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G = C3×5- 1+2  order 375 = 3·53

Direct product of C3 and 5- 1+2

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

Aliases: C3×5- 1+2, C75⋊C5, C25⋊C15, C52.C15, C15.2C52, (C5×C15).C5, C5.2(C5×C15), SmallGroup(375,5)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — C3×5- 1+2
Chief series
C1 — C5 — C52 — 5- 1+2 — C3×5- 1+2
Lower central
C1 — C5 — C3×5- 1+2
Upper central
C1 — C15 — C3×5- 1+2

Generators and relations for C3×5- 1+2
 G = < a,b,c | a3=b25=c5=1, ab=ba, ac=ca, cbc-1=b6 >

C1
C3
C5
5C5
C15
5C15
C25
C25
C25
C52
C25
C25
C75
C75
C75
C75
C75
C5×C15
5- 1+2
C3×5- 1+2

Smallest permutation representation of C3×5- 1+2
►On 75 points
Generators in S75
(1 65 50)(2 66 26)(3 67 27)(4 68 28)(5 69 29)(6 70 30)(7 71 31)(8 72 32)(9 73 33)(10 74 34)(11 75 35)(12 51 36)(13 52 37)(14 53 38)(15 54 39)(16 55 40)(17 56 41)(18 57 42)(19 58 43)(20 59 44)(21 60 45)(22 61 46)(23 62 47)(24 63 48)(25 64 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 6 11 16 21)(3 23 18 13 8)(4 19 9 24 14)(5 15 25 10 20)(27 47 42 37 32)(28 43 33 48 38)(29 39 49 34 44)(30 35 40 45 50)(52 72 67 62 57)(53 68 58 73 63)(54 64 74 59 69)(55 60 65 70 75)
 
G:=sub<Sym(75)| (1,65,50)(2,66,26)(3,67,27)(4,68,28)(5,69,29)(6,70,30)(7,71,31)(8,72,32)(9,73,33)(10,74,34)(11,75,35)(12,51,36)(13,52,37)(14,53,38)(15,54,39)(16,55,40)(17,56,41)(18,57,42)(19,58,43)(20,59,44)(21,60,45)(22,61,46)(23,62,47)(24,63,48)(25,64,49), (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), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(27,47,42,37,32)(28,43,33,48,38)(29,39,49,34,44)(30,35,40,45,50)(52,72,67,62,57)(53,68,58,73,63)(54,64,74,59,69)(55,60,65,70,75)>;
 
G:=Group( (1,65,50)(2,66,26)(3,67,27)(4,68,28)(5,69,29)(6,70,30)(7,71,31)(8,72,32)(9,73,33)(10,74,34)(11,75,35)(12,51,36)(13,52,37)(14,53,38)(15,54,39)(16,55,40)(17,56,41)(18,57,42)(19,58,43)(20,59,44)(21,60,45)(22,61,46)(23,62,47)(24,63,48)(25,64,49), (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), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(27,47,42,37,32)(28,43,33,48,38)(29,39,49,34,44)(30,35,40,45,50)(52,72,67,62,57)(53,68,58,73,63)(54,64,74,59,69)(55,60,65,70,75) );
 
G=PermutationGroup([[(1,65,50),(2,66,26),(3,67,27),(4,68,28),(5,69,29),(6,70,30),(7,71,31),(8,72,32),(9,73,33),(10,74,34),(11,75,35),(12,51,36),(13,52,37),(14,53,38),(15,54,39),(16,55,40),(17,56,41),(18,57,42),(19,58,43),(20,59,44),(21,60,45),(22,61,46),(23,62,47),(24,63,48),(25,64,49)], [(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)], [(1,6,11,16,21),(3,23,18,13,8),(4,19,9,24,14),(5,15,25,10,20),(27,47,42,37,32),(28,43,33,48,38),(29,39,49,34,44),(30,35,40,45,50),(52,72,67,62,57),(53,68,58,73,63),(54,64,74,59,69),(55,60,65,70,75)]])
 

87 conjugacy classes

class 1 3A3B5A5B5C5D5E5F5G5H15A···15H15I···15P25A···25T75A···75AN
order1335555555515···1515···1525···2575···75
size111111155551···15···55···55···5

87 irreducible representations

dim11111155
type+
imageC1C3C5C5C15C155- 1+2C3×5- 1+2
kernelC3×5- 1+25- 1+2C75C5×C15C25C52C3C1
# reps1220440848

Matrix representation of C3×5- 1+2 ►in GL5(𝔽151)

320000
032000
003200
000320
000032
,
1435900142
870100
92006420
132000132
70008
,
19000150
05900142
0064078
000819
00001

G:=sub<GL(5,GF(151))| [32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[143,87,92,132,7,59,0,0,0,0,0,1,0,0,0,0,0,64,0,0,142,0,20,132,8],[19,0,0,0,0,0,59,0,0,0,0,0,64,0,0,0,0,0,8,0,150,142,78,19,1] >;
 

C3×5- 1+2 in GAP, Magma, Sage, TeX 

C_3\times 5_-^{1+2}
 
% in TeX
 
G:=Group("C3xES-(5,1)");
 
// GroupNames label
 
G:=SmallGroup(375,5);
 
// by ID
 
G=gap.SmallGroup(375,5);
 
# by ID
 
G:=PCGroup([4,-3,-5,-5,-5,205,1266]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^25=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C3×5- 1+2 in TeX

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