direct product, metabelian, supersoluble, monomial, A-group
Aliases: C32×Dic9, C33.6Dic3, C9⋊3(C3×C12), (C32×C9)⋊5C4, (C3×C9)⋊14C12, C2.(C32×D9), C18.4(C3×C6), C6.10(C3×D9), (C3×C6).11D9, (C3×C18).26C6, C6.1(S3×C32), (C32×C18).3C2, (C32×C6).15S3, C3.1(C32×Dic3), C32.15(C3×Dic3), (C3×C6).33(C3×S3), SmallGroup(324,90)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C32×Dic9 |
Generators and relations for C32×Dic9
G = < a,b,c,d | a3=b3=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 178 in 86 conjugacy classes, 42 normal (14 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, C33, Dic9, C3×Dic3, C3×C12, C3×C18, C3×C18, C32×C6, C32×C9, C3×Dic9, C32×Dic3, C32×C18, C32×Dic9
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, D9, C3×S3, C3×C6, Dic9, C3×Dic3, C3×C12, C3×D9, S3×C32, C3×Dic9, C32×Dic3, C32×D9, C32×Dic9
►On 108 points
Generators in S108
(1 45 21)(2 46 22)(3 47 23)(4 48 24)(5 49 25)(6 50 26)(7 51 27)(8 52 28)(9 53 29)(10 54 30)(11 37 31)(12 38 32)(13 39 33)(14 40 34)(15 41 35)(16 42 36)(17 43 19)(18 44 20)(55 101 89)(56 102 90)(57 103 73)(58 104 74)(59 105 75)(60 106 76)(61 107 77)(62 108 78)(63 91 79)(64 92 80)(65 93 81)(66 94 82)(67 95 83)(68 96 84)(69 97 85)(70 98 86)(71 99 87)(72 100 88)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(73 79 85)(74 80 86)(75 81 87)(76 82 88)(77 83 89)(78 84 90)(91 97 103)(92 98 104)(93 99 105)(94 100 106)(95 101 107)(96 102 108)
(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 72 10 63)(2 71 11 62)(3 70 12 61)(4 69 13 60)(5 68 14 59)(6 67 15 58)(7 66 16 57)(8 65 17 56)(9 64 18 55)(19 90 28 81)(20 89 29 80)(21 88 30 79)(22 87 31 78)(23 86 32 77)(24 85 33 76)(25 84 34 75)(26 83 35 74)(27 82 36 73)(37 108 46 99)(38 107 47 98)(39 106 48 97)(40 105 49 96)(41 104 50 95)(42 103 51 94)(43 102 52 93)(44 101 53 92)(45 100 54 91)
G:=sub<Sym(108)| (1,45,21)(2,46,22)(3,47,23)(4,48,24)(5,49,25)(6,50,26)(7,51,27)(8,52,28)(9,53,29)(10,54,30)(11,37,31)(12,38,32)(13,39,33)(14,40,34)(15,41,35)(16,42,36)(17,43,19)(18,44,20)(55,101,89)(56,102,90)(57,103,73)(58,104,74)(59,105,75)(60,106,76)(61,107,77)(62,108,78)(63,91,79)(64,92,80)(65,93,81)(66,94,82)(67,95,83)(68,96,84)(69,97,85)(70,98,86)(71,99,87)(72,100,88), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72)(73,79,85)(74,80,86)(75,81,87)(76,82,88)(77,83,89)(78,84,90)(91,97,103)(92,98,104)(93,99,105)(94,100,106)(95,101,107)(96,102,108), (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), (1,72,10,63)(2,71,11,62)(3,70,12,61)(4,69,13,60)(5,68,14,59)(6,67,15,58)(7,66,16,57)(8,65,17,56)(9,64,18,55)(19,90,28,81)(20,89,29,80)(21,88,30,79)(22,87,31,78)(23,86,32,77)(24,85,33,76)(25,84,34,75)(26,83,35,74)(27,82,36,73)(37,108,46,99)(38,107,47,98)(39,106,48,97)(40,105,49,96)(41,104,50,95)(42,103,51,94)(43,102,52,93)(44,101,53,92)(45,100,54,91)>;
G:=Group( (1,45,21)(2,46,22)(3,47,23)(4,48,24)(5,49,25)(6,50,26)(7,51,27)(8,52,28)(9,53,29)(10,54,30)(11,37,31)(12,38,32)(13,39,33)(14,40,34)(15,41,35)(16,42,36)(17,43,19)(18,44,20)(55,101,89)(56,102,90)(57,103,73)(58,104,74)(59,105,75)(60,106,76)(61,107,77)(62,108,78)(63,91,79)(64,92,80)(65,93,81)(66,94,82)(67,95,83)(68,96,84)(69,97,85)(70,98,86)(71,99,87)(72,100,88), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72)(73,79,85)(74,80,86)(75,81,87)(76,82,88)(77,83,89)(78,84,90)(91,97,103)(92,98,104)(93,99,105)(94,100,106)(95,101,107)(96,102,108), (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), (1,72,10,63)(2,71,11,62)(3,70,12,61)(4,69,13,60)(5,68,14,59)(6,67,15,58)(7,66,16,57)(8,65,17,56)(9,64,18,55)(19,90,28,81)(20,89,29,80)(21,88,30,79)(22,87,31,78)(23,86,32,77)(24,85,33,76)(25,84,34,75)(26,83,35,74)(27,82,36,73)(37,108,46,99)(38,107,47,98)(39,106,48,97)(40,105,49,96)(41,104,50,95)(42,103,51,94)(43,102,52,93)(44,101,53,92)(45,100,54,91) );
G=PermutationGroup([[(1,45,21),(2,46,22),(3,47,23),(4,48,24),(5,49,25),(6,50,26),(7,51,27),(8,52,28),(9,53,29),(10,54,30),(11,37,31),(12,38,32),(13,39,33),(14,40,34),(15,41,35),(16,42,36),(17,43,19),(18,44,20),(55,101,89),(56,102,90),(57,103,73),(58,104,74),(59,105,75),(60,106,76),(61,107,77),(62,108,78),(63,91,79),(64,92,80),(65,93,81),(66,94,82),(67,95,83),(68,96,84),(69,97,85),(70,98,86),(71,99,87),(72,100,88)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72),(73,79,85),(74,80,86),(75,81,87),(76,82,88),(77,83,89),(78,84,90),(91,97,103),(92,98,104),(93,99,105),(94,100,106),(95,101,107),(96,102,108)], [(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)], [(1,72,10,63),(2,71,11,62),(3,70,12,61),(4,69,13,60),(5,68,14,59),(6,67,15,58),(7,66,16,57),(8,65,17,56),(9,64,18,55),(19,90,28,81),(20,89,29,80),(21,88,30,79),(22,87,31,78),(23,86,32,77),(24,85,33,76),(25,84,34,75),(26,83,35,74),(27,82,36,73),(37,108,46,99),(38,107,47,98),(39,106,48,97),(40,105,49,96),(41,104,50,95),(42,103,51,94),(43,102,52,93),(44,101,53,92),(45,100,54,91)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 9A | ··· | 9AA | 12A | ··· | 12P | 18A | ··· | 18AA |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | S3 | Dic3 | D9 | C3×S3 | Dic9 | C3×Dic3 | C3×D9 | C3×Dic9 |
| kernel | C32×Dic9 | C32×C18 | C3×Dic9 | C32×C9 | C3×C18 | C3×C9 | C32×C6 | C33 | C3×C6 | C3×C6 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 8 | 16 | 1 | 1 | 3 | 8 | 3 | 8 | 24 | 24 |
Matrix representation of C32×Dic9 ►in GL5(𝔽37)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 26 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 |
| 0 | 0 | 0 | 0 | 26 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 20 | 34 |
| 31 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 25 | 9 |
| 0 | 0 | 0 | 17 | 12 |
G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[26,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26],[36,0,0,0,0,0,0,36,0,0,0,1,36,0,0,0,0,0,12,20,0,0,0,0,34],[31,0,0,0,0,0,1,0,0,0,0,1,36,0,0,0,0,0,25,17,0,0,0,9,12] >;
C32×Dic9 in GAP, Magma, Sage, TeX
C_3^2\times {\rm Dic}_9 % in TeX
G:=Group("C3^2xDic9"); // GroupNames label
G:=SmallGroup(324,90);
// by ID
G=gap.SmallGroup(324,90);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,5404,208,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations