Aliases: He5⋊C3, C5.(C52⋊C3), SmallGroup(375,2)
Series: Derived ►Chief ►Lower central ►Upper central
| He5 — He5⋊C3 |
Generators and relations for He5⋊C3
G = < a,b,c,d | a5=b5=c5=d3=1, cac-1=dcd-1=ab=ba, dad-1=a-1c-1, bc=cb, bd=db >
Character table of He5⋊C3
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 5I | 5J | 5K | 5L | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 25 | 25 | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | |
| ρ1 | 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 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | ζ53+2ζ5 | 1-√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 2ζ52+ζ5 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ5 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | 1-√5/2 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 2ζ52+ζ5 | 1+√5/2 | 2ζ54+ζ52 | ζ53+2ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ6 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | 2ζ52+ζ5 | 1+√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 2ζ54+ζ52 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ7 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | 1+√5/2 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | ζ53+2ζ5 | 1-√5/2 | 2ζ52+ζ5 | ζ54+2ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ8 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | 2ζ54+ζ52 | 1-√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | ζ54+2ζ53 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ9 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | 1-√5/2 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | ζ54+2ζ53 | 1+√5/2 | ζ53+2ζ5 | 2ζ54+ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ10 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | ζ54+2ζ53 | 1+√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | ζ53+2ζ5 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ11 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | 1+√5/2 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 2ζ54+ζ52 | 1-√5/2 | ζ54+2ζ53 | 2ζ52+ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C52⋊C3 |
| ρ12 | 5 | -1 | -1 | 5ζ5 | 5ζ53 | 5ζ52 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | complex faithful |
| ρ13 | 5 | -1 | -1 | 5ζ52 | 5ζ5 | 5ζ54 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | complex faithful |
| ρ14 | 5 | -1 | -1 | 5ζ53 | 5ζ54 | 5ζ5 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | complex faithful |
| ρ15 | 5 | -1 | -1 | 5ζ54 | 5ζ52 | 5ζ53 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | complex faithful |
| ρ16 | 5 | ζ65 | ζ6 | 5ζ53 | 5ζ54 | 5ζ5 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ32ζ52 | -ζ32ζ53 | -ζ32ζ54 | -ζ32ζ5 | -ζ3ζ5 | -ζ3ζ52 | -ζ3ζ53 | -ζ3ζ54 | complex faithful |
| ρ17 | 5 | ζ6 | ζ65 | 5ζ53 | 5ζ54 | 5ζ5 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ3ζ52 | -ζ3ζ53 | -ζ3ζ54 | -ζ3ζ5 | -ζ32ζ5 | -ζ32ζ52 | -ζ32ζ53 | -ζ32ζ54 | complex faithful |
| ρ18 | 5 | ζ6 | ζ65 | 5ζ52 | 5ζ5 | 5ζ54 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ3ζ53 | -ζ3ζ52 | -ζ3ζ5 | -ζ3ζ54 | -ζ32ζ54 | -ζ32ζ53 | -ζ32ζ52 | -ζ32ζ5 | complex faithful |
| ρ19 | 5 | ζ65 | ζ6 | 5ζ5 | 5ζ53 | 5ζ52 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ32ζ54 | -ζ32ζ5 | -ζ32ζ53 | -ζ32ζ52 | -ζ3ζ52 | -ζ3ζ54 | -ζ3ζ5 | -ζ3ζ53 | complex faithful |
| ρ20 | 5 | ζ65 | ζ6 | 5ζ54 | 5ζ52 | 5ζ53 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ32ζ5 | -ζ32ζ54 | -ζ32ζ52 | -ζ32ζ53 | -ζ3ζ53 | -ζ3ζ5 | -ζ3ζ54 | -ζ3ζ52 | complex faithful |
| ρ21 | 5 | ζ65 | ζ6 | 5ζ52 | 5ζ5 | 5ζ54 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ32ζ53 | -ζ32ζ52 | -ζ32ζ5 | -ζ32ζ54 | -ζ3ζ54 | -ζ3ζ53 | -ζ3ζ52 | -ζ3ζ5 | complex faithful |
| ρ22 | 5 | ζ6 | ζ65 | 5ζ54 | 5ζ52 | 5ζ53 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ3ζ5 | -ζ3ζ54 | -ζ3ζ52 | -ζ3ζ53 | -ζ32ζ53 | -ζ32ζ5 | -ζ32ζ54 | -ζ32ζ52 | complex faithful |
| ρ23 | 5 | ζ6 | ζ65 | 5ζ5 | 5ζ53 | 5ζ52 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ3ζ54 | -ζ3ζ5 | -ζ3ζ53 | -ζ3ζ52 | -ζ32ζ52 | -ζ32ζ54 | -ζ32ζ5 | -ζ32ζ53 | complex faithful |
►On 75 points
Generators in S75
(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 73 74 75)
(1 45 28 36 32)(2 41 29 37 33)(3 42 30 38 34)(4 43 26 39 35)(5 44 27 40 31)(6 24 71 14 16)(7 25 72 15 17)(8 21 73 11 18)(9 22 74 12 19)(10 23 75 13 20)(46 60 67 52 65)(47 56 68 53 61)(48 57 69 54 62)(49 58 70 55 63)(50 59 66 51 64)
(1 2 34 26 31)(3 39 5 45 41)(4 27 36 37 30)(6 73 75 7 22)(8 10 15 19 14)(9 16 21 23 17)(11 13 25 74 24)(12 71 18 20 72)(28 29 42 35 44)(32 33 38 43 40)(46 63 47 51 54)(48 67 58 68 50)(49 56 64 62 60)(52 70 53 59 57)(55 61 66 69 65)
(1 16 64)(2 8 46)(3 20 68)(4 25 48)(5 22 63)(6 50 45)(7 62 35)(9 55 31)(10 53 42)(11 52 37)(12 58 27)(13 56 34)(14 51 32)(15 69 26)(17 54 39)(18 65 33)(19 70 40)(21 60 41)(23 61 30)(24 59 28)(29 73 67)(36 71 66)(38 75 47)(43 72 57)(44 74 49)
G:=sub<Sym(75)| (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)(71,72,73,74,75), (1,45,28,36,32)(2,41,29,37,33)(3,42,30,38,34)(4,43,26,39,35)(5,44,27,40,31)(6,24,71,14,16)(7,25,72,15,17)(8,21,73,11,18)(9,22,74,12,19)(10,23,75,13,20)(46,60,67,52,65)(47,56,68,53,61)(48,57,69,54,62)(49,58,70,55,63)(50,59,66,51,64), (1,2,34,26,31)(3,39,5,45,41)(4,27,36,37,30)(6,73,75,7,22)(8,10,15,19,14)(9,16,21,23,17)(11,13,25,74,24)(12,71,18,20,72)(28,29,42,35,44)(32,33,38,43,40)(46,63,47,51,54)(48,67,58,68,50)(49,56,64,62,60)(52,70,53,59,57)(55,61,66,69,65), (1,16,64)(2,8,46)(3,20,68)(4,25,48)(5,22,63)(6,50,45)(7,62,35)(9,55,31)(10,53,42)(11,52,37)(12,58,27)(13,56,34)(14,51,32)(15,69,26)(17,54,39)(18,65,33)(19,70,40)(21,60,41)(23,61,30)(24,59,28)(29,73,67)(36,71,66)(38,75,47)(43,72,57)(44,74,49)>;
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)(71,72,73,74,75), (1,45,28,36,32)(2,41,29,37,33)(3,42,30,38,34)(4,43,26,39,35)(5,44,27,40,31)(6,24,71,14,16)(7,25,72,15,17)(8,21,73,11,18)(9,22,74,12,19)(10,23,75,13,20)(46,60,67,52,65)(47,56,68,53,61)(48,57,69,54,62)(49,58,70,55,63)(50,59,66,51,64), (1,2,34,26,31)(3,39,5,45,41)(4,27,36,37,30)(6,73,75,7,22)(8,10,15,19,14)(9,16,21,23,17)(11,13,25,74,24)(12,71,18,20,72)(28,29,42,35,44)(32,33,38,43,40)(46,63,47,51,54)(48,67,58,68,50)(49,56,64,62,60)(52,70,53,59,57)(55,61,66,69,65), (1,16,64)(2,8,46)(3,20,68)(4,25,48)(5,22,63)(6,50,45)(7,62,35)(9,55,31)(10,53,42)(11,52,37)(12,58,27)(13,56,34)(14,51,32)(15,69,26)(17,54,39)(18,65,33)(19,70,40)(21,60,41)(23,61,30)(24,59,28)(29,73,67)(36,71,66)(38,75,47)(43,72,57)(44,74,49) );
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),(71,72,73,74,75)], [(1,45,28,36,32),(2,41,29,37,33),(3,42,30,38,34),(4,43,26,39,35),(5,44,27,40,31),(6,24,71,14,16),(7,25,72,15,17),(8,21,73,11,18),(9,22,74,12,19),(10,23,75,13,20),(46,60,67,52,65),(47,56,68,53,61),(48,57,69,54,62),(49,58,70,55,63),(50,59,66,51,64)], [(1,2,34,26,31),(3,39,5,45,41),(4,27,36,37,30),(6,73,75,7,22),(8,10,15,19,14),(9,16,21,23,17),(11,13,25,74,24),(12,71,18,20,72),(28,29,42,35,44),(32,33,38,43,40),(46,63,47,51,54),(48,67,58,68,50),(49,56,64,62,60),(52,70,53,59,57),(55,61,66,69,65)], [(1,16,64),(2,8,46),(3,20,68),(4,25,48),(5,22,63),(6,50,45),(7,62,35),(9,55,31),(10,53,42),(11,52,37),(12,58,27),(13,56,34),(14,51,32),(15,69,26),(17,54,39),(18,65,33),(19,70,40),(21,60,41),(23,61,30),(24,59,28),(29,73,67),(36,71,66),(38,75,47),(43,72,57),(44,74,49)]])
Matrix representation of He5⋊C3 ►in GL5(𝔽31)
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 28 | 14 | 25 | 25 | 14 |
| 19 | 28 | 25 | 28 | 19 |
| 28 | 19 | 19 | 28 | 25 |
| 14 | 28 | 14 | 25 | 25 |
| 14 | 14 | 19 | 7 | 19 |
G:=sub<GL(5,GF(31))| [0,0,0,0,1,4,0,0,0,0,0,16,0,0,0,0,0,2,0,0,0,0,0,8,0],[8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0],[28,19,28,14,14,14,28,19,28,14,25,25,19,14,19,25,28,28,25,7,14,19,25,25,19] >;
He5⋊C3 in GAP, Magma, Sage, TeX
{\rm He}_5\rtimes C_3 % in TeX
G:=Group("He5:C3"); // GroupNames label
G:=SmallGroup(375,2);
// by ID
G=gap.SmallGroup(375,2);
# by ID
G:=PCGroup([4,-3,-5,5,-5,385,686,366]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^3=1,c*a*c^-1=d*c*d^-1=a*b=b*a,d*a*d^-1=a^-1*c^-1,b*c=c*b,b*d=d*b>;
// generators/relations
Export
Subgroup lattice of He5⋊C3 in TeX
Character table of He5⋊C3 in TeX