non-abelian, supersoluble, monomial
Aliases: He3:7SD16, (C3xC24):4S3, (C8xHe3):4C2, He3:4Q8:7C2, (C3xC12).46D6, (C3xC6).23D12, C24.12(C3:S3), C8:2(He3:C2), (C2xHe3).22D4, He3:5D4.3C2, C3.2(C24:2S3), C32:6(C24:C2), C2.3(He3:5D4), C6.27(C12:S3), (C4xHe3).35C22, C12.79(C2xC3:S3), C4.8(C2xHe3:C2), SmallGroup(432,175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for He3:7SD16
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d3 >
Subgroups: 549 in 110 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, SD16, C3xS3, C3xC6, C24, C24, Dic6, D12, C3xD4, C3xQ8, He3, C3xDic3, C3xC12, S3xC6, C24:C2, C3xSD16, He3:C2, C2xHe3, C3xC24, C3xDic6, C3xD12, He3:3C4, C4xHe3, C2xHe3:C2, C3xC24:C2, C8xHe3, He3:4Q8, He3:5D4, He3:7SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, C3:S3, D12, C2xC3:S3, C24:C2, He3:C2, C12:S3, C2xHe3:C2, C24:2S3, He3:5D4, He3:7SD16
►On 72 points
Generators in S72
(1 68 56)(2 69 49)(3 70 50)(4 71 51)(5 72 52)(6 65 53)(7 66 54)(8 67 55)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 41 27)(18 42 28)(19 43 29)(20 44 30)(21 45 31)(22 46 32)(23 47 25)(24 48 26)
(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 56 39)(18 49 40)(19 50 33)(20 51 34)(21 52 35)(22 53 36)(23 54 37)(24 55 38)(25 66 61)(26 67 62)(27 68 63)(28 69 64)(29 70 57)(30 71 58)(31 72 59)(32 65 60)
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 31 21)(6 32 22)(7 25 23)(8 26 24)(9 70 50)(10 71 51)(11 72 52)(12 65 53)(13 66 54)(14 67 55)(15 68 56)(16 69 49)(33 43 57)(34 44 58)(35 45 59)(36 46 60)(37 47 61)(38 48 62)(39 41 63)(40 42 64)
(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 27)(18 30)(19 25)(20 28)(21 31)(22 26)(23 29)(24 32)(33 61)(34 64)(35 59)(36 62)(37 57)(38 60)(39 63)(40 58)(42 44)(43 47)(46 48)(49 71)(50 66)(51 69)(52 72)(53 67)(54 70)(55 65)(56 68)
G:=sub<Sym(72)| (1,68,56)(2,69,49)(3,70,50)(4,71,51)(5,72,52)(6,65,53)(7,66,54)(8,67,55)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,41,27)(18,42,28)(19,43,29)(20,44,30)(21,45,31)(22,46,32)(23,47,25)(24,48,26), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,56,39)(18,49,40)(19,50,33)(20,51,34)(21,52,35)(22,53,36)(23,54,37)(24,55,38)(25,66,61)(26,67,62)(27,68,63)(28,69,64)(29,70,57)(30,71,58)(31,72,59)(32,65,60), (1,27,17)(2,28,18)(3,29,19)(4,30,20)(5,31,21)(6,32,22)(7,25,23)(8,26,24)(9,70,50)(10,71,51)(11,72,52)(12,65,53)(13,66,54)(14,67,55)(15,68,56)(16,69,49)(33,43,57)(34,44,58)(35,45,59)(36,46,60)(37,47,61)(38,48,62)(39,41,63)(40,42,64), (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)(17,27)(18,30)(19,25)(20,28)(21,31)(22,26)(23,29)(24,32)(33,61)(34,64)(35,59)(36,62)(37,57)(38,60)(39,63)(40,58)(42,44)(43,47)(46,48)(49,71)(50,66)(51,69)(52,72)(53,67)(54,70)(55,65)(56,68)>;
G:=Group( (1,68,56)(2,69,49)(3,70,50)(4,71,51)(5,72,52)(6,65,53)(7,66,54)(8,67,55)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,41,27)(18,42,28)(19,43,29)(20,44,30)(21,45,31)(22,46,32)(23,47,25)(24,48,26), (1,15,41)(2,16,42)(3,9,43)(4,10,44)(5,11,45)(6,12,46)(7,13,47)(8,14,48)(17,56,39)(18,49,40)(19,50,33)(20,51,34)(21,52,35)(22,53,36)(23,54,37)(24,55,38)(25,66,61)(26,67,62)(27,68,63)(28,69,64)(29,70,57)(30,71,58)(31,72,59)(32,65,60), (1,27,17)(2,28,18)(3,29,19)(4,30,20)(5,31,21)(6,32,22)(7,25,23)(8,26,24)(9,70,50)(10,71,51)(11,72,52)(12,65,53)(13,66,54)(14,67,55)(15,68,56)(16,69,49)(33,43,57)(34,44,58)(35,45,59)(36,46,60)(37,47,61)(38,48,62)(39,41,63)(40,42,64), (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)(17,27)(18,30)(19,25)(20,28)(21,31)(22,26)(23,29)(24,32)(33,61)(34,64)(35,59)(36,62)(37,57)(38,60)(39,63)(40,58)(42,44)(43,47)(46,48)(49,71)(50,66)(51,69)(52,72)(53,67)(54,70)(55,65)(56,68) );
G=PermutationGroup([[(1,68,56),(2,69,49),(3,70,50),(4,71,51),(5,72,52),(6,65,53),(7,66,54),(8,67,55),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,41,27),(18,42,28),(19,43,29),(20,44,30),(21,45,31),(22,46,32),(23,47,25),(24,48,26)], [(1,15,41),(2,16,42),(3,9,43),(4,10,44),(5,11,45),(6,12,46),(7,13,47),(8,14,48),(17,56,39),(18,49,40),(19,50,33),(20,51,34),(21,52,35),(22,53,36),(23,54,37),(24,55,38),(25,66,61),(26,67,62),(27,68,63),(28,69,64),(29,70,57),(30,71,58),(31,72,59),(32,65,60)], [(1,27,17),(2,28,18),(3,29,19),(4,30,20),(5,31,21),(6,32,22),(7,25,23),(8,26,24),(9,70,50),(10,71,51),(11,72,52),(12,65,53),(13,66,54),(14,67,55),(15,68,56),(16,69,49),(33,43,57),(34,44,58),(35,45,59),(36,46,60),(37,47,61),(38,48,62),(39,41,63),(40,42,64)], [(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),(17,27),(18,30),(19,25),(20,28),(21,31),(22,26),(23,29),(24,32),(33,61),(34,64),(35,59),(36,62),(37,57),(38,60),(39,63),(40,58),(42,44),(43,47),(46,48),(49,71),(50,66),(51,69),(52,72),(53,67),(54,70),(55,65),(56,68)]])
53 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 | ··· | 12J | 12K | 12L | 24A | 24B | 24C | 24D | 24E | ··· | 24T |
| 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 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 36 | 1 | 1 | 6 | 6 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 36 | 36 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | SD16 | D12 | C24:C2 | He3:C2 | C2xHe3:C2 | He3:5D4 | He3:7SD16 |
| kernel | He3:7SD16 | C8xHe3 | He3:4Q8 | He3:5D4 | C3xC24 | C2xHe3 | C3xC12 | He3 | C3xC6 | C32 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 16 | 4 | 4 | 2 | 4 |
Matrix representation of He3:7SD16 ►in GL5(F73)
| 0 | 1 | 0 | 0 | 0 |
| 72 | 72 | 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 |
| 72 | 72 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 8 | 0 | 0 |
| 25 | 62 | 0 | 0 | 0 |
| 11 | 36 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(73))| [0,72,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[72,1,0,0,0,72,0,0,0,0,0,0,0,0,8,0,0,64,0,0,0,0,0,1,0],[25,11,0,0,0,62,36,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
He3:7SD16 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_7{\rm SD}_{16} % in TeX
G:=Group("He3:7SD16"); // GroupNames label
G:=SmallGroup(432,175);
// by ID
G=gap.SmallGroup(432,175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,1124,4037,537]);
// 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,a*d=d*a,e*a*e=a^-1,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^3>;
// generators/relations