direct product, metabelian, supersoluble, monomial, A-group
Aliases: C6×C33⋊C2, C33⋊20D6, C34⋊9C22, (C32×C6)⋊8S3, (C32×C6)⋊9C6, (C33×C6)⋊3C2, C33⋊13(C2×C6), C32⋊11(S3×C6), C6⋊(C3×C3⋊S3), C3⋊2(C6×C3⋊S3), (C3×C6)⋊6(C3×S3), (C3×C6)⋊3(C3⋊S3), C32⋊8(C2×C3⋊S3), SmallGroup(324,174)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C6×C33⋊C2 |
Generators and relations for C6×C33⋊C2
G = < a,b,c,d,e | a6=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 1528 in 436 conjugacy classes, 118 normal (10 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C33, C33, C33, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C33⋊C2, C32×C6, C32×C6, C32×C6, C34, C6×C3⋊S3, C2×C33⋊C2, C3×C33⋊C2, C33×C6, C6×C33⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, C3⋊S3, S3×C6, C2×C3⋊S3, C3×C3⋊S3, C33⋊C2, C6×C3⋊S3, C2×C33⋊C2, C3×C33⋊C2, C6×C33⋊C2
►On 108 points
Generators in S108
(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)
(1 101 41)(2 102 42)(3 97 37)(4 98 38)(5 99 39)(6 100 40)(7 68 92)(8 69 93)(9 70 94)(10 71 95)(11 72 96)(12 67 91)(13 107 90)(14 108 85)(15 103 86)(16 104 87)(17 105 88)(18 106 89)(19 63 52)(20 64 53)(21 65 54)(22 66 49)(23 61 50)(24 62 51)(25 47 81)(26 48 82)(27 43 83)(28 44 84)(29 45 79)(30 46 80)(31 74 57)(32 75 58)(33 76 59)(34 77 60)(35 78 55)(36 73 56)
(1 57 91)(2 58 92)(3 59 93)(4 60 94)(5 55 95)(6 56 96)(7 102 32)(8 97 33)(9 98 34)(10 99 35)(11 100 36)(12 101 31)(13 64 80)(14 65 81)(15 66 82)(16 61 83)(17 62 84)(18 63 79)(19 45 89)(20 46 90)(21 47 85)(22 48 86)(23 43 87)(24 44 88)(25 108 54)(26 103 49)(27 104 50)(28 105 51)(29 106 52)(30 107 53)(37 76 69)(38 77 70)(39 78 71)(40 73 72)(41 74 67)(42 75 68)
(1 78 8)(2 73 9)(3 74 10)(4 75 11)(5 76 12)(6 77 7)(13 22 28)(14 23 29)(15 24 30)(16 19 25)(17 20 26)(18 21 27)(31 95 37)(32 96 38)(33 91 39)(34 92 40)(35 93 41)(36 94 42)(43 106 65)(44 107 66)(45 108 61)(46 103 62)(47 104 63)(48 105 64)(49 84 90)(50 79 85)(51 80 86)(52 81 87)(53 82 88)(54 83 89)(55 69 101)(56 70 102)(57 71 97)(58 72 98)(59 67 99)(60 68 100)
(1 107)(2 108)(3 103)(4 104)(5 105)(6 106)(7 65)(8 66)(9 61)(10 62)(11 63)(12 64)(13 101)(14 102)(15 97)(16 98)(17 99)(18 100)(19 72)(20 67)(21 68)(22 69)(23 70)(24 71)(25 58)(26 59)(27 60)(28 55)(29 56)(30 57)(31 80)(32 81)(33 82)(34 83)(35 84)(36 79)(37 86)(38 87)(39 88)(40 89)(41 90)(42 85)(43 77)(44 78)(45 73)(46 74)(47 75)(48 76)(49 93)(50 94)(51 95)(52 96)(53 91)(54 92)
G:=sub<Sym(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), (1,101,41)(2,102,42)(3,97,37)(4,98,38)(5,99,39)(6,100,40)(7,68,92)(8,69,93)(9,70,94)(10,71,95)(11,72,96)(12,67,91)(13,107,90)(14,108,85)(15,103,86)(16,104,87)(17,105,88)(18,106,89)(19,63,52)(20,64,53)(21,65,54)(22,66,49)(23,61,50)(24,62,51)(25,47,81)(26,48,82)(27,43,83)(28,44,84)(29,45,79)(30,46,80)(31,74,57)(32,75,58)(33,76,59)(34,77,60)(35,78,55)(36,73,56), (1,57,91)(2,58,92)(3,59,93)(4,60,94)(5,55,95)(6,56,96)(7,102,32)(8,97,33)(9,98,34)(10,99,35)(11,100,36)(12,101,31)(13,64,80)(14,65,81)(15,66,82)(16,61,83)(17,62,84)(18,63,79)(19,45,89)(20,46,90)(21,47,85)(22,48,86)(23,43,87)(24,44,88)(25,108,54)(26,103,49)(27,104,50)(28,105,51)(29,106,52)(30,107,53)(37,76,69)(38,77,70)(39,78,71)(40,73,72)(41,74,67)(42,75,68), (1,78,8)(2,73,9)(3,74,10)(4,75,11)(5,76,12)(6,77,7)(13,22,28)(14,23,29)(15,24,30)(16,19,25)(17,20,26)(18,21,27)(31,95,37)(32,96,38)(33,91,39)(34,92,40)(35,93,41)(36,94,42)(43,106,65)(44,107,66)(45,108,61)(46,103,62)(47,104,63)(48,105,64)(49,84,90)(50,79,85)(51,80,86)(52,81,87)(53,82,88)(54,83,89)(55,69,101)(56,70,102)(57,71,97)(58,72,98)(59,67,99)(60,68,100), (1,107)(2,108)(3,103)(4,104)(5,105)(6,106)(7,65)(8,66)(9,61)(10,62)(11,63)(12,64)(13,101)(14,102)(15,97)(16,98)(17,99)(18,100)(19,72)(20,67)(21,68)(22,69)(23,70)(24,71)(25,58)(26,59)(27,60)(28,55)(29,56)(30,57)(31,80)(32,81)(33,82)(34,83)(35,84)(36,79)(37,86)(38,87)(39,88)(40,89)(41,90)(42,85)(43,77)(44,78)(45,73)(46,74)(47,75)(48,76)(49,93)(50,94)(51,95)(52,96)(53,91)(54,92)>;
G:=Group( (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), (1,101,41)(2,102,42)(3,97,37)(4,98,38)(5,99,39)(6,100,40)(7,68,92)(8,69,93)(9,70,94)(10,71,95)(11,72,96)(12,67,91)(13,107,90)(14,108,85)(15,103,86)(16,104,87)(17,105,88)(18,106,89)(19,63,52)(20,64,53)(21,65,54)(22,66,49)(23,61,50)(24,62,51)(25,47,81)(26,48,82)(27,43,83)(28,44,84)(29,45,79)(30,46,80)(31,74,57)(32,75,58)(33,76,59)(34,77,60)(35,78,55)(36,73,56), (1,57,91)(2,58,92)(3,59,93)(4,60,94)(5,55,95)(6,56,96)(7,102,32)(8,97,33)(9,98,34)(10,99,35)(11,100,36)(12,101,31)(13,64,80)(14,65,81)(15,66,82)(16,61,83)(17,62,84)(18,63,79)(19,45,89)(20,46,90)(21,47,85)(22,48,86)(23,43,87)(24,44,88)(25,108,54)(26,103,49)(27,104,50)(28,105,51)(29,106,52)(30,107,53)(37,76,69)(38,77,70)(39,78,71)(40,73,72)(41,74,67)(42,75,68), (1,78,8)(2,73,9)(3,74,10)(4,75,11)(5,76,12)(6,77,7)(13,22,28)(14,23,29)(15,24,30)(16,19,25)(17,20,26)(18,21,27)(31,95,37)(32,96,38)(33,91,39)(34,92,40)(35,93,41)(36,94,42)(43,106,65)(44,107,66)(45,108,61)(46,103,62)(47,104,63)(48,105,64)(49,84,90)(50,79,85)(51,80,86)(52,81,87)(53,82,88)(54,83,89)(55,69,101)(56,70,102)(57,71,97)(58,72,98)(59,67,99)(60,68,100), (1,107)(2,108)(3,103)(4,104)(5,105)(6,106)(7,65)(8,66)(9,61)(10,62)(11,63)(12,64)(13,101)(14,102)(15,97)(16,98)(17,99)(18,100)(19,72)(20,67)(21,68)(22,69)(23,70)(24,71)(25,58)(26,59)(27,60)(28,55)(29,56)(30,57)(31,80)(32,81)(33,82)(34,83)(35,84)(36,79)(37,86)(38,87)(39,88)(40,89)(41,90)(42,85)(43,77)(44,78)(45,73)(46,74)(47,75)(48,76)(49,93)(50,94)(51,95)(52,96)(53,91)(54,92) );
G=PermutationGroup([[(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)], [(1,101,41),(2,102,42),(3,97,37),(4,98,38),(5,99,39),(6,100,40),(7,68,92),(8,69,93),(9,70,94),(10,71,95),(11,72,96),(12,67,91),(13,107,90),(14,108,85),(15,103,86),(16,104,87),(17,105,88),(18,106,89),(19,63,52),(20,64,53),(21,65,54),(22,66,49),(23,61,50),(24,62,51),(25,47,81),(26,48,82),(27,43,83),(28,44,84),(29,45,79),(30,46,80),(31,74,57),(32,75,58),(33,76,59),(34,77,60),(35,78,55),(36,73,56)], [(1,57,91),(2,58,92),(3,59,93),(4,60,94),(5,55,95),(6,56,96),(7,102,32),(8,97,33),(9,98,34),(10,99,35),(11,100,36),(12,101,31),(13,64,80),(14,65,81),(15,66,82),(16,61,83),(17,62,84),(18,63,79),(19,45,89),(20,46,90),(21,47,85),(22,48,86),(23,43,87),(24,44,88),(25,108,54),(26,103,49),(27,104,50),(28,105,51),(29,106,52),(30,107,53),(37,76,69),(38,77,70),(39,78,71),(40,73,72),(41,74,67),(42,75,68)], [(1,78,8),(2,73,9),(3,74,10),(4,75,11),(5,76,12),(6,77,7),(13,22,28),(14,23,29),(15,24,30),(16,19,25),(17,20,26),(18,21,27),(31,95,37),(32,96,38),(33,91,39),(34,92,40),(35,93,41),(36,94,42),(43,106,65),(44,107,66),(45,108,61),(46,103,62),(47,104,63),(48,105,64),(49,84,90),(50,79,85),(51,80,86),(52,81,87),(53,82,88),(54,83,89),(55,69,101),(56,70,102),(57,71,97),(58,72,98),(59,67,99),(60,68,100)], [(1,107),(2,108),(3,103),(4,104),(5,105),(6,106),(7,65),(8,66),(9,61),(10,62),(11,63),(12,64),(13,101),(14,102),(15,97),(16,98),(17,99),(18,100),(19,72),(20,67),(21,68),(22,69),(23,70),(24,71),(25,58),(26,59),(27,60),(28,55),(29,56),(30,57),(31,80),(32,81),(33,82),(34,83),(35,84),(36,79),(37,86),(38,87),(39,88),(40,89),(41,90),(42,85),(43,77),(44,78),(45,73),(46,74),(47,75),(48,76),(49,93),(50,94),(51,95),(52,96),(53,91),(54,92)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3AO | 6A | 6B | 6C | ··· | 6AO | 6AP | 6AQ | 6AR | 6AS |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 |
| size | 1 | 1 | 27 | 27 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 27 | 27 | 27 | 27 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 |
| kernel | C6×C33⋊C2 | C3×C33⋊C2 | C33×C6 | C2×C33⋊C2 | C33⋊C2 | C32×C6 | C32×C6 | C33 | C3×C6 | C32 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 13 | 13 | 26 | 26 |
Matrix representation of C6×C33⋊C2 ►in GL6(𝔽7)
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(7))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,3,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C6×C33⋊C2 in GAP, Magma, Sage, TeX
C_6\times C_3^3\rtimes C_2
% in TeX
G:=Group("C6xC3^3:C2"); // GroupNames label
G:=SmallGroup(324,174);
// by ID
G=gap.SmallGroup(324,174);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations