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G = C3xHe5order 375 = 3·53

Direct product of C3 and He5

direct product, metabelian, nilpotent (class 2), monomial, 5-elementary

Aliases: C3xHe5, C52:C15, C15.1C52, (C5xC15):C5, C5.1(C5xC15), SmallGroup(375,4)

Series: Derived Chief Lower central Upper central

C1C5 — C3xHe5
C1C5C52He5 — C3xHe5
C1C5 — C3xHe5
C1C15 — C3xHe5

Generators and relations for C3xHe5
 G = < a,b,c,d | a3=b5=c5=d5=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 78 in 30 conjugacy classes, 18 normal (6 characteristic)
Quotients: C1, C3, C5, C15, C52, C5xC15, He5, C3xHe5
5C5
5C5
5C5
5C5
5C5
5C5
5C15
5C15
5C15
5C15
5C15
5C15

Smallest permutation representation of C3xHe5
On 75 points
Generators in S75
(1 11 57)(2 12 58)(3 13 59)(4 14 60)(5 15 56)(6 51 33)(7 52 34)(8 53 35)(9 54 31)(10 55 32)(16 63 38)(17 64 39)(18 65 40)(19 61 36)(20 62 37)(21 69 44)(22 70 45)(23 66 41)(24 67 42)(25 68 43)(26 75 50)(27 71 46)(28 72 47)(29 73 48)(30 74 49)
(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 43 29 39 33)(2 44 30 40 34)(3 45 26 36 35)(4 41 27 37 31)(5 42 28 38 32)(6 11 25 73 17)(7 12 21 74 18)(8 13 22 75 19)(9 14 23 71 20)(10 15 24 72 16)(46 62 54 60 66)(47 63 55 56 67)(48 64 51 57 68)(49 65 52 58 69)(50 61 53 59 70)
(1 2 35 27 32)(3 37 5 43 44)(4 28 39 40 26)(6 7 19 23 16)(8 71 10 11 12)(9 24 73 74 22)(13 20 15 25 21)(14 72 17 18 75)(29 30 45 31 42)(33 34 36 41 38)(46 55 57 58 53)(47 64 65 50 60)(48 49 70 54 67)(51 52 61 66 63)(56 68 69 59 62)

G:=sub<Sym(75)| (1,11,57)(2,12,58)(3,13,59)(4,14,60)(5,15,56)(6,51,33)(7,52,34)(8,53,35)(9,54,31)(10,55,32)(16,63,38)(17,64,39)(18,65,40)(19,61,36)(20,62,37)(21,69,44)(22,70,45)(23,66,41)(24,67,42)(25,68,43)(26,75,50)(27,71,46)(28,72,47)(29,73,48)(30,74,49), (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), (1,43,29,39,33)(2,44,30,40,34)(3,45,26,36,35)(4,41,27,37,31)(5,42,28,38,32)(6,11,25,73,17)(7,12,21,74,18)(8,13,22,75,19)(9,14,23,71,20)(10,15,24,72,16)(46,62,54,60,66)(47,63,55,56,67)(48,64,51,57,68)(49,65,52,58,69)(50,61,53,59,70), (1,2,35,27,32)(3,37,5,43,44)(4,28,39,40,26)(6,7,19,23,16)(8,71,10,11,12)(9,24,73,74,22)(13,20,15,25,21)(14,72,17,18,75)(29,30,45,31,42)(33,34,36,41,38)(46,55,57,58,53)(47,64,65,50,60)(48,49,70,54,67)(51,52,61,66,63)(56,68,69,59,62)>;

G:=Group( (1,11,57)(2,12,58)(3,13,59)(4,14,60)(5,15,56)(6,51,33)(7,52,34)(8,53,35)(9,54,31)(10,55,32)(16,63,38)(17,64,39)(18,65,40)(19,61,36)(20,62,37)(21,69,44)(22,70,45)(23,66,41)(24,67,42)(25,68,43)(26,75,50)(27,71,46)(28,72,47)(29,73,48)(30,74,49), (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), (1,43,29,39,33)(2,44,30,40,34)(3,45,26,36,35)(4,41,27,37,31)(5,42,28,38,32)(6,11,25,73,17)(7,12,21,74,18)(8,13,22,75,19)(9,14,23,71,20)(10,15,24,72,16)(46,62,54,60,66)(47,63,55,56,67)(48,64,51,57,68)(49,65,52,58,69)(50,61,53,59,70), (1,2,35,27,32)(3,37,5,43,44)(4,28,39,40,26)(6,7,19,23,16)(8,71,10,11,12)(9,24,73,74,22)(13,20,15,25,21)(14,72,17,18,75)(29,30,45,31,42)(33,34,36,41,38)(46,55,57,58,53)(47,64,65,50,60)(48,49,70,54,67)(51,52,61,66,63)(56,68,69,59,62) );

G=PermutationGroup([[(1,11,57),(2,12,58),(3,13,59),(4,14,60),(5,15,56),(6,51,33),(7,52,34),(8,53,35),(9,54,31),(10,55,32),(16,63,38),(17,64,39),(18,65,40),(19,61,36),(20,62,37),(21,69,44),(22,70,45),(23,66,41),(24,67,42),(25,68,43),(26,75,50),(27,71,46),(28,72,47),(29,73,48),(30,74,49)], [(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)], [(1,43,29,39,33),(2,44,30,40,34),(3,45,26,36,35),(4,41,27,37,31),(5,42,28,38,32),(6,11,25,73,17),(7,12,21,74,18),(8,13,22,75,19),(9,14,23,71,20),(10,15,24,72,16),(46,62,54,60,66),(47,63,55,56,67),(48,64,51,57,68),(49,65,52,58,69),(50,61,53,59,70)], [(1,2,35,27,32),(3,37,5,43,44),(4,28,39,40,26),(6,7,19,23,16),(8,71,10,11,12),(9,24,73,74,22),(13,20,15,25,21),(14,72,17,18,75),(29,30,45,31,42),(33,34,36,41,38),(46,55,57,58,53),(47,64,65,50,60),(48,49,70,54,67),(51,52,61,66,63),(56,68,69,59,62)]])

87 conjugacy classes

class 1 3A3B5A5B5C5D5E···5AB15A···15H15I···15BD
order13355555···515···1515···15
size11111115···51···15···5

87 irreducible representations

dim111155
type+
imageC1C3C5C15He5C3xHe5
kernelC3xHe5He5C5xC15C52C3C1
# reps12244848

Matrix representation of C3xHe5 in GL5(F31)

50000
05000
00500
00050
00005
,
21202615
00200
000160
00004
715232729
,
40000
04000
00400
00040
00004
,
20000
00100
00010
00001
715232729

G:=sub<GL(5,GF(31))| [5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[2,0,0,0,7,1,0,0,0,15,20,2,0,0,23,26,0,16,0,27,15,0,0,4,29],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,7,0,0,0,0,15,0,1,0,0,23,0,0,1,0,27,0,0,0,1,29] >;

C3xHe5 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_5
% in TeX

G:=Group("C3xHe5");
// GroupNames label

G:=SmallGroup(375,4);
// by ID

G=gap.SmallGroup(375,4);
# by ID

G:=PCGroup([4,-3,-5,-5,-5,366]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^5=d^5=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

Export

Subgroup lattice of C3xHe5 in TeX

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