Aliases: He3⋊2C8, C2.(He3⋊C4), (C2×He3).C4, C6.2(C32⋊C4), C3.(C32⋊2C8), He3⋊3C4.1C2, SmallGroup(216,25)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — He3⋊2C8 |
Generators and relations for He3⋊2C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, cac-1=ab-1, dad-1=abc, bc=cb, bd=db, dcd-1=ac-1 >
Character table of He3⋊2C8
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 9 | 9 | 1 | 1 | 12 | 12 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | i | i | i | -i | -i | i | -i | -i | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | -i | -i | -i | i | i | -i | i | i | linear of order 4 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ8 | i | -i | i | -i | ζ87 | ζ87 | ζ83 | ζ8 | ζ8 | ζ83 | ζ85 | ζ85 | linear of order 8 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ83 | -i | i | -i | i | ζ85 | ζ85 | ζ8 | ζ83 | ζ83 | ζ8 | ζ87 | ζ87 | linear of order 8 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ85 | i | -i | i | -i | ζ83 | ζ83 | ζ87 | ζ85 | ζ85 | ζ87 | ζ8 | ζ8 | linear of order 8 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ87 | -i | i | -i | i | ζ8 | ζ8 | ζ85 | ζ87 | ζ87 | ζ85 | ζ83 | ζ83 | linear of order 8 |
| ρ9 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -1 | -1 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from He3⋊C4 |
| ρ10 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | complex lifted from He3⋊C4 |
| ρ11 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | complex lifted from He3⋊C4 |
| ρ12 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -1 | -1 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from He3⋊C4 |
| ρ13 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | i | -i | -i | i | ζ3 | ζ32 | ζ32 | ζ3 | ζ43ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | complex lifted from He3⋊C4 |
| ρ14 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -i | i | i | -i | ζ3 | ζ32 | ζ32 | ζ3 | ζ4ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | complex lifted from He3⋊C4 |
| ρ15 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | i | -i | -i | i | ζ32 | ζ3 | ζ3 | ζ32 | ζ43ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | complex lifted from He3⋊C4 |
| ρ16 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -i | i | i | -i | ζ32 | ζ3 | ζ3 | ζ32 | ζ4ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | complex lifted from He3⋊C4 |
| ρ17 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -i | i | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | ζ85 | ζ83 | ζ87 | ζ8 | ζ86ζ32 | ζ82ζ3 | ζ86ζ3 | ζ82ζ32 | ζ87ζ32 | ζ87ζ3 | ζ83ζ3 | ζ8ζ32 | ζ8ζ3 | ζ83ζ32 | ζ85ζ32 | ζ85ζ3 | complex faithful |
| ρ18 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -i | i | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | ζ85 | ζ83 | ζ87 | ζ8 | ζ86ζ3 | ζ82ζ32 | ζ86ζ32 | ζ82ζ3 | ζ87ζ3 | ζ87ζ32 | ζ83ζ32 | ζ8ζ3 | ζ8ζ32 | ζ83ζ3 | ζ85ζ3 | ζ85ζ32 | complex faithful |
| ρ19 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | i | -i | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | ζ87 | ζ8 | ζ85 | ζ83 | ζ82ζ3 | ζ86ζ32 | ζ82ζ32 | ζ86ζ3 | ζ85ζ3 | ζ85ζ32 | ζ8ζ32 | ζ83ζ3 | ζ83ζ32 | ζ8ζ3 | ζ87ζ3 | ζ87ζ32 | complex faithful |
| ρ20 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | i | -i | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | ζ83 | ζ85 | ζ8 | ζ87 | ζ82ζ32 | ζ86ζ3 | ζ82ζ3 | ζ86ζ32 | ζ8ζ32 | ζ8ζ3 | ζ85ζ3 | ζ87ζ32 | ζ87ζ3 | ζ85ζ32 | ζ83ζ32 | ζ83ζ3 | complex faithful |
| ρ21 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | i | -i | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | ζ87 | ζ8 | ζ85 | ζ83 | ζ82ζ32 | ζ86ζ3 | ζ82ζ3 | ζ86ζ32 | ζ85ζ32 | ζ85ζ3 | ζ8ζ3 | ζ83ζ32 | ζ83ζ3 | ζ8ζ32 | ζ87ζ32 | ζ87ζ3 | complex faithful |
| ρ22 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | i | -i | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | ζ83 | ζ85 | ζ8 | ζ87 | ζ82ζ3 | ζ86ζ32 | ζ82ζ32 | ζ86ζ3 | ζ8ζ3 | ζ8ζ32 | ζ85ζ32 | ζ87ζ3 | ζ87ζ32 | ζ85ζ3 | ζ83ζ3 | ζ83ζ32 | complex faithful |
| ρ23 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -i | i | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | ζ8 | ζ87 | ζ83 | ζ85 | ζ86ζ32 | ζ82ζ3 | ζ86ζ3 | ζ82ζ32 | ζ83ζ32 | ζ83ζ3 | ζ87ζ3 | ζ85ζ32 | ζ85ζ3 | ζ87ζ32 | ζ8ζ32 | ζ8ζ3 | complex faithful |
| ρ24 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -i | i | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | ζ8 | ζ87 | ζ83 | ζ85 | ζ86ζ3 | ζ82ζ32 | ζ86ζ32 | ζ82ζ3 | ζ83ζ3 | ζ83ζ32 | ζ87ζ32 | ζ85ζ3 | ζ85ζ32 | ζ87ζ3 | ζ8ζ3 | ζ8ζ32 | complex faithful |
| ρ25 | 4 | 4 | 4 | 4 | 1 | -2 | 0 | 0 | 4 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ26 | 4 | 4 | 4 | 4 | -2 | 1 | 0 | 0 | 4 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ27 | 4 | -4 | 4 | 4 | -2 | 1 | 0 | 0 | -4 | -4 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ28 | 4 | -4 | 4 | 4 | 1 | -2 | 0 | 0 | -4 | -4 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2C8, Schur index 2 |
►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)
(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), (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), (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)], [(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)]])
He3⋊2C8 is a maximal subgroup of
He3⋊C16 He3⋊2SD16 He3⋊D8 He3⋊Q16 He3⋊2(C2×C8) He3⋊1M4(2) He3⋊4M4(2)
He3⋊2C8 is a maximal quotient of He3⋊2C16
Matrix representation of He3⋊2C8 ►in GL3(𝔽73) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 64 |
| 8 | 8 | 1 |
| 8 | 1 | 8 |
| 64 | 1 | 1 |
G:=sub<GL(3,GF(73))| [0,0,1,1,0,0,0,1,0],[64,0,0,0,64,0,0,0,64],[1,0,0,0,8,0,0,0,64],[8,8,64,8,1,1,1,8,1] >;
He3⋊2C8 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_2C_8 % in TeX
G:=Group("He3:2C8"); // GroupNames label
G:=SmallGroup(216,25);
// by ID
G=gap.SmallGroup(216,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-3,12,31,1347,201,1924,1810,382]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations
Export
Subgroup lattice of He3⋊2C8 in TeX
Character table of He3⋊2C8 in TeX