direct product, metabelian, supersoluble, monomial
Aliases: C3×S3×SD16, C24⋊21D6, C8⋊5(S3×C6), (S3×D4).C6, (S3×C8)⋊4C6, C24⋊5(C2×C6), Q8⋊3(S3×C6), (S3×Q8)⋊4C6, C24⋊C2⋊5C6, D4.S3⋊3C6, (C3×Q8)⋊14D6, D4.2(S3×C6), C3⋊2(C6×SD16), C6.30(C6×D4), (S3×C24)⋊10C2, Q8⋊2S3⋊1C6, (C3×SD16)⋊3C6, Dic6⋊2(C2×C6), (C3×D4).25D6, D12.2(C2×C6), D6.13(C3×D4), (S3×C6).49D4, C6.190(S3×D4), (C3×C24)⋊16C22, C12.4(C22×C6), Dic3.4(C3×D4), C32⋊15(C2×SD16), (C3×C12).75C23, (C3×Dic3).31D4, (C32×SD16)⋊5C2, (Q8×C32)⋊5C22, (S3×C12).49C22, C12.155(C22×S3), (C3×Dic6)⋊12C22, (C3×D12).26C22, (D4×C32).12C22, C3⋊C8⋊6(C2×C6), C4.4(S3×C2×C6), (C3×S3×Q8)⋊4C2, (C3×S3×D4).2C2, C2.18(C3×S3×D4), (C3×Q8)⋊2(C2×C6), (C3×C3⋊C8)⋊38C22, (C4×S3).9(C2×C6), (C3×D4).2(C2×C6), (C3×C24⋊C2)⋊13C2, (C3×D4.S3)⋊14C2, (C3×C6).218(C2×D4), (C3×Q8⋊2S3)⋊13C2, SmallGroup(288,684)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3×SD16
G = < a,b,c,d,e | a3=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 402 in 146 conjugacy classes, 58 normal (54 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, SD16, SD16, C2×D4, C2×Q8, C3×S3, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C2×SD16, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C2×C24, C3×SD16, C3×SD16, S3×D4, S3×Q8, C6×D4, C6×Q8, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×C3⋊D4, D4×C32, Q8×C32, S3×C2×C6, S3×SD16, C6×SD16, S3×C24, C3×C24⋊C2, C3×D4.S3, C3×Q8⋊2S3, C32×SD16, C3×S3×D4, C3×S3×Q8, C3×S3×SD16
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, SD16, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C2×SD16, S3×C6, C3×SD16, S3×D4, C6×D4, S3×C2×C6, S3×SD16, C6×SD16, C3×S3×D4, C3×S3×SD16
►On 48 points
(1 31 40)(2 32 33)(3 25 34)(4 26 35)(5 27 36)(6 28 37)(7 29 38)(8 30 39)(9 48 23)(10 41 24)(11 42 17)(12 43 18)(13 44 19)(14 45 20)(15 46 21)(16 47 22)
(1 31 40)(2 32 33)(3 25 34)(4 26 35)(5 27 36)(6 28 37)(7 29 38)(8 30 39)(9 23 48)(10 24 41)(11 17 42)(12 18 43)(13 19 44)(14 20 45)(15 21 46)(16 22 47)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 44)(26 45)(27 46)(28 47)(29 48)(30 41)(31 42)(32 43)
(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)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)(25 29)(26 32)(28 30)(33 35)(34 38)(37 39)(41 47)(43 45)(44 48)
G:=sub<Sym(48)| (1,31,40)(2,32,33)(3,25,34)(4,26,35)(5,27,36)(6,28,37)(7,29,38)(8,30,39)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (1,31,40)(2,32,33)(3,25,34)(4,26,35)(5,27,36)(6,28,37)(7,29,38)(8,30,39)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48)>;
G:=Group( (1,31,40)(2,32,33)(3,25,34)(4,26,35)(5,27,36)(6,28,37)(7,29,38)(8,30,39)(9,48,23)(10,41,24)(11,42,17)(12,43,18)(13,44,19)(14,45,20)(15,46,21)(16,47,22), (1,31,40)(2,32,33)(3,25,34)(4,26,35)(5,27,36)(6,28,37)(7,29,38)(8,30,39)(9,23,48)(10,24,41)(11,17,42)(12,18,43)(13,19,44)(14,20,45)(15,21,46)(16,22,47), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,40)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,44)(26,45)(27,46)(28,47)(29,48)(30,41)(31,42)(32,43), (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), (2,4)(3,7)(6,8)(9,13)(10,16)(12,14)(18,20)(19,23)(22,24)(25,29)(26,32)(28,30)(33,35)(34,38)(37,39)(41,47)(43,45)(44,48) );
G=PermutationGroup([[(1,31,40),(2,32,33),(3,25,34),(4,26,35),(5,27,36),(6,28,37),(7,29,38),(8,30,39),(9,48,23),(10,41,24),(11,42,17),(12,43,18),(13,44,19),(14,45,20),(15,46,21),(16,47,22)], [(1,31,40),(2,32,33),(3,25,34),(4,26,35),(5,27,36),(6,28,37),(7,29,38),(8,30,39),(9,23,48),(10,24,41),(11,17,42),(12,18,43),(13,19,44),(14,20,45),(15,21,46),(16,22,47)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,40),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,44),(26,45),(27,46),(28,47),(29,48),(30,41),(31,42),(32,43)], [(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)], [(2,4),(3,7),(6,8),(9,13),(10,16),(12,14),(18,20),(19,23),(22,24),(25,29),(26,32),(28,30),(33,35),(34,38),(37,39),(41,47),(43,45),(44,48)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 8 | 8 | 8 | 12 | 12 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
63 irreducible representations
| dim | 1 | 1 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | S3×C6 | C3×SD16 | S3×D4 | S3×SD16 | C3×S3×D4 | C3×S3×SD16 |
| kernel | C3×S3×SD16 | S3×C24 | C3×C24⋊C2 | C3×D4.S3 | C3×Q8⋊2S3 | C32×SD16 | C3×S3×D4 | C3×S3×Q8 | S3×SD16 | S3×C8 | C24⋊C2 | D4.S3 | Q8⋊2S3 | C3×SD16 | S3×D4 | S3×Q8 | C3×SD16 | C3×Dic3 | S3×C6 | C24 | C3×D4 | C3×Q8 | C3×S3 | SD16 | Dic3 | D6 | C8 | D4 | Q8 | S3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×S3×SD16 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 64 | 0 | 0 | 0 |
| 65 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 7 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 6 | 12 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[64,65,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,7,72,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,6,0,0,61,12],[72,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
C3×S3×SD16 in GAP, Magma, Sage, TeX
C_3\times S_3\times {\rm SD}_{16} % in TeX
G:=Group("C3xS3xSD16"); // GroupNames label
G:=SmallGroup(288,684);
// by ID
G=gap.SmallGroup(288,684);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,303,268,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=e^2=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=d^3>;
// generators/relations