Aliases: He3⋊2(C2×C8), He3⋊C2⋊2C8, He3⋊2C8⋊5C2, C4.3(He3⋊C4), (C4×He3).2C4, C12.8(C32⋊C4), He3⋊3C4.7C22, C2.1(C2×He3⋊C4), C3.(C3⋊S3⋊3C8), C6.15(C2×C32⋊C4), (C2×He3).1(C2×C4), (C2×He3⋊C2).5C4, (C4×He3⋊C2).5C2, SmallGroup(432,273)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊2(C2×C8) |
Generators and relations for He3⋊2(C2×C8)
G = < a,b,c,d,e | a3=b3=c3=d2=e8=1, ab=ba, cac-1=ab-1, dad=a-1b, eae-1=abc, bc=cb, bd=db, be=eb, dcd=c-1, ece-1=ac-1, de=ed >
Subgroups: 329 in 65 conjugacy classes, 17 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C24, C4×S3, C2×C12, He3, C3×Dic3, C3×C12, S3×C6, C2×C24, He3⋊C2, C2×He3, S3×C12, He3⋊3C4, C4×He3, C2×He3⋊C2, He3⋊2C8, C4×He3⋊C2, He3⋊2(C2×C8)
Quotients: C1, C2, C4, C22, C8, C2×C4, C2×C8, C32⋊C4, C2×C32⋊C4, He3⋊C4, C3⋊S3⋊3C8, C2×He3⋊C4, He3⋊2(C2×C8)
►On 72 points
Generators in S72
(1 68 11)(2 19 45)(3 51 20)(4 14 58)(5 72 15)(6 23 41)(7 55 24)(8 10 62)(9 61 29)(12 35 69)(13 57 25)(16 39 65)(17 33 56)(18 44 31)(21 37 52)(22 48 27)(26 47 71)(28 60 54)(30 43 67)(32 64 50)(34 63 49)(36 46 70)(38 59 53)(40 42 66)
(1 34 31)(2 35 32)(3 36 25)(4 37 26)(5 38 27)(6 39 28)(7 40 29)(8 33 30)(9 55 42)(10 56 43)(11 49 44)(12 50 45)(13 51 46)(14 52 47)(15 53 48)(16 54 41)(17 67 62)(18 68 63)(19 69 64)(20 70 57)(21 71 58)(22 72 59)(23 65 60)(24 66 61)
(2 19 12)(4 14 21)(6 23 16)(8 10 17)(9 55 42)(11 44 49)(13 51 46)(15 48 53)(18 68 63)(20 57 70)(22 72 59)(24 61 66)(26 47 58)(28 60 41)(30 43 62)(32 64 45)(33 56 67)(35 69 50)(37 52 71)(39 65 54)
(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)(49 68)(50 69)(51 70)(52 71)(53 72)(54 65)(55 66)(56 67)
(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)
G:=sub<Sym(72)| (1,68,11)(2,19,45)(3,51,20)(4,14,58)(5,72,15)(6,23,41)(7,55,24)(8,10,62)(9,61,29)(12,35,69)(13,57,25)(16,39,65)(17,33,56)(18,44,31)(21,37,52)(22,48,27)(26,47,71)(28,60,54)(30,43,67)(32,64,50)(34,63,49)(36,46,70)(38,59,53)(40,42,66), (1,34,31)(2,35,32)(3,36,25)(4,37,26)(5,38,27)(6,39,28)(7,40,29)(8,33,30)(9,55,42)(10,56,43)(11,49,44)(12,50,45)(13,51,46)(14,52,47)(15,53,48)(16,54,41)(17,67,62)(18,68,63)(19,69,64)(20,70,57)(21,71,58)(22,72,59)(23,65,60)(24,66,61), (2,19,12)(4,14,21)(6,23,16)(8,10,17)(9,55,42)(11,44,49)(13,51,46)(15,48,53)(18,68,63)(20,57,70)(22,72,59)(24,61,66)(26,47,58)(28,60,41)(30,43,62)(32,64,45)(33,56,67)(35,69,50)(37,52,71)(39,65,54), (9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59)(49,68)(50,69)(51,70)(52,71)(53,72)(54,65)(55,66)(56,67), (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)>;
G:=Group( (1,68,11)(2,19,45)(3,51,20)(4,14,58)(5,72,15)(6,23,41)(7,55,24)(8,10,62)(9,61,29)(12,35,69)(13,57,25)(16,39,65)(17,33,56)(18,44,31)(21,37,52)(22,48,27)(26,47,71)(28,60,54)(30,43,67)(32,64,50)(34,63,49)(36,46,70)(38,59,53)(40,42,66), (1,34,31)(2,35,32)(3,36,25)(4,37,26)(5,38,27)(6,39,28)(7,40,29)(8,33,30)(9,55,42)(10,56,43)(11,49,44)(12,50,45)(13,51,46)(14,52,47)(15,53,48)(16,54,41)(17,67,62)(18,68,63)(19,69,64)(20,70,57)(21,71,58)(22,72,59)(23,65,60)(24,66,61), (2,19,12)(4,14,21)(6,23,16)(8,10,17)(9,55,42)(11,44,49)(13,51,46)(15,48,53)(18,68,63)(20,57,70)(22,72,59)(24,61,66)(26,47,58)(28,60,41)(30,43,62)(32,64,45)(33,56,67)(35,69,50)(37,52,71)(39,65,54), (9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(41,60)(42,61)(43,62)(44,63)(45,64)(46,57)(47,58)(48,59)(49,68)(50,69)(51,70)(52,71)(53,72)(54,65)(55,66)(56,67), (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) );
G=PermutationGroup([[(1,68,11),(2,19,45),(3,51,20),(4,14,58),(5,72,15),(6,23,41),(7,55,24),(8,10,62),(9,61,29),(12,35,69),(13,57,25),(16,39,65),(17,33,56),(18,44,31),(21,37,52),(22,48,27),(26,47,71),(28,60,54),(30,43,67),(32,64,50),(34,63,49),(36,46,70),(38,59,53),(40,42,66)], [(1,34,31),(2,35,32),(3,36,25),(4,37,26),(5,38,27),(6,39,28),(7,40,29),(8,33,30),(9,55,42),(10,56,43),(11,49,44),(12,50,45),(13,51,46),(14,52,47),(15,53,48),(16,54,41),(17,67,62),(18,68,63),(19,69,64),(20,70,57),(21,71,58),(22,72,59),(23,65,60),(24,66,61)], [(2,19,12),(4,14,21),(6,23,16),(8,10,17),(9,55,42),(11,44,49),(13,51,46),(15,48,53),(18,68,63),(20,57,70),(22,72,59),(24,61,66),(26,47,58),(28,60,41),(30,43,62),(32,64,45),(33,56,67),(35,69,50),(37,52,71),(39,65,54)], [(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(41,60),(42,61),(43,62),(44,63),(45,64),(46,57),(47,58),(48,59),(49,68),(50,69),(51,70),(52,71),(53,72),(54,65),(55,66),(56,67)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 12 | 12 | 1 | 1 | 9 | 9 | 1 | 1 | 9 | 9 | 9 | 9 | 12 | 12 | 9 | ··· | 9 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 9 | ··· | 9 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | He3⋊C4 | C2×He3⋊C4 | He3⋊2(C2×C8) | C32⋊C4 | C2×C32⋊C4 | C3⋊S3⋊3C8 |
| kernel | He3⋊2(C2×C8) | He3⋊2C8 | C4×He3⋊C2 | C4×He3 | C2×He3⋊C2 | He3⋊C2 | C4 | C2 | C1 | C12 | C6 | C3 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 8 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of He3⋊2(C2×C8) ►in GL3(𝔽73) generated by
| 64 | 0 | 10 |
| 64 | 0 | 9 |
| 65 | 8 | 9 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 56 |
| 72 | 0 | 65 |
| 8 | 64 | 65 |
| 1 | 0 | 0 |
| 1 | 0 | 8 |
| 9 | 64 | 0 |
| 66 | 17 | 63 |
| 0 | 17 | 56 |
| 66 | 17 | 0 |
G:=sub<GL(3,GF(73))| [64,64,65,0,0,8,10,9,9],[8,0,0,0,8,0,0,0,8],[8,72,8,0,0,64,56,65,65],[1,1,9,0,0,64,0,8,0],[66,0,66,17,17,17,63,56,0] >;
He3⋊2(C2×C8) in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_2(C_2\times C_8) % in TeX
G:=Group("He3:2(C2xC8)"); // GroupNames label
G:=SmallGroup(432,273);
// by ID
G=gap.SmallGroup(432,273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,64,58,3924,298,5381,2539,537]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^2=e^8=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1*b,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,e*c*e^-1=a*c^-1,d*e=e*d>;
// generators/relations