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## G = C3×C3⋊Q32order 288 = 25·32

### Direct product of C3 and C3⋊Q32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C3⋊Q32
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×Dic12 — C3×C3⋊Q32
 Lower central C3 — C6 — C12 — C24 — C3×C3⋊Q32
 Upper central C1 — C6 — C12 — C24 — C3×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 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F ··· 12M 12N 12O 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 8 8 12 12 12 12 12 12 ··· 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 1 1 2 2 2 2 8 24 1 1 2 2 2 2 2 2 2 4 4 4 8 ··· 8 24 24 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + - image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 Q32 S3×C6 C3×D8 C3×C3⋊D4 C3×Q32 D4⋊S3 C3⋊Q32 C3×D4⋊S3 C3×C3⋊Q32 kernel C3×C3⋊Q32 C3×C3⋊C16 C3×Dic12 C32×Q16 C3⋊Q32 C3⋊C16 Dic12 C3×Q16 C3×Q16 C3×C12 C24 C3×C6 Q16 C12 C12 C32 C8 C6 C4 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 4 4 8 1 2 2 4

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

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 2 5 1 1 3 3 3 6 4 3 4 6 2 2 6 3
,
 0 2 1 5 0 3 6 1 2 1 2 1 6 1 2 2
,
 6 5 3 0 0 0 6 2 4 4 5 6 2 5 6 3
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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