direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C9⋊Dic3, C32⋊4Dic9, C33.7Dic3, C3⋊(C3×Dic9), (C32×C9)⋊6C4, (C3×C6).9D9, C6.7(C3×D9), (C3×C9)⋊15C12, C6.8(C9⋊S3), C18.7(C3×S3), C9⋊3(C3×Dic3), (C3×C18).27C6, (C3×C18).24S3, (C3×C9)⋊10Dic3, (C32×C18).4C2, (C32×C6).16S3, C32.9(C3⋊Dic3), C32.16(C3×Dic3), C2.(C3×C9⋊S3), C6.1(C3×C3⋊S3), (C3×C6).34(C3×S3), C3.1(C3×C3⋊Dic3), (C3×C6).19(C3⋊S3), SmallGroup(324,96)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C3×C9⋊Dic3 |
Generators and relations for C3×C9⋊Dic3
G = < a,b,c,d | a3=b9=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 252 in 90 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C9, C32, C32, C32, Dic3, C12, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, C33, Dic9, C3×Dic3, C3⋊Dic3, C3×C18, C3×C18, C3×C18, C32×C6, C32×C9, C3×Dic9, C9⋊Dic3, C3×C3⋊Dic3, C32×C18, C3×C9⋊Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, D9, C3×S3, C3⋊S3, Dic9, C3×Dic3, C3⋊Dic3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×Dic9, C9⋊Dic3, C3×C3⋊Dic3, C3×C9⋊S3, C3×C9⋊Dic3
►On 108 points
Generators in S108
(1 29 37)(2 30 38)(3 31 39)(4 32 40)(5 33 41)(6 34 42)(7 35 43)(8 36 44)(9 28 45)(10 104 19)(11 105 20)(12 106 21)(13 107 22)(14 108 23)(15 100 24)(16 101 25)(17 102 26)(18 103 27)(46 60 69)(47 61 70)(48 62 71)(49 63 72)(50 55 64)(51 56 65)(52 57 66)(53 58 67)(54 59 68)(73 83 94)(74 84 95)(75 85 96)(76 86 97)(77 87 98)(78 88 99)(79 89 91)(80 90 92)(81 82 93)
(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 97 98 99)(100 101 102 103 104 105 106 107 108)
(1 48 35 59 40 65)(2 49 36 60 41 66)(3 50 28 61 42 67)(4 51 29 62 43 68)(5 52 30 63 44 69)(6 53 31 55 45 70)(7 54 32 56 37 71)(8 46 33 57 38 72)(9 47 34 58 39 64)(10 80 25 98 107 84)(11 81 26 99 108 85)(12 73 27 91 100 86)(13 74 19 92 101 87)(14 75 20 93 102 88)(15 76 21 94 103 89)(16 77 22 95 104 90)(17 78 23 96 105 82)(18 79 24 97 106 83)
(1 23 59 82)(2 22 60 90)(3 21 61 89)(4 20 62 88)(5 19 63 87)(6 27 55 86)(7 26 56 85)(8 25 57 84)(9 24 58 83)(10 72 98 33)(11 71 99 32)(12 70 91 31)(13 69 92 30)(14 68 93 29)(15 67 94 28)(16 66 95 36)(17 65 96 35)(18 64 97 34)(37 108 54 81)(38 107 46 80)(39 106 47 79)(40 105 48 78)(41 104 49 77)(42 103 50 76)(43 102 51 75)(44 101 52 74)(45 100 53 73)
G:=sub<Sym(108)| (1,29,37)(2,30,38)(3,31,39)(4,32,40)(5,33,41)(6,34,42)(7,35,43)(8,36,44)(9,28,45)(10,104,19)(11,105,20)(12,106,21)(13,107,22)(14,108,23)(15,100,24)(16,101,25)(17,102,26)(18,103,27)(46,60,69)(47,61,70)(48,62,71)(49,63,72)(50,55,64)(51,56,65)(52,57,66)(53,58,67)(54,59,68)(73,83,94)(74,84,95)(75,85,96)(76,86,97)(77,87,98)(78,88,99)(79,89,91)(80,90,92)(81,82,93), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,48,35,59,40,65)(2,49,36,60,41,66)(3,50,28,61,42,67)(4,51,29,62,43,68)(5,52,30,63,44,69)(6,53,31,55,45,70)(7,54,32,56,37,71)(8,46,33,57,38,72)(9,47,34,58,39,64)(10,80,25,98,107,84)(11,81,26,99,108,85)(12,73,27,91,100,86)(13,74,19,92,101,87)(14,75,20,93,102,88)(15,76,21,94,103,89)(16,77,22,95,104,90)(17,78,23,96,105,82)(18,79,24,97,106,83), (1,23,59,82)(2,22,60,90)(3,21,61,89)(4,20,62,88)(5,19,63,87)(6,27,55,86)(7,26,56,85)(8,25,57,84)(9,24,58,83)(10,72,98,33)(11,71,99,32)(12,70,91,31)(13,69,92,30)(14,68,93,29)(15,67,94,28)(16,66,95,36)(17,65,96,35)(18,64,97,34)(37,108,54,81)(38,107,46,80)(39,106,47,79)(40,105,48,78)(41,104,49,77)(42,103,50,76)(43,102,51,75)(44,101,52,74)(45,100,53,73)>;
G:=Group( (1,29,37)(2,30,38)(3,31,39)(4,32,40)(5,33,41)(6,34,42)(7,35,43)(8,36,44)(9,28,45)(10,104,19)(11,105,20)(12,106,21)(13,107,22)(14,108,23)(15,100,24)(16,101,25)(17,102,26)(18,103,27)(46,60,69)(47,61,70)(48,62,71)(49,63,72)(50,55,64)(51,56,65)(52,57,66)(53,58,67)(54,59,68)(73,83,94)(74,84,95)(75,85,96)(76,86,97)(77,87,98)(78,88,99)(79,89,91)(80,90,92)(81,82,93), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108), (1,48,35,59,40,65)(2,49,36,60,41,66)(3,50,28,61,42,67)(4,51,29,62,43,68)(5,52,30,63,44,69)(6,53,31,55,45,70)(7,54,32,56,37,71)(8,46,33,57,38,72)(9,47,34,58,39,64)(10,80,25,98,107,84)(11,81,26,99,108,85)(12,73,27,91,100,86)(13,74,19,92,101,87)(14,75,20,93,102,88)(15,76,21,94,103,89)(16,77,22,95,104,90)(17,78,23,96,105,82)(18,79,24,97,106,83), (1,23,59,82)(2,22,60,90)(3,21,61,89)(4,20,62,88)(5,19,63,87)(6,27,55,86)(7,26,56,85)(8,25,57,84)(9,24,58,83)(10,72,98,33)(11,71,99,32)(12,70,91,31)(13,69,92,30)(14,68,93,29)(15,67,94,28)(16,66,95,36)(17,65,96,35)(18,64,97,34)(37,108,54,81)(38,107,46,80)(39,106,47,79)(40,105,48,78)(41,104,49,77)(42,103,50,76)(43,102,51,75)(44,101,52,74)(45,100,53,73) );
G=PermutationGroup([[(1,29,37),(2,30,38),(3,31,39),(4,32,40),(5,33,41),(6,34,42),(7,35,43),(8,36,44),(9,28,45),(10,104,19),(11,105,20),(12,106,21),(13,107,22),(14,108,23),(15,100,24),(16,101,25),(17,102,26),(18,103,27),(46,60,69),(47,61,70),(48,62,71),(49,63,72),(50,55,64),(51,56,65),(52,57,66),(53,58,67),(54,59,68),(73,83,94),(74,84,95),(75,85,96),(76,86,97),(77,87,98),(78,88,99),(79,89,91),(80,90,92),(81,82,93)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108)], [(1,48,35,59,40,65),(2,49,36,60,41,66),(3,50,28,61,42,67),(4,51,29,62,43,68),(5,52,30,63,44,69),(6,53,31,55,45,70),(7,54,32,56,37,71),(8,46,33,57,38,72),(9,47,34,58,39,64),(10,80,25,98,107,84),(11,81,26,99,108,85),(12,73,27,91,100,86),(13,74,19,92,101,87),(14,75,20,93,102,88),(15,76,21,94,103,89),(16,77,22,95,104,90),(17,78,23,96,105,82),(18,79,24,97,106,83)], [(1,23,59,82),(2,22,60,90),(3,21,61,89),(4,20,62,88),(5,19,63,87),(6,27,55,86),(7,26,56,85),(8,25,57,84),(9,24,58,83),(10,72,98,33),(11,71,99,32),(12,70,91,31),(13,69,92,30),(14,68,93,29),(15,67,94,28),(16,66,95,36),(17,65,96,35),(18,64,97,34),(37,108,54,81),(38,107,46,80),(39,106,47,79),(40,105,48,78),(41,104,49,77),(42,103,50,76),(43,102,51,75),(44,101,52,74),(45,100,53,73)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | 6B | 6C | ··· | 6N | 9A | ··· | 9AA | 12A | 12B | 12C | 12D | 18A | ··· | 18AA |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 27 | 27 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 27 | 27 | 27 | 27 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | S3 | Dic3 | Dic3 | C3×S3 | D9 | C3×S3 | C3×Dic3 | Dic9 | C3×Dic3 | C3×D9 | C3×Dic9 |
| kernel | C3×C9⋊Dic3 | C32×C18 | C9⋊Dic3 | C32×C9 | C3×C18 | C3×C9 | C3×C18 | C32×C6 | C3×C9 | C33 | C18 | C3×C6 | C3×C6 | C9 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 3 | 1 | 3 | 1 | 6 | 9 | 2 | 6 | 9 | 2 | 18 | 18 |
Matrix representation of C3×C9⋊Dic3 ►in GL5(𝔽37)
| 10 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 26 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 33 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(37))| [10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,16,0,0,0,0,0,7,0,0,0,0,0,9,0,0,0,0,0,33],[36,0,0,0,0,0,10,0,0,0,0,0,26,0,0,0,0,0,1,0,0,0,0,0,1],[31,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;
C3×C9⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times C_9\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xC9:Dic3"); // GroupNames label
G:=SmallGroup(324,96);
// by ID
G=gap.SmallGroup(324,96);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,3171,453,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations