non-abelian, supersoluble, monomial
Aliases: He3:11SD16, (Q8xHe3):2C2, He3:4C8:7C2, (C3xC12).18D6, (Q8xC32):2S3, (C2xHe3).36D4, He3:5D4.2C2, Q8:2(He3:C2), C2.6(He3:7D4), C32:6(Q8:2S3), C6.42(C32:7D4), (C4xHe3).14C22, C3.2(C32:11SD16), C12.46(C2xC3:S3), C4.3(C2xHe3:C2), (C3xC6).37(C3:D4), (C3xQ8).14(C3:S3), SmallGroup(432,196)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3:11SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, dad-1=eae=a-1, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d3 >
Subgroups: 501 in 110 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C12, C12, D6, C2xC6, SD16, C3xS3, C3xC6, C3:C8, C24, D12, C3xD4, C3xQ8, C3xQ8, He3, C3xC12, C3xC12, S3xC6, Q8:2S3, C3xSD16, He3:C2, C2xHe3, C3xC3:C8, C3xD12, Q8xC32, C4xHe3, C4xHe3, C2xHe3:C2, C3xQ8:2S3, He3:4C8, He3:5D4, Q8xHe3, He3:11SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3:S3, C3:D4, C2xC3:S3, Q8:2S3, He3:C2, C32:7D4, C2xHe3:C2, C32:11SD16, He3:7D4, He3:11SD16
►On 72 points
Generators in S72
(1 56 35)(2 36 49)(3 50 37)(4 38 51)(5 52 39)(6 40 53)(7 54 33)(8 34 55)(9 27 57)(10 58 28)(11 29 59)(12 60 30)(13 31 61)(14 62 32)(15 25 63)(16 64 26)(17 67 48)(18 41 68)(19 69 42)(20 43 70)(21 71 44)(22 45 72)(23 65 46)(24 47 66)
(1 15 41)(2 16 42)(3 9 43)(4 10 44)(5 11 45)(6 12 46)(7 13 47)(8 14 48)(17 34 62)(18 35 63)(19 36 64)(20 37 57)(21 38 58)(22 39 59)(23 40 60)(24 33 61)(25 68 56)(26 69 49)(27 70 50)(28 71 51)(29 72 52)(30 65 53)(31 66 54)(32 67 55)
(1 68 18)(2 19 69)(3 70 20)(4 21 71)(5 72 22)(6 23 65)(7 66 24)(8 17 67)(9 50 37)(10 38 51)(11 52 39)(12 40 53)(13 54 33)(14 34 55)(15 56 35)(16 36 49)(25 63 41)(26 42 64)(27 57 43)(28 44 58)(29 59 45)(30 46 60)(31 61 47)(32 48 62)
(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)(9 13)(10 16)(12 14)(17 65)(18 68)(19 71)(20 66)(21 69)(22 72)(23 67)(24 70)(25 63)(26 58)(27 61)(28 64)(29 59)(30 62)(31 57)(32 60)(33 50)(34 53)(35 56)(36 51)(37 54)(38 49)(39 52)(40 55)(42 44)(43 47)(46 48)
G:=sub<Sym(72)| (1,56,35)(2,36,49)(3,50,37)(4,38,51)(5,52,39)(6,40,53)(7,54,33)(8,34,55)(9,27,57)(10,58,28)(11,29,59)(12,60,30)(13,31,61)(14,62,32)(15,25,63)(16,64,26)(17,67,48)(18,41,68)(19,69,42)(20,43,70)(21,71,44)(22,45,72)(23,65,46)(24,47,66), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,34,62)(18,35,63)(19,36,64)(20,37,57)(21,38,58)(22,39,59)(23,40,60)(24,33,61)(25,68,56)(26,69,49)(27,70,50)(28,71,51)(29,72,52)(30,65,53)(31,66,54)(32,67,55), (1,68,18)(2,19,69)(3,70,20)(4,21,71)(5,72,22)(6,23,65)(7,66,24)(8,17,67)(9,50,37)(10,38,51)(11,52,39)(12,40,53)(13,54,33)(14,34,55)(15,56,35)(16,36,49)(25,63,41)(26,42,64)(27,57,43)(28,44,58)(29,59,45)(30,46,60)(31,61,47)(32,48,62), (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)(9,13)(10,16)(12,14)(17,65)(18,68)(19,71)(20,66)(21,69)(22,72)(23,67)(24,70)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(33,50)(34,53)(35,56)(36,51)(37,54)(38,49)(39,52)(40,55)(42,44)(43,47)(46,48)>;
G:=Group( (1,56,35)(2,36,49)(3,50,37)(4,38,51)(5,52,39)(6,40,53)(7,54,33)(8,34,55)(9,27,57)(10,58,28)(11,29,59)(12,60,30)(13,31,61)(14,62,32)(15,25,63)(16,64,26)(17,67,48)(18,41,68)(19,69,42)(20,43,70)(21,71,44)(22,45,72)(23,65,46)(24,47,66), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,34,62)(18,35,63)(19,36,64)(20,37,57)(21,38,58)(22,39,59)(23,40,60)(24,33,61)(25,68,56)(26,69,49)(27,70,50)(28,71,51)(29,72,52)(30,65,53)(31,66,54)(32,67,55), (1,68,18)(2,19,69)(3,70,20)(4,21,71)(5,72,22)(6,23,65)(7,66,24)(8,17,67)(9,50,37)(10,38,51)(11,52,39)(12,40,53)(13,54,33)(14,34,55)(15,56,35)(16,36,49)(25,63,41)(26,42,64)(27,57,43)(28,44,58)(29,59,45)(30,46,60)(31,61,47)(32,48,62), (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)(9,13)(10,16)(12,14)(17,65)(18,68)(19,71)(20,66)(21,69)(22,72)(23,67)(24,70)(25,63)(26,58)(27,61)(28,64)(29,59)(30,62)(31,57)(32,60)(33,50)(34,53)(35,56)(36,51)(37,54)(38,49)(39,52)(40,55)(42,44)(43,47)(46,48) );
G=PermutationGroup([[(1,56,35),(2,36,49),(3,50,37),(4,38,51),(5,52,39),(6,40,53),(7,54,33),(8,34,55),(9,27,57),(10,58,28),(11,29,59),(12,60,30),(13,31,61),(14,62,32),(15,25,63),(16,64,26),(17,67,48),(18,41,68),(19,69,42),(20,43,70),(21,71,44),(22,45,72),(23,65,46),(24,47,66)], [(1,15,41),(2,16,42),(3,9,43),(4,10,44),(5,11,45),(6,12,46),(7,13,47),(8,14,48),(17,34,62),(18,35,63),(19,36,64),(20,37,57),(21,38,58),(22,39,59),(23,40,60),(24,33,61),(25,68,56),(26,69,49),(27,70,50),(28,71,51),(29,72,52),(30,65,53),(31,66,54),(32,67,55)], [(1,68,18),(2,19,69),(3,70,20),(4,21,71),(5,72,22),(6,23,65),(7,66,24),(8,17,67),(9,50,37),(10,38,51),(11,52,39),(12,40,53),(13,54,33),(14,34,55),(15,56,35),(16,36,49),(25,63,41),(26,42,64),(27,57,43),(28,44,58),(29,59,45),(30,46,60),(31,61,47),(32,48,62)], [(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),(9,13),(10,16),(12,14),(17,65),(18,68),(19,71),(20,66),(21,69),(22,72),(23,67),(24,70),(25,63),(26,58),(27,61),(28,64),(29,59),(30,62),(31,57),(32,60),(33,50),(34,53),(35,56),(36,51),(37,54),(38,49),(39,52),(40,55),(42,44),(43,47),(46,48)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 36 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 4 | 1 | 1 | 6 | 6 | 6 | 6 | 36 | 36 | 18 | 18 | 2 | 2 | 4 | 4 | 12 | ··· | 12 | 18 | 18 | 18 | 18 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | SD16 | C3:D4 | He3:C2 | C2xHe3:C2 | Q8:2S3 | He3:7D4 | He3:11SD16 |
| kernel | He3:11SD16 | He3:4C8 | He3:5D4 | Q8xHe3 | Q8xC32 | C2xHe3 | C3xC12 | He3 | C3xC6 | Q8 | C4 | C32 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of He3:11SD16 ►in GL5(F73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 45 | 61 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 44 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,64,0,0,0,0,0,1,0],[0,45,0,0,0,13,61,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0],[1,44,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
He3:11SD16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_{11}{\rm SD}_{16} % in TeX
G:=Group("He3:11SD16"); // GroupNames label
G:=SmallGroup(432,196);
// by ID
G=gap.SmallGroup(432,196);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,1124,4037,537]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=e*a*e=a^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^3>;
// generators/relations