direct product, metabelian, supersoluble, monomial
Aliases: SD16×C3⋊S3, C24⋊10D6, (C3×Q8)⋊7D6, C3⋊5(S3×SD16), C24⋊2S3⋊9C2, (C3×SD16)⋊3S3, (C3×D4).16D6, C6.121(S3×D4), (C3×C24)⋊17C22, C3⋊Dic3.49D4, C32⋊16(C2×SD16), C32⋊11SD16⋊5C2, C32⋊9SD16⋊7C2, (C3×C12).94C23, C12.90(C22×S3), (C32×SD16)⋊7C2, (Q8×C32)⋊7C22, C32⋊4Q8⋊8C22, C32⋊4C8⋊21C22, C12⋊S3.17C22, (D4×C32).17C22, C8⋊5(C2×C3⋊S3), (C8×C3⋊S3)⋊8C2, Q8⋊2(C2×C3⋊S3), (Q8×C3⋊S3)⋊3C2, D4.2(C2×C3⋊S3), (D4×C3⋊S3).3C2, C2.18(D4×C3⋊S3), (C2×C3⋊S3).73D4, C4.4(C22×C3⋊S3), (C3×C6).242(C2×D4), (C4×C3⋊S3).72C22, SmallGroup(288,770)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C3⋊S3
G = < a,b,c,d,e | a8=b2=c3=d3=e2=1, bab=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 948 in 204 conjugacy classes, 57 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, SD16, SD16, C2×D4, C2×Q8, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, C2×SD16, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C32⋊4C8, C3×C24, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, Q8×C32, C22×C3⋊S3, S3×SD16, C8×C3⋊S3, C24⋊2S3, C32⋊9SD16, C32⋊11SD16, C32×SD16, D4×C3⋊S3, Q8×C3⋊S3, SD16×C3⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊S3, C22×S3, C2×SD16, C2×C3⋊S3, S3×D4, C22×C3⋊S3, S3×SD16, D4×C3⋊S3, SD16×C3⋊S3
►On 72 points
(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)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(49 51)(50 54)(53 55)(57 61)(58 64)(60 62)(65 67)(66 70)(69 71)
(1 52 22)(2 53 23)(3 54 24)(4 55 17)(5 56 18)(6 49 19)(7 50 20)(8 51 21)(9 68 63)(10 69 64)(11 70 57)(12 71 58)(13 72 59)(14 65 60)(15 66 61)(16 67 62)(25 42 38)(26 43 39)(27 44 40)(28 45 33)(29 46 34)(30 47 35)(31 48 36)(32 41 37)
(1 47 68)(2 48 69)(3 41 70)(4 42 71)(5 43 72)(6 44 65)(7 45 66)(8 46 67)(9 22 30)(10 23 31)(11 24 32)(12 17 25)(13 18 26)(14 19 27)(15 20 28)(16 21 29)(33 61 50)(34 62 51)(35 63 52)(36 64 53)(37 57 54)(38 58 55)(39 59 56)(40 60 49)
(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 57)(41 70)(42 71)(43 72)(44 65)(45 66)(46 67)(47 68)(48 69)
G:=sub<Sym(72)| (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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,51)(50,54)(53,55)(57,61)(58,64)(60,62)(65,67)(66,70)(69,71), (1,52,22)(2,53,23)(3,54,24)(4,55,17)(5,56,18)(6,49,19)(7,50,20)(8,51,21)(9,68,63)(10,69,64)(11,70,57)(12,71,58)(13,72,59)(14,65,60)(15,66,61)(16,67,62)(25,42,38)(26,43,39)(27,44,40)(28,45,33)(29,46,34)(30,47,35)(31,48,36)(32,41,37), (1,47,68)(2,48,69)(3,41,70)(4,42,71)(5,43,72)(6,44,65)(7,45,66)(8,46,67)(9,22,30)(10,23,31)(11,24,32)(12,17,25)(13,18,26)(14,19,27)(15,20,28)(16,21,29)(33,61,50)(34,62,51)(35,63,52)(36,64,53)(37,57,54)(38,58,55)(39,59,56)(40,60,49), (9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69)>;
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), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(49,51)(50,54)(53,55)(57,61)(58,64)(60,62)(65,67)(66,70)(69,71), (1,52,22)(2,53,23)(3,54,24)(4,55,17)(5,56,18)(6,49,19)(7,50,20)(8,51,21)(9,68,63)(10,69,64)(11,70,57)(12,71,58)(13,72,59)(14,65,60)(15,66,61)(16,67,62)(25,42,38)(26,43,39)(27,44,40)(28,45,33)(29,46,34)(30,47,35)(31,48,36)(32,41,37), (1,47,68)(2,48,69)(3,41,70)(4,42,71)(5,43,72)(6,44,65)(7,45,66)(8,46,67)(9,22,30)(10,23,31)(11,24,32)(12,17,25)(13,18,26)(14,19,27)(15,20,28)(16,21,29)(33,61,50)(34,62,51)(35,63,52)(36,64,53)(37,57,54)(38,58,55)(39,59,56)(40,60,49), (9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,57)(41,70)(42,71)(43,72)(44,65)(45,66)(46,67)(47,68)(48,69) );
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)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,37),(34,40),(36,38),(41,45),(42,48),(44,46),(49,51),(50,54),(53,55),(57,61),(58,64),(60,62),(65,67),(66,70),(69,71)], [(1,52,22),(2,53,23),(3,54,24),(4,55,17),(5,56,18),(6,49,19),(7,50,20),(8,51,21),(9,68,63),(10,69,64),(11,70,57),(12,71,58),(13,72,59),(14,65,60),(15,66,61),(16,67,62),(25,42,38),(26,43,39),(27,44,40),(28,45,33),(29,46,34),(30,47,35),(31,48,36),(32,41,37)], [(1,47,68),(2,48,69),(3,41,70),(4,42,71),(5,43,72),(6,44,65),(7,45,66),(8,46,67),(9,22,30),(10,23,31),(11,24,32),(12,17,25),(13,18,26),(14,19,27),(15,20,28),(16,21,29),(33,61,50),(34,62,51),(35,63,52),(36,64,53),(37,57,54),(38,58,55),(39,59,56),(40,60,49)], [(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,33),(16,34),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,57),(41,70),(42,71),(43,72),(44,65),(45,66),(46,67),(47,68),(48,69)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 9 | 9 | 36 | 2 | 2 | 2 | 2 | 2 | 4 | 18 | 36 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | SD16 | S3×D4 | S3×SD16 |
| kernel | SD16×C3⋊S3 | C8×C3⋊S3 | C24⋊2S3 | C32⋊9SD16 | C32⋊11SD16 | C32×SD16 | D4×C3⋊S3 | Q8×C3⋊S3 | C3×SD16 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×D4 | C3×Q8 | C3⋊S3 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of SD16×C3⋊S3 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 69 | 0 | 0 |
| 0 | 0 | 18 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 48 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,18,0,0,0,0,69,61,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72,72] >;
SD16×C3⋊S3 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_3\rtimes S_3 % in TeX
G:=Group("SD16xC3:S3"); // GroupNames label
G:=SmallGroup(288,770);
// by ID
G=gap.SmallGroup(288,770);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,100,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations