direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×S3×C3⋊C8, (C3×S3)⋊C24, C3⋊3(S3×C24), C33⋊8(C2×C8), C12.101S32, (S3×C6).3C12, (S3×C12).8C6, C6.28(S3×C12), C12.44(S3×C6), (S3×C32)⋊3C8, C32⋊5(C2×C24), C32⋊14(S3×C8), C6.1(C6×Dic3), (S3×C12).14S3, C32⋊4C8⋊13C6, (C3×C12).178D6, D6.2(C3×Dic3), (S3×C6).8Dic3, C6.32(S3×Dic3), (C3×Dic3).2C12, (C3×Dic3).8Dic3, Dic3.2(C3×Dic3), (C32×Dic3).5C4, (C32×C12).60C22, C3⋊1(C6×C3⋊C8), (C3×C3⋊C8)⋊5C6, C4.13(C3×S32), (S3×C3×C6).3C4, C32⋊9(C2×C3⋊C8), (C32×C3⋊C8)⋊8C2, (S3×C3×C12).6C2, C2.1(C3×S3×Dic3), (C4×S3).3(C3×S3), (C3×C6).85(C4×S3), (C3×C6).17(C2×C12), (C3×C12).61(C2×C6), (C3×C32⋊4C8)⋊1C2, (C32×C6).22(C2×C4), (C3×C6).45(C2×Dic3), SmallGroup(432,414)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×S3×C3⋊C8 |
Generators and relations for C3×S3×C3⋊C8
G = < a,b,c,d,e | a3=b3=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 304 in 126 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, S3×C8, C2×C3⋊C8, C2×C24, S3×C32, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C32⋊4C8, C3×C24, S3×C12, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C3⋊C8, S3×C24, C6×C3⋊C8, C32×C3⋊C8, C3×C32⋊4C8, S3×C3×C12, C3×S3×C3⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S32, S3×C6, S3×C8, C2×C3⋊C8, C2×C24, C3×C3⋊C8, S3×Dic3, S3×C12, C6×Dic3, C3×S32, S3×C3⋊C8, S3×C24, C6×C3⋊C8, C3×S3×Dic3, C3×S3×C3⋊C8
►On 48 points
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 41 20)(10 42 21)(11 43 22)(12 44 23)(13 45 24)(14 46 17)(15 47 18)(16 48 19)
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 20 41)(10 21 42)(11 22 43)(12 23 44)(13 24 45)(14 17 46)(15 18 47)(16 19 48)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 35 30)(2 31 36)(3 37 32)(4 25 38)(5 39 26)(6 27 40)(7 33 28)(8 29 34)(9 41 20)(10 21 42)(11 43 22)(12 23 44)(13 45 24)(14 17 46)(15 47 18)(16 19 48)
(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)
G:=sub<Sym(48)| (1,30,35)(2,31,36)(3,32,37)(4,25,38)(5,26,39)(6,27,40)(7,28,33)(8,29,34)(9,41,20)(10,42,21)(11,43,22)(12,44,23)(13,45,24)(14,46,17)(15,47,18)(16,48,19), (1,30,35)(2,31,36)(3,32,37)(4,25,38)(5,26,39)(6,27,40)(7,28,33)(8,29,34)(9,20,41)(10,21,42)(11,22,43)(12,23,44)(13,24,45)(14,17,46)(15,18,47)(16,19,48), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (1,35,30)(2,31,36)(3,37,32)(4,25,38)(5,39,26)(6,27,40)(7,33,28)(8,29,34)(9,41,20)(10,21,42)(11,43,22)(12,23,44)(13,45,24)(14,17,46)(15,47,18)(16,19,48), (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)>;
G:=Group( (1,30,35)(2,31,36)(3,32,37)(4,25,38)(5,26,39)(6,27,40)(7,28,33)(8,29,34)(9,41,20)(10,42,21)(11,43,22)(12,44,23)(13,45,24)(14,46,17)(15,47,18)(16,48,19), (1,30,35)(2,31,36)(3,32,37)(4,25,38)(5,26,39)(6,27,40)(7,28,33)(8,29,34)(9,20,41)(10,21,42)(11,22,43)(12,23,44)(13,24,45)(14,17,46)(15,18,47)(16,19,48), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,33)(24,34)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (1,35,30)(2,31,36)(3,37,32)(4,25,38)(5,39,26)(6,27,40)(7,33,28)(8,29,34)(9,41,20)(10,21,42)(11,43,22)(12,23,44)(13,45,24)(14,17,46)(15,47,18)(16,19,48), (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) );
G=PermutationGroup([[(1,30,35),(2,31,36),(3,32,37),(4,25,38),(5,26,39),(6,27,40),(7,28,33),(8,29,34),(9,41,20),(10,42,21),(11,43,22),(12,44,23),(13,45,24),(14,46,17),(15,47,18),(16,48,19)], [(1,30,35),(2,31,36),(3,32,37),(4,25,38),(5,26,39),(6,27,40),(7,28,33),(8,29,34),(9,20,41),(10,21,42),(11,22,43),(12,23,44),(13,24,45),(14,17,46),(15,18,47),(16,19,48)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,33),(24,34),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(1,35,30),(2,31,36),(3,37,32),(4,25,38),(5,39,26),(6,27,40),(7,33,28),(8,29,34),(9,41,20),(10,21,42),(11,43,22),(12,23,44),(13,45,24),(14,17,46),(15,47,18),(16,19,48)], [(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)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | ··· | 6U | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | 12R | 12S | 12T | 12U | ··· | 12Z | 12AA | ··· | 12AF | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | 1 | 3 | 3 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | |||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3×S3 | C3⋊C8 | C4×S3 | C3×Dic3 | S3×C6 | C3×Dic3 | S3×C8 | C3×C3⋊C8 | S3×C12 | S3×C24 | S32 | S3×Dic3 | C3×S32 | S3×C3⋊C8 | C3×S3×Dic3 | C3×S3×C3⋊C8 |
| kernel | C3×S3×C3⋊C8 | C32×C3⋊C8 | C3×C32⋊4C8 | S3×C3×C12 | S3×C3⋊C8 | C32×Dic3 | S3×C3×C6 | C3×C3⋊C8 | C32⋊4C8 | S3×C12 | S3×C32 | C3×Dic3 | S3×C6 | C3×S3 | C3×C3⋊C8 | S3×C12 | C3×Dic3 | C3×C12 | S3×C6 | C3⋊C8 | C4×S3 | C3×S3 | C3×C6 | Dic3 | C12 | D6 | C32 | S3 | C6 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 8 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×S3×C3⋊C8 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,72,0,0,1,0],[51,0,0,0,0,51,0,0,0,0,0,1,0,0,1,0] >;
C3×S3×C3⋊C8 in GAP, Magma, Sage, TeX
C_3\times S_3\times C_3\rtimes C_8
% in TeX
G:=Group("C3xS3xC3:C8"); // GroupNames label
G:=SmallGroup(432,414);
// by ID
G=gap.SmallGroup(432,414);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations