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G = C322Q32order 288 = 25·32

1st semidirect product of C32 and Q32 acting via Q32/C8=C22

metabelian, supersoluble, monomial

Aliases: C322Q32, C24.17D6, Dic12.2S3, C8.15S32, (C3×C6).11D8, C32(C3⋊Q32), (C3×C12).26D4, C6.10(D4⋊S3), C4.3(D6⋊S3), C12.23(C3⋊D4), C24.S3.1C2, (C3×C24).11C22, (C3×Dic12).2C2, C2.5(C322D8), SmallGroup(288,198)

Series: Derived Chief Lower central Upper central

C1C3×C24 — C322Q32
C1C3C32C3×C6C3×C12C3×C24C3×Dic12 — C322Q32
C32C3×C6C3×C12C3×C24 — C322Q32
C1C2C4C8

Generators and relations for C322Q32
 G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=c-1 >

2C3
12C4
12C4
2C6
6Q8
6Q8
2C12
4Dic3
4Dic3
12C12
12C12
3Q16
3Q16
9C16
2C24
2Dic6
2Dic6
6C3×Q8
6C3×Q8
4C3×Dic3
4C3×Dic3
9Q32
3C3×Q16
3C3×Q16
3C3⋊C16
3C3⋊C16
6C3⋊C16
2C3×Dic6
2C3×Dic6
3C3⋊Q32
3C3⋊Q32

Smallest permutation representation of C322Q32
On 96 points
Generators in S96
(1 18 43)(2 44 19)(3 20 45)(4 46 21)(5 22 47)(6 48 23)(7 24 33)(8 34 25)(9 26 35)(10 36 27)(11 28 37)(12 38 29)(13 30 39)(14 40 31)(15 32 41)(16 42 17)(49 70 83)(50 84 71)(51 72 85)(52 86 73)(53 74 87)(54 88 75)(55 76 89)(56 90 77)(57 78 91)(58 92 79)(59 80 93)(60 94 65)(61 66 95)(62 96 67)(63 68 81)(64 82 69)
(1 43 18)(2 19 44)(3 45 20)(4 21 46)(5 47 22)(6 23 48)(7 33 24)(8 25 34)(9 35 26)(10 27 36)(11 37 28)(12 29 38)(13 39 30)(14 31 40)(15 41 32)(16 17 42)(49 70 83)(50 84 71)(51 72 85)(52 86 73)(53 74 87)(54 88 75)(55 76 89)(56 90 77)(57 78 91)(58 92 79)(59 80 93)(60 94 65)(61 66 95)(62 96 67)(63 68 81)(64 82 69)
(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 82 9 90)(2 81 10 89)(3 96 11 88)(4 95 12 87)(5 94 13 86)(6 93 14 85)(7 92 15 84)(8 91 16 83)(17 49 25 57)(18 64 26 56)(19 63 27 55)(20 62 28 54)(21 61 29 53)(22 60 30 52)(23 59 31 51)(24 58 32 50)(33 79 41 71)(34 78 42 70)(35 77 43 69)(36 76 44 68)(37 75 45 67)(38 74 46 66)(39 73 47 65)(40 72 48 80)

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

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

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

33 conjugacy classes

class 1  2 3A3B3C4A4B4C6A6B6C8A8B12A12B12C12D12E12F12G12H16A16B16C16D24A···24H
order123334446668812121212121212121616161624···24
size112242242422422444424242424181818184···4

33 irreducible representations

dim111222222444444
type+++++++-++--
imageC1C2C2S3D4D6D8C3⋊D4Q32S32D4⋊S3D6⋊S3C3⋊Q32C322D8C322Q32
kernelC322Q32C24.S3C3×Dic12Dic12C3×C12C24C3×C6C12C32C8C6C4C3C2C1
# reps112212244121424

Matrix representation of C322Q32 in GL6(𝔽97)

100000
010000
001000
000100
0000096
0000196
,
100000
010000
0009600
0019600
000010
000001
,
73930000
2690000
000100
001000
000001
000010
,
0170000
5700000
0096000
0009600
000001
000010

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[73,2,0,0,0,0,93,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,57,0,0,0,0,17,0,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C322Q32 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2Q_{32}
% in TeX

G:=Group("C3^2:2Q32");
// GroupNames label

G:=SmallGroup(288,198);
// by ID

G=gap.SmallGroup(288,198);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,120,254,135,142,675,346,80,1356,9414]);
// 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=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C322Q32 in TeX

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