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G = He5⋊C3order 375 = 3·53

The semidirect product of He5 and C3 acting faithfully

non-abelian, soluble

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

Series: Derived Chief Lower central Upper central

C1C5He5 — He5⋊C3
C1C5He5 — He5⋊C3
He5 — He5⋊C3
C1C5

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 13A3B5A5B5C5D5E5F5G5H5I5J5K5L15A15B15C15D15E15F15G15H
 size 12525111115151515151515152525252525252525
ρ111111111111111111111111    trivial
ρ21ζ32ζ3111111111111ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32    linear of order 3
ρ31ζ3ζ32111111111111ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3    linear of order 3
ρ43003333ζ53+2ζ51-5/25452ζ54+2ζ531-5/25251+5/21+5/200000000    complex lifted from C52⋊C3
ρ530033331-5/2ζ54+2ζ531-5/21+5/25251+5/25452ζ53+2ζ500000000    complex lifted from C52⋊C3
ρ630033335251+5/2ζ54+2ζ53ζ53+2ζ51+5/254521-5/21-5/200000000    complex lifted from C52⋊C3
ρ730033331+5/254521+5/21-5/2ζ53+2ζ51-5/2525ζ54+2ζ5300000000    complex lifted from C52⋊C3
ρ8300333354521-5/2ζ53+2ζ55251-5/2ζ54+2ζ531+5/21+5/200000000    complex lifted from C52⋊C3
ρ930033331-5/25251-5/21+5/2ζ54+2ζ531+5/2ζ53+2ζ5545200000000    complex lifted from C52⋊C3
ρ103003333ζ54+2ζ531+5/252554521+5/2ζ53+2ζ51-5/21-5/200000000    complex lifted from C52⋊C3
ρ1130033331+5/2ζ53+2ζ51+5/21-5/254521-5/2ζ54+2ζ5352500000000    complex lifted from C52⋊C3
ρ125-1-155352540000000054553525254553    complex faithful
ρ135-1-152554530000000053525545453525    complex faithful
ρ145-1-153545520000000052535455525354    complex faithful
ρ155-1-154525350000000055452535355452    complex faithful
ρ165ζ65ζ653545520000000032ζ5232ζ5332ζ5432ζ53ζ53ζ523ζ533ζ54    complex faithful
ρ175ζ6ζ655354552000000003ζ523ζ533ζ543ζ532ζ532ζ5232ζ5332ζ54    complex faithful
ρ185ζ6ζ655255453000000003ζ533ζ523ζ53ζ5432ζ5432ζ5332ζ5232ζ5    complex faithful
ρ195ζ65ζ655352540000000032ζ5432ζ532ζ5332ζ523ζ523ζ543ζ53ζ53    complex faithful
ρ205ζ65ζ654525350000000032ζ532ζ5432ζ5232ζ533ζ533ζ53ζ543ζ52    complex faithful
ρ215ζ65ζ652554530000000032ζ5332ζ5232ζ532ζ543ζ543ζ533ζ523ζ5    complex faithful
ρ225ζ6ζ655452535000000003ζ53ζ543ζ523ζ5332ζ5332ζ532ζ5432ζ52    complex faithful
ρ235ζ6ζ655535254000000003ζ543ζ53ζ533ζ5232ζ5232ζ5432ζ532ζ53    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)

04000
001600
00020
00008
10000
,
80000
08000
00800
00080
00008
,
01000
00100
00010
00001
10000
,
2814252514
1928252819
2819192825
1428142525
141419719

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

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