direct product, metabelian, supersoluble, monomial
Aliases: C3×C3⋊Q32, C32⋊6Q32, C24.55D6, Dic12.2C6, C3⋊C16.C6, C8.7(S3×C6), C3⋊2(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
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, C3, C4, C4, C6, C6, C8, Q8, C32, Dic3, C12, C12, C16, Q16, Q16, C3×C6, C24, C24, Dic6, C3×Q8, Q32, C3×Dic3, C3×C12, C3×C12, C3⋊C16, C48, Dic12, C3×Q16, C3×Q16, C3×C24, C3×Dic6, Q8×C32, C3⋊Q32, C3×Q32, C3×C3⋊C16, C3×Dic12, C32×Q16, C3×C3⋊Q32
Quotients: C1, C2, C3, C22, S3, C6, 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
►On 96 points
Generators in S96
(1 89 38)(2 90 39)(3 91 40)(4 92 41)(5 93 42)(6 94 43)(7 95 44)(8 96 45)(9 81 46)(10 82 47)(11 83 48)(12 84 33)(13 85 34)(14 86 35)(15 87 36)(16 88 37)(17 80 56)(18 65 57)(19 66 58)(20 67 59)(21 68 60)(22 69 61)(23 70 62)(24 71 63)(25 72 64)(26 73 49)(27 74 50)(28 75 51)(29 76 52)(30 77 53)(31 78 54)(32 79 55)
(1 89 38)(2 39 90)(3 91 40)(4 41 92)(5 93 42)(6 43 94)(7 95 44)(8 45 96)(9 81 46)(10 47 82)(11 83 48)(12 33 84)(13 85 34)(14 35 86)(15 87 36)(16 37 88)(17 80 56)(18 57 65)(19 66 58)(20 59 67)(21 68 60)(22 61 69)(23 70 62)(24 63 71)(25 72 64)(26 49 73)(27 74 50)(28 51 75)(29 76 52)(30 53 77)(31 78 54)(32 55 79)
(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 80 9 72)(2 79 10 71)(3 78 11 70)(4 77 12 69)(5 76 13 68)(6 75 14 67)(7 74 15 66)(8 73 16 65)(17 46 25 38)(18 45 26 37)(19 44 27 36)(20 43 28 35)(21 42 29 34)(22 41 30 33)(23 40 31 48)(24 39 32 47)(49 88 57 96)(50 87 58 95)(51 86 59 94)(52 85 60 93)(53 84 61 92)(54 83 62 91)(55 82 63 90)(56 81 64 89)
G:=sub<Sym(96)| (1,89,38)(2,90,39)(3,91,40)(4,92,41)(5,93,42)(6,94,43)(7,95,44)(8,96,45)(9,81,46)(10,82,47)(11,83,48)(12,84,33)(13,85,34)(14,86,35)(15,87,36)(16,88,37)(17,80,56)(18,65,57)(19,66,58)(20,67,59)(21,68,60)(22,69,61)(23,70,62)(24,71,63)(25,72,64)(26,73,49)(27,74,50)(28,75,51)(29,76,52)(30,77,53)(31,78,54)(32,79,55), (1,89,38)(2,39,90)(3,91,40)(4,41,92)(5,93,42)(6,43,94)(7,95,44)(8,45,96)(9,81,46)(10,47,82)(11,83,48)(12,33,84)(13,85,34)(14,35,86)(15,87,36)(16,37,88)(17,80,56)(18,57,65)(19,66,58)(20,59,67)(21,68,60)(22,61,69)(23,70,62)(24,63,71)(25,72,64)(26,49,73)(27,74,50)(28,51,75)(29,76,52)(30,53,77)(31,78,54)(32,55,79), (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), (1,80,9,72)(2,79,10,71)(3,78,11,70)(4,77,12,69)(5,76,13,68)(6,75,14,67)(7,74,15,66)(8,73,16,65)(17,46,25,38)(18,45,26,37)(19,44,27,36)(20,43,28,35)(21,42,29,34)(22,41,30,33)(23,40,31,48)(24,39,32,47)(49,88,57,96)(50,87,58,95)(51,86,59,94)(52,85,60,93)(53,84,61,92)(54,83,62,91)(55,82,63,90)(56,81,64,89)>;
G:=Group( (1,89,38)(2,90,39)(3,91,40)(4,92,41)(5,93,42)(6,94,43)(7,95,44)(8,96,45)(9,81,46)(10,82,47)(11,83,48)(12,84,33)(13,85,34)(14,86,35)(15,87,36)(16,88,37)(17,80,56)(18,65,57)(19,66,58)(20,67,59)(21,68,60)(22,69,61)(23,70,62)(24,71,63)(25,72,64)(26,73,49)(27,74,50)(28,75,51)(29,76,52)(30,77,53)(31,78,54)(32,79,55), (1,89,38)(2,39,90)(3,91,40)(4,41,92)(5,93,42)(6,43,94)(7,95,44)(8,45,96)(9,81,46)(10,47,82)(11,83,48)(12,33,84)(13,85,34)(14,35,86)(15,87,36)(16,37,88)(17,80,56)(18,57,65)(19,66,58)(20,59,67)(21,68,60)(22,61,69)(23,70,62)(24,63,71)(25,72,64)(26,49,73)(27,74,50)(28,51,75)(29,76,52)(30,53,77)(31,78,54)(32,55,79), (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), (1,80,9,72)(2,79,10,71)(3,78,11,70)(4,77,12,69)(5,76,13,68)(6,75,14,67)(7,74,15,66)(8,73,16,65)(17,46,25,38)(18,45,26,37)(19,44,27,36)(20,43,28,35)(21,42,29,34)(22,41,30,33)(23,40,31,48)(24,39,32,47)(49,88,57,96)(50,87,58,95)(51,86,59,94)(52,85,60,93)(53,84,61,92)(54,83,62,91)(55,82,63,90)(56,81,64,89) );
G=PermutationGroup([[(1,89,38),(2,90,39),(3,91,40),(4,92,41),(5,93,42),(6,94,43),(7,95,44),(8,96,45),(9,81,46),(10,82,47),(11,83,48),(12,84,33),(13,85,34),(14,86,35),(15,87,36),(16,88,37),(17,80,56),(18,65,57),(19,66,58),(20,67,59),(21,68,60),(22,69,61),(23,70,62),(24,71,63),(25,72,64),(26,73,49),(27,74,50),(28,75,51),(29,76,52),(30,77,53),(31,78,54),(32,79,55)], [(1,89,38),(2,39,90),(3,91,40),(4,41,92),(5,93,42),(6,43,94),(7,95,44),(8,45,96),(9,81,46),(10,47,82),(11,83,48),(12,33,84),(13,85,34),(14,35,86),(15,87,36),(16,37,88),(17,80,56),(18,57,65),(19,66,58),(20,59,67),(21,68,60),(22,61,69),(23,70,62),(24,63,71),(25,72,64),(26,49,73),(27,74,50),(28,51,75),(29,76,52),(30,53,77),(31,78,54),(32,55,79)], [(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)], [(1,80,9,72),(2,79,10,71),(3,78,11,70),(4,77,12,69),(5,76,13,68),(6,75,14,67),(7,74,15,66),(8,73,16,65),(17,46,25,38),(18,45,26,37),(19,44,27,36),(20,43,28,35),(21,42,29,34),(22,41,30,33),(23,40,31,48),(24,39,32,47),(49,88,57,96),(50,87,58,95),(51,86,59,94),(52,85,60,93),(53,84,61,92),(54,83,62,91),(55,82,63,90),(56,81,64,89)]])
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