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G = C32⋊Q32order 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), C322Q16.C2, C322C16.2C2, C324C8.3C22, SmallGroup(288,384)

Series: Derived Chief Lower central Upper central

C1C32C324C8 — C32⋊Q32
C1C32C3×C6C3×C12C324C8C322Q16 — C32⋊Q32
C32C3×C6C3×C12C324C8 — C32⋊Q32
C1C2C4

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 >

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

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
ρ62222-2002200-2-2000022-2-2    orthogonal lifted from D8
ρ72222-2002200-2-20000-2-222    orthogonal lifted from D8
ρ82-222000-2-2-2200000016716ζ16716165163ζ165163    symplectic lifted from Q32, Schur index 2
ρ92-222000-2-22-2000000ζ16516316516316716ζ16716    symplectic lifted from Q32, Schur index 2
ρ102-222000-2-22-2000000165163ζ165163ζ1671616716    symplectic lifted from Q32, Schur index 2
ρ112-222000-2-2-22000000ζ1671616716ζ165163165163    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 69 44)(4 46 71)(6 73 48)(8 34 75)(10 77 36)(12 38 79)(14 65 40)(16 42 67)(17 95 57)(19 59 81)(21 83 61)(23 63 85)(25 87 49)(27 51 89)(29 91 53)(31 55 93)
(1 68 43)(3 45 70)(5 72 47)(7 33 74)(9 76 35)(11 37 78)(13 80 39)(15 41 66)(18 58 96)(20 82 60)(22 62 84)(24 86 64)(26 50 88)(28 90 52)(30 54 92)(32 94 56)
(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 59 9 51)(2 58 10 50)(3 57 11 49)(4 56 12 64)(5 55 13 63)(6 54 14 62)(7 53 15 61)(8 52 16 60)(17 37 25 45)(18 36 26 44)(19 35 27 43)(20 34 28 42)(21 33 29 41)(22 48 30 40)(23 47 31 39)(24 46 32 38)(65 84 73 92)(66 83 74 91)(67 82 75 90)(68 81 76 89)(69 96 77 88)(70 95 78 87)(71 94 79 86)(72 93 80 85)

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

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

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

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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