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## G = He5⋊C3order 375 = 3·53

### The semidirect product of He5 and C3 acting faithfully

Aliases: He5⋊C3, C5.(C52⋊C3), SmallGroup(375,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — He5 — He5⋊C3
 Chief series C1 — C5 — He5 — He5⋊C3
 Lower central He5 — He5⋊C3
 Upper central C1 — C5

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 >

25C3
15C5
15C5
25C15
3C52
3C52

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

Smallest permutation representation of He5⋊C3
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

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