metabelian, supersoluble, monomial, A-group
Aliases: C32⋊4D9, C33.7S3, C9⋊(C3⋊S3), C3⋊(C9⋊S3), (C3×C9)⋊11S3, (C32×C9)⋊5C2, C3.(C33⋊C2), C32.10(C3⋊S3), SmallGroup(162,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×C9 — C32⋊4D9 |
Generators and relations for C32⋊4D9
G = < a,b,c,d | a3=b3=c9=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 720 in 100 conjugacy classes, 51 normal (5 characteristic)
C1, C2, C3, C3, S3, C9, C32, D9, C3⋊S3, C3×C9, C33, C9⋊S3, C33⋊C2, C32×C9, C32⋊4D9
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, C33⋊C2, C32⋊4D9
►On 81 points
Generators in S81
(1 25 47)(2 26 48)(3 27 49)(4 19 50)(5 20 51)(6 21 52)(7 22 53)(8 23 54)(9 24 46)(10 37 67)(11 38 68)(12 39 69)(13 40 70)(14 41 71)(15 42 72)(16 43 64)(17 44 65)(18 45 66)(28 77 58)(29 78 59)(30 79 60)(31 80 61)(32 81 62)(33 73 63)(34 74 55)(35 75 56)(36 76 57)
(1 77 68)(2 78 69)(3 79 70)(4 80 71)(5 81 72)(6 73 64)(7 74 65)(8 75 66)(9 76 67)(10 24 57)(11 25 58)(12 26 59)(13 27 60)(14 19 61)(15 20 62)(16 21 63)(17 22 55)(18 23 56)(28 38 47)(29 39 48)(30 40 49)(31 41 50)(32 42 51)(33 43 52)(34 44 53)(35 45 54)(36 37 46)
(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)
(1 9)(2 8)(3 7)(4 6)(10 28)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 54)(27 53)(37 58)(38 57)(39 56)(40 55)(41 63)(42 62)(43 61)(44 60)(45 59)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(72 81)
G:=sub<Sym(81)| (1,25,47)(2,26,48)(3,27,49)(4,19,50)(5,20,51)(6,21,52)(7,22,53)(8,23,54)(9,24,46)(10,37,67)(11,38,68)(12,39,69)(13,40,70)(14,41,71)(15,42,72)(16,43,64)(17,44,65)(18,45,66)(28,77,58)(29,78,59)(30,79,60)(31,80,61)(32,81,62)(33,73,63)(34,74,55)(35,75,56)(36,76,57), (1,77,68)(2,78,69)(3,79,70)(4,80,71)(5,81,72)(6,73,64)(7,74,65)(8,75,66)(9,76,67)(10,24,57)(11,25,58)(12,26,59)(13,27,60)(14,19,61)(15,20,62)(16,21,63)(17,22,55)(18,23,56)(28,38,47)(29,39,48)(30,40,49)(31,41,50)(32,42,51)(33,43,52)(34,44,53)(35,45,54)(36,37,46), (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), (1,9)(2,8)(3,7)(4,6)(10,28)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53)(37,58)(38,57)(39,56)(40,55)(41,63)(42,62)(43,61)(44,60)(45,59)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,81)>;
G:=Group( (1,25,47)(2,26,48)(3,27,49)(4,19,50)(5,20,51)(6,21,52)(7,22,53)(8,23,54)(9,24,46)(10,37,67)(11,38,68)(12,39,69)(13,40,70)(14,41,71)(15,42,72)(16,43,64)(17,44,65)(18,45,66)(28,77,58)(29,78,59)(30,79,60)(31,80,61)(32,81,62)(33,73,63)(34,74,55)(35,75,56)(36,76,57), (1,77,68)(2,78,69)(3,79,70)(4,80,71)(5,81,72)(6,73,64)(7,74,65)(8,75,66)(9,76,67)(10,24,57)(11,25,58)(12,26,59)(13,27,60)(14,19,61)(15,20,62)(16,21,63)(17,22,55)(18,23,56)(28,38,47)(29,39,48)(30,40,49)(31,41,50)(32,42,51)(33,43,52)(34,44,53)(35,45,54)(36,37,46), (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), (1,9)(2,8)(3,7)(4,6)(10,28)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,54)(27,53)(37,58)(38,57)(39,56)(40,55)(41,63)(42,62)(43,61)(44,60)(45,59)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(72,81) );
G=PermutationGroup([[(1,25,47),(2,26,48),(3,27,49),(4,19,50),(5,20,51),(6,21,52),(7,22,53),(8,23,54),(9,24,46),(10,37,67),(11,38,68),(12,39,69),(13,40,70),(14,41,71),(15,42,72),(16,43,64),(17,44,65),(18,45,66),(28,77,58),(29,78,59),(30,79,60),(31,80,61),(32,81,62),(33,73,63),(34,74,55),(35,75,56),(36,76,57)], [(1,77,68),(2,78,69),(3,79,70),(4,80,71),(5,81,72),(6,73,64),(7,74,65),(8,75,66),(9,76,67),(10,24,57),(11,25,58),(12,26,59),(13,27,60),(14,19,61),(15,20,62),(16,21,63),(17,22,55),(18,23,56),(28,38,47),(29,39,48),(30,40,49),(31,41,50),(32,42,51),(33,43,52),(34,44,53),(35,45,54),(36,37,46)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,28),(11,36),(12,35),(13,34),(14,33),(15,32),(16,31),(17,30),(18,29),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,54),(27,53),(37,58),(38,57),(39,56),(40,55),(41,63),(42,62),(43,61),(44,60),(45,59),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(72,81)]])
C32⋊4D9 is a maximal subgroup of
D9×C3⋊S3 S3×C9⋊S3 He3⋊D9 C33⋊D9 He3⋊3D9 C9⋊He3⋊2C2 (C32×C9)⋊C6 C32⋊4D9⋊C3 He3⋊C3⋊3S3 C3≀C3.S3 C92⋊8S3 C32⋊4D27 C34.11S3 C9○He3⋊3S3 C33⋊9D9
C32⋊4D9 is a maximal quotient of
C32⋊5Dic9 C92⋊8S3 C33⋊6D9 He3⋊4D9 C32⋊4D27 C33⋊9D9
42 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3M | 9A | ··· | 9AA |
| order | 1 | 2 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 81 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | S3 | S3 | D9 |
| kernel | C32⋊4D9 | C32×C9 | C3×C9 | C33 | C32 |
| # reps | 1 | 1 | 12 | 1 | 27 |
Matrix representation of C32⋊4D9 ►in GL6(𝔽19)
| 18 | 18 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 7 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 17 |
| 0 | 0 | 0 | 0 | 12 | 5 |
G:=sub<GL(6,GF(19))| [18,1,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[18,1,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,12,0,0,0,0,7,5],[18,0,0,0,0,0,18,1,0,0,0,0,0,0,18,1,0,0,0,0,0,1,0,0,0,0,0,0,14,12,0,0,0,0,17,5] >;
C32⋊4D9 in GAP, Magma, Sage, TeX
C_3^2\rtimes_4D_9
% in TeX
G:=Group("C3^2:4D9"); // GroupNames label
G:=SmallGroup(162,45);
// by ID
G=gap.SmallGroup(162,45);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,581,546,182,723,2704]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^9=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations