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G = C3xC5:D5order 150 = 2·3·52

Direct product of C3 and C5:D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3xC5:D5, C15:2D5, C52:4C6, C5:(C3xD5), (C5xC15):4C2, SmallGroup(150,9)

Series: Derived Chief Lower central Upper central

C1C52 — C3xC5:D5
C1C5C52C5xC15 — C3xC5:D5
C52 — C3xC5:D5
C1C3

Generators and relations for C3xC5:D5
 G = < a,b,c,d | a3=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 128 in 32 conjugacy classes, 18 normal (6 characteristic)
Quotients: C1, C2, C3, C6, D5, C3xD5, C5:D5, C3xC5:D5
25C2
25C6
5D5
5D5
5D5
5D5
5D5
5D5
5C3xD5
5C3xD5
5C3xD5
5C3xD5
5C3xD5
5C3xD5

Smallest permutation representation of C3xC5:D5
On 75 points
Generators in S75
(1 54 29)(2 55 30)(3 51 26)(4 52 27)(5 53 28)(6 56 31)(7 57 32)(8 58 33)(9 59 34)(10 60 35)(11 61 36)(12 62 37)(13 63 38)(14 64 39)(15 65 40)(16 66 41)(17 67 42)(18 68 43)(19 69 44)(20 70 45)(21 71 46)(22 72 47)(23 73 48)(24 74 49)(25 75 50)
(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 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)
(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)(26 32)(27 31)(28 35)(29 34)(30 33)(36 47)(37 46)(38 50)(39 49)(40 48)(41 42)(43 45)(51 57)(52 56)(53 60)(54 59)(55 58)(61 72)(62 71)(63 75)(64 74)(65 73)(66 67)(68 70)

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

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

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

C3xC5:D5 is a maximal subgroup of   C15:F5  C15:2F5  C3xD52  D15:D5  C52:C18

42 conjugacy classes

class 1  2 3A3B5A···5L6A6B15A···15X
order12335···56615···15
size125112···225252···2

42 irreducible representations

dim111122
type+++
imageC1C2C3C6D5C3xD5
kernelC3xC5:D5C5xC15C5:D5C52C15C5
# reps11221224

Matrix representation of C3xC5:D5 in GL4(F31) generated by

25000
02500
00250
00025
,
12100
30000
0001
003018
,
301900
121900
0010
0001
,
193000
191200
003018
0001
G:=sub<GL(4,GF(31))| [25,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[12,30,0,0,1,0,0,0,0,0,0,30,0,0,1,18],[30,12,0,0,19,19,0,0,0,0,1,0,0,0,0,1],[19,19,0,0,30,12,0,0,0,0,30,0,0,0,18,1] >;

C3xC5:D5 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes D_5
% in TeX

G:=Group("C3xC5:D5");
// GroupNames label

G:=SmallGroup(150,9);
// by ID

G=gap.SmallGroup(150,9);
# by ID

G:=PCGroup([4,-2,-3,-5,-5,290,1923]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C3xC5:D5 in TeX

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