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G = C3×C3⋊Q32order 288 = 25·32

Direct product of C3 and C3⋊Q32

direct product, metabelian, supersoluble, monomial

Aliases: C3×C3⋊Q32, C326Q32, C24.55D6, Dic12.2C6, C3⋊C16.C6, C8.7(S3×C6), C32(C3×Q32), Q16.(C3×S3), C24.5(C2×C6), (C3×C6).33D8, C12.6(C3×D4), C6.11(C3×D8), (C3×C12).44D4, (C3×Q16).7S3, (C3×Q16).1C6, C6.33(D4⋊S3), C12.86(C3⋊D4), (C3×C24).16C22, (C3×Dic12).4C2, (C32×Q16).1C2, (C3×C3⋊C16).2C2, C2.7(C3×D4⋊S3), C4.4(C3×C3⋊D4), SmallGroup(288,263)

Series: Derived Chief Lower central Upper central

C1C24 — C3×C3⋊Q32
C1C3C6C12C24C3×C24C3×Dic12 — C3×C3⋊Q32
C3C6C12C24 — C3×C3⋊Q32
C1C6C12C24C3×Q16

Generators and relations for C3×C3⋊Q32
 G = < a,b,c,d | a3=b3=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 138 in 57 conjugacy classes, 26 normal (all characteristic)
C1, C2, C3 [×2], C3, C4, C4 [×2], C6 [×2], C6, C8, Q8 [×2], C32, Dic3, C12 [×2], C12 [×6], C16, Q16, Q16, C3×C6, C24 [×2], C24, Dic6, C3×Q8 [×5], Q32, C3×Dic3, C3×C12, C3×C12, C3⋊C16, C48, Dic12, C3×Q16 [×2], C3×Q16 [×2], C3×C24, C3×Dic6, Q8×C32, C3⋊Q32, C3×Q32, C3×C3⋊C16, C3×Dic12, C32×Q16, C3×C3⋊Q32
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, Q32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, C3⋊Q32, C3×Q32, C3×D4⋊S3, C3×C3⋊Q32

Smallest permutation representation of C3×C3⋊Q32
On 96 points
Generators in S96
(1 36 94)(2 37 95)(3 38 96)(4 39 81)(5 40 82)(6 41 83)(7 42 84)(8 43 85)(9 44 86)(10 45 87)(11 46 88)(12 47 89)(13 48 90)(14 33 91)(15 34 92)(16 35 93)(17 54 75)(18 55 76)(19 56 77)(20 57 78)(21 58 79)(22 59 80)(23 60 65)(24 61 66)(25 62 67)(26 63 68)(27 64 69)(28 49 70)(29 50 71)(30 51 72)(31 52 73)(32 53 74)
(1 36 94)(2 95 37)(3 38 96)(4 81 39)(5 40 82)(6 83 41)(7 42 84)(8 85 43)(9 44 86)(10 87 45)(11 46 88)(12 89 47)(13 48 90)(14 91 33)(15 34 92)(16 93 35)(17 75 54)(18 55 76)(19 77 56)(20 57 78)(21 79 58)(22 59 80)(23 65 60)(24 61 66)(25 67 62)(26 63 68)(27 69 64)(28 49 70)(29 71 50)(30 51 72)(31 73 52)(32 53 74)
(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 30 9 22)(2 29 10 21)(3 28 11 20)(4 27 12 19)(5 26 13 18)(6 25 14 17)(7 24 15 32)(8 23 16 31)(33 54 41 62)(34 53 42 61)(35 52 43 60)(36 51 44 59)(37 50 45 58)(38 49 46 57)(39 64 47 56)(40 63 48 55)(65 93 73 85)(66 92 74 84)(67 91 75 83)(68 90 76 82)(69 89 77 81)(70 88 78 96)(71 87 79 95)(72 86 80 94)

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

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

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

54 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B12A12B12C12D12E12F···12M12N12O16A16B16C16D24A24B24C24D24E···24J48A···48H
order12333334446666688121212121212···121212161616162424242424···2448···48
size111122228241122222224448···82424666622224···46···6

54 irreducible representations

dim111111112222222222224444
type++++++++-+-
imageC1C2C2C2C3C6C6C6S3D4D6D8C3×S3C3⋊D4C3×D4Q32S3×C6C3×D8C3×C3⋊D4C3×Q32D4⋊S3C3⋊Q32C3×D4⋊S3C3×C3⋊Q32
kernelC3×C3⋊Q32C3×C3⋊C16C3×Dic12C32×Q16C3⋊Q32C3⋊C16Dic12C3×Q16C3×Q16C3×C12C24C3×C6Q16C12C12C32C8C6C4C3C6C3C2C1
# reps111122221112222424481224

Matrix representation of C3×C3⋊Q32 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
2511
3336
4346
2263
,
0215
0361
2121
6122
,
6530
0062
4456
2563
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[2,3,4,2,5,3,3,2,1,3,4,6,1,6,6,3],[0,0,2,6,2,3,1,1,1,6,2,2,5,1,1,2],[6,0,4,2,5,0,4,5,3,6,5,6,0,2,6,3] >;

C3×C3⋊Q32 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes Q_{32}
% in TeX

G:=Group("C3xC3:Q32");
// GroupNames label

G:=SmallGroup(288,263);
// by ID

G=gap.SmallGroup(288,263);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,197,344,1011,514,192,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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