75 87)

(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)

(1 78 9 70)(2 77 10 69)(3 76 11 68)(4 75 12 67)(5 74 13 66)(6 73 14 65)(7 72 15 80)(8 71 16 79)(17 41 25 33)(18 40 26 48)(19 39 27 47)(20 38 28 46)(21 37 29 45)(22 36 30 44)(23 35 31 43)(24 34 32 42)(49 85 57 93)(50 84 58 92)(51 83 59 91)(52 82 60 90)(53 81 61 89)(54 96 62 88)(55 95 63 87)(56 94 64 86)
 
G:=sub<Sym(96)| (2,25,61)(4,63,27)(6,29,49)(8,51,31)(10,17,53)(12,55,19)(14,21,57)(16,59,23)(34,90,78)(36,80,92)(38,94,66)(40,68,96)(42,82,70)(44,72,84)(46,86,74)(48,76,88), (1,24,60)(3,62,26)(5,28,64)(7,50,30)(9,32,52)(11,54,18)(13,20,56)(15,58,22)(33,89,77)(35,79,91)(37,93,65)(39,67,95)(41,81,69)(43,71,83)(45,85,73)(47,75,87), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,78,9,70)(2,77,10,69)(3,76,11,68)(4,75,12,67)(5,74,13,66)(6,73,14,65)(7,72,15,80)(8,71,16,79)(17,41,25,33)(18,40,26,48)(19,39,27,47)(20,38,28,46)(21,37,29,45)(22,36,30,44)(23,35,31,43)(24,34,32,42)(49,85,57,93)(50,84,58,92)(51,83,59,91)(52,82,60,90)(53,81,61,89)(54,96,62,88)(55,95,63,87)(56,94,64,86)>;
 
G:=Group( (2,25,61)(4,63,27)(6,29,49)(8,51,31)(10,17,53)(12,55,19)(14,21,57)(16,59,23)(34,90,78)(36,80,92)(38,94,66)(40,68,96)(42,82,70)(44,72,84)(46,86,74)(48,76,88), (1,24,60)(3,62,26)(5,28,64)(7,50,30)(9,32,52)(11,54,18)(13,20,56)(15,58,22)(33,89,77)(35,79,91)(37,93,65)(39,67,95)(41,81,69)(43,71,83)(45,85,73)(47,75,87), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,78,9,70)(2,77,10,69)(3,76,11,68)(4,75,12,67)(5,74,13,66)(6,73,14,65)(7,72,15,80)(8,71,16,79)(17,41,25,33)(18,40,26,48)(19,39,27,47)(20,38,28,46)(21,37,29,45)(22,36,30,44)(23,35,31,43)(24,34,32,42)(49,85,57,93)(50,84,58,92)(51,83,59,91)(52,82,60,90)(53,81,61,89)(54,96,62,88)(55,95,63,87)(56,94,64,86) );
 
G=PermutationGroup([[(2,25,61),(4,63,27),(6,29,49),(8,51,31),(10,17,53),(12,55,19),(14,21,57),(16,59,23),(34,90,78),(36,80,92),(38,94,66),(40,68,96),(42,82,70),(44,72,84),(46,86,74),(48,76,88)], [(1,24,60),(3,62,26),(5,28,64),(7,50,30),(9,32,52),(11,54,18),(13,20,56),(15,58,22),(33,89,77),(35,79,91),(37,93,65),(39,67,95),(41,81,69),(43,71,83),(45,85,73),(47,75,87)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,78,9,70),(2,77,10,69),(3,76,11,68),(4,75,12,67),(5,74,13,66),(6,73,14,65),(7,72,15,80),(8,71,16,79),(17,41,25,33),(18,40,26,48),(19,39,27,47),(20,38,28,46),(21,37,29,45),(22,36,30,44),(23,35,31,43),(24,34,32,42),(49,85,57,93),(50,84,58,92),(51,83,59,91),(52,82,60,90),(53,81,61,89),(54,96,62,88),(55,95,63,87),(56,94,64,86)]])
 

Matrix representation of C32⋊Q32 ►in GL6(𝔽97)

100000
010000
001000
000100
0000961
0000960
,
100000
010000
0096100
0096000
000010
000001
,
95710000
26950000
000010
0000196
00411500
00825600
,
40400000
40570000
000010
000001
001000
000100

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[95,26,0,0,0,0,71,95,0,0,0,0,0,0,0,0,41,82,0,0,0,0,15,56,0,0,1,1,0,0,0,0,0,96,0,0],[40,40,0,0,0,0,40,57,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C32⋊Q32 in GAP, Magma, Sage, TeX 

C_3^2\rtimes Q_{32}
 
% in TeX
 
G:=Group("C3^2:Q32");
 
// GroupNames label
 
G:=SmallGroup(288,384);
 
// by ID
 
G=gap.SmallGroup(288,384);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,112,85,120,254,135,142,675,346,80,2693,2028,691,797,2372]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,c*a*c^-1=d*a*d^-1=b,c*b*c^-1=a^-1,d*b*d^-1=a,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊Q32 in TeX
Character table of C32⋊Q32 in TeX

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