metabelian, soluble, monomial, A-group
Aliases: C3⋊S3⋊3C16, (C3×C24).2C4, C32⋊3(C2×C16), C8.5(C32⋊C4), C3⋊Dic3.6C8, C32⋊2C16⋊5C2, C32⋊4C8.32C22, (C2×C3⋊S3).6C8, (C8×C3⋊S3).6C2, (C3×C6).8(C2×C8), (C4×C3⋊S3).10C4, (C3×C12).8(C2×C4), C4.15(C2×C32⋊C4), C2.1(C3⋊S3⋊3C8), SmallGroup(288,412)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3⋊3C16 |
Generators and relations for C3⋊S3⋊3C16
G = < a,b,c,d | a3=b3=c2=d16=1, ab=ba, cac=a-1, dad-1=ab-1, cbc=b-1, dbd-1=a-1b-1, cd=dc >
9C2
9C2
2C3
2C3
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
9C8
2C12
2C12
6Dic3
6D6
6Dic3
6D6
9C16
9C16
2C24
2C24
Smallest permutation representation of C3⋊S3⋊3C16
►On 48 points
Generators in S48
(1 20 43)(2 21 44)(3 45 22)(4 46 23)(5 24 47)(6 25 48)(7 33 26)(8 34 27)(9 28 35)(10 29 36)(11 37 30)(12 38 31)(13 32 39)(14 17 40)(15 41 18)(16 42 19)
(2 44 21)(4 23 46)(6 48 25)(8 27 34)(10 36 29)(12 31 38)(14 40 17)(16 19 42)
(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(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)| (1,20,43)(2,21,44)(3,45,22)(4,46,23)(5,24,47)(6,25,48)(7,33,26)(8,34,27)(9,28,35)(10,29,36)(11,37,30)(12,38,31)(13,32,39)(14,17,40)(15,41,18)(16,42,19), (2,44,21)(4,23,46)(6,48,25)(8,27,34)(10,36,29)(12,31,38)(14,40,17)(16,19,42), (17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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( (1,20,43)(2,21,44)(3,45,22)(4,46,23)(5,24,47)(6,25,48)(7,33,26)(8,34,27)(9,28,35)(10,29,36)(11,37,30)(12,38,31)(13,32,39)(14,17,40)(15,41,18)(16,42,19), (2,44,21)(4,23,46)(6,48,25)(8,27,34)(10,36,29)(12,31,38)(14,40,17)(16,19,42), (17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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([(1,20,43),(2,21,44),(3,45,22),(4,46,23),(5,24,47),(6,25,48),(7,33,26),(8,34,27),(9,28,35),(10,29,36),(11,37,30),(12,38,31),(13,32,39),(14,17,40),(15,41,18),(16,42,19)], [(2,44,21),(4,23,46),(6,48,25),(8,27,34),(10,36,29),(12,31,38),(14,40,17),(16,19,42)], [(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 16A | ··· | 16P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 1 | 1 | 9 | 9 | 4 | 4 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 4 | 4 | 4 | 4 | 9 | ··· | 9 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C32⋊C4 | C2×C32⋊C4 | C3⋊S3⋊3C8 | C3⋊S3⋊3C16 |
| kernel | C3⋊S3⋊3C16 | C32⋊2C16 | C8×C3⋊S3 | C3×C24 | C4×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊S3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of C3⋊S3⋊3C16 ►in GL4(𝔽97) generated by
| 0 | 1 | 0 | 64 |
| 96 | 96 | 33 | 0 |
| 0 | 0 | 0 | 96 |
| 0 | 0 | 1 | 96 |
| 1 | 0 | 64 | 0 |
| 0 | 1 | 64 | 0 |
| 0 | 0 | 96 | 1 |
| 0 | 0 | 96 | 0 |
| 0 | 1 | 0 | 64 |
| 1 | 0 | 0 | 64 |
| 0 | 0 | 1 | 96 |
| 0 | 0 | 0 | 96 |
| 22 | 0 | 81 | 82 |
| 22 | 0 | 81 | 81 |
| 66 | 33 | 75 | 75 |
| 33 | 64 | 0 | 0 |
G:=sub<GL(4,GF(97))| [0,96,0,0,1,96,0,0,0,33,0,1,64,0,96,96],[1,0,0,0,0,1,0,0,64,64,96,96,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,64,64,96,96],[22,22,66,33,0,0,33,64,81,81,75,0,82,81,75,0] >;
C3⋊S3⋊3C16 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\rtimes_3C_{16} % in TeX
G:=Group("C3:S3:3C16"); // GroupNames label
G:=SmallGroup(288,412);
// by ID
G=gap.SmallGroup(288,412);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,64,58,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^2=d^16=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=a^-1*b^-1,c*d=d*c>;
// generators/relations
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