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 25 39)(2 26 40)(3 41 27)(4 42 28)(5 29 43)(6 30 44)(7 45 31)(8 46 32)(9 17 47)(10 18 48)(11 33 19)(12 34 20)(13 21 35)(14 22 36)(15 37 23)(16 38 24)
(2 40 26)(4 28 42)(6 44 30)(8 32 46)(10 48 18)(12 20 34)(14 36 22)(16 24 38)
(17 47)(18 48)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 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)
G:=sub<Sym(48)| (1,25,39)(2,26,40)(3,41,27)(4,42,28)(5,29,43)(6,30,44)(7,45,31)(8,46,32)(9,17,47)(10,18,48)(11,33,19)(12,34,20)(13,21,35)(14,22,36)(15,37,23)(16,38,24), (2,40,26)(4,28,42)(6,44,30)(8,32,46)(10,48,18)(12,20,34)(14,36,22)(16,24,38), (17,47)(18,48)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (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)>;
G:=Group( (1,25,39)(2,26,40)(3,41,27)(4,42,28)(5,29,43)(6,30,44)(7,45,31)(8,46,32)(9,17,47)(10,18,48)(11,33,19)(12,34,20)(13,21,35)(14,22,36)(15,37,23)(16,38,24), (2,40,26)(4,28,42)(6,44,30)(8,32,46)(10,48,18)(12,20,34)(14,36,22)(16,24,38), (17,47)(18,48)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46), (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) );
G=PermutationGroup([[(1,25,39),(2,26,40),(3,41,27),(4,42,28),(5,29,43),(6,30,44),(7,45,31),(8,46,32),(9,17,47),(10,18,48),(11,33,19),(12,34,20),(13,21,35),(14,22,36),(15,37,23),(16,38,24)], [(2,40,26),(4,28,42),(6,44,30),(8,32,46),(10,48,18),(12,20,34),(14,36,22),(16,24,38)], [(17,47),(18,48),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46)], [(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)]])
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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