direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C3⋊D15, C33⋊2D5, C32⋊3D15, C3⋊(C3×D15), C15⋊1(C3×S3), (C3×C15)⋊6S3, (C3×C15)⋊5C6, C15⋊2(C3⋊S3), (C32×C15)⋊3C2, C32⋊4(C3×D5), C5⋊(C3×C3⋊S3), SmallGroup(270,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C3×C3⋊D15 |
Generators and relations for C3×C3⋊D15
G = < a,b,c,d | a3=b3=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 320 in 64 conjugacy classes, 26 normal (10 characteristic)
C1, C2, C3, C3, C3, C5, S3, C6, C32, C32, C32, D5, C15, C15, C15, C3×S3, C3⋊S3, C33, C3×D5, D15, C3×C15, C3×C15, C3×C15, C3×C3⋊S3, C3×D15, C3⋊D15, C32×C15, C3×C3⋊D15
Quotients: C1, C2, C3, S3, C6, D5, C3×S3, C3⋊S3, C3×D5, D15, C3×C3⋊S3, C3×D15, C3⋊D15, C3×C3⋊D15
►On 90 points
Generators in S90
(1 24 45)(2 25 31)(3 26 32)(4 27 33)(5 28 34)(6 29 35)(7 30 36)(8 16 37)(9 17 38)(10 18 39)(11 19 40)(12 20 41)(13 21 42)(14 22 43)(15 23 44)(46 73 81)(47 74 82)(48 75 83)(49 61 84)(50 62 85)(51 63 86)(52 64 87)(53 65 88)(54 66 89)(55 67 90)(56 68 76)(57 69 77)(58 70 78)(59 71 79)(60 72 80)
(1 35 19)(2 36 20)(3 37 21)(4 38 22)(5 39 23)(6 40 24)(7 41 25)(8 42 26)(9 43 27)(10 44 28)(11 45 29)(12 31 30)(13 32 16)(14 33 17)(15 34 18)(46 63 76)(47 64 77)(48 65 78)(49 66 79)(50 67 80)(51 68 81)(52 69 82)(53 70 83)(54 71 84)(55 72 85)(56 73 86)(57 74 87)(58 75 88)(59 61 89)(60 62 90)
(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)
(1 48)(2 47)(3 46)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)
G:=sub<Sym(90)| (1,24,45)(2,25,31)(3,26,32)(4,27,33)(5,28,34)(6,29,35)(7,30,36)(8,16,37)(9,17,38)(10,18,39)(11,19,40)(12,20,41)(13,21,42)(14,22,43)(15,23,44)(46,73,81)(47,74,82)(48,75,83)(49,61,84)(50,62,85)(51,63,86)(52,64,87)(53,65,88)(54,66,89)(55,67,90)(56,68,76)(57,69,77)(58,70,78)(59,71,79)(60,72,80), (1,35,19)(2,36,20)(3,37,21)(4,38,22)(5,39,23)(6,40,24)(7,41,25)(8,42,26)(9,43,27)(10,44,28)(11,45,29)(12,31,30)(13,32,16)(14,33,17)(15,34,18)(46,63,76)(47,64,77)(48,65,78)(49,66,79)(50,67,80)(51,68,81)(52,69,82)(53,70,83)(54,71,84)(55,72,85)(56,73,86)(57,74,87)(58,75,88)(59,61,89)(60,62,90), (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), (1,48)(2,47)(3,46)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,84)(45,83)>;
G:=Group( (1,24,45)(2,25,31)(3,26,32)(4,27,33)(5,28,34)(6,29,35)(7,30,36)(8,16,37)(9,17,38)(10,18,39)(11,19,40)(12,20,41)(13,21,42)(14,22,43)(15,23,44)(46,73,81)(47,74,82)(48,75,83)(49,61,84)(50,62,85)(51,63,86)(52,64,87)(53,65,88)(54,66,89)(55,67,90)(56,68,76)(57,69,77)(58,70,78)(59,71,79)(60,72,80), (1,35,19)(2,36,20)(3,37,21)(4,38,22)(5,39,23)(6,40,24)(7,41,25)(8,42,26)(9,43,27)(10,44,28)(11,45,29)(12,31,30)(13,32,16)(14,33,17)(15,34,18)(46,63,76)(47,64,77)(48,65,78)(49,66,79)(50,67,80)(51,68,81)(52,69,82)(53,70,83)(54,71,84)(55,72,85)(56,73,86)(57,74,87)(58,75,88)(59,61,89)(60,62,90), (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), (1,48)(2,47)(3,46)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,53)(12,52)(13,51)(14,50)(15,49)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,84)(45,83) );
G=PermutationGroup([[(1,24,45),(2,25,31),(3,26,32),(4,27,33),(5,28,34),(6,29,35),(7,30,36),(8,16,37),(9,17,38),(10,18,39),(11,19,40),(12,20,41),(13,21,42),(14,22,43),(15,23,44),(46,73,81),(47,74,82),(48,75,83),(49,61,84),(50,62,85),(51,63,86),(52,64,87),(53,65,88),(54,66,89),(55,67,90),(56,68,76),(57,69,77),(58,70,78),(59,71,79),(60,72,80)], [(1,35,19),(2,36,20),(3,37,21),(4,38,22),(5,39,23),(6,40,24),(7,41,25),(8,42,26),(9,43,27),(10,44,28),(11,45,29),(12,31,30),(13,32,16),(14,33,17),(15,34,18),(46,63,76),(47,64,77),(48,65,78),(49,66,79),(50,67,80),(51,68,81),(52,69,82),(53,70,83),(54,71,84),(55,72,85),(56,73,86),(57,74,87),(58,75,88),(59,61,89),(60,62,90)], [(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)], [(1,48),(2,47),(3,46),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,53),(12,52),(13,51),(14,50),(15,49),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,75),(25,74),(26,73),(27,72),(28,71),(29,70),(30,69),(31,82),(32,81),(33,80),(34,79),(35,78),(36,77),(37,76),(38,90),(39,89),(40,88),(41,87),(42,86),(43,85),(44,84),(45,83)]])
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 5A | 5B | 6A | 6B | 15A | ··· | 15AZ |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 5 | 5 | 6 | 6 | 15 | ··· | 15 |
| size | 1 | 45 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 45 | 45 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | D5 | C3×S3 | C3×D5 | D15 | C3×D15 |
| kernel | C3×C3⋊D15 | C32×C15 | C3⋊D15 | C3×C15 | C3×C15 | C33 | C15 | C32 | C32 | C3 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of C3×C3⋊D15 ►in GL4(𝔽31) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 25 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 5 |
| 19 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 18 | 0 | 0 |
| 19 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(31))| [25,0,0,0,0,25,0,0,0,0,5,0,0,0,0,5],[25,0,0,0,0,5,0,0,0,0,25,0,0,0,0,5],[19,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,19,0,0,18,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C3⋊D15 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes D_{15} % in TeX
G:=Group("C3xC3:D15"); // GroupNames label
G:=SmallGroup(270,27);
// by ID
G=gap.SmallGroup(270,27);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-5,182,723,5404]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations