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G = S3×C6×D5order 360 = 23·32·5

Direct product of C6, S3 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — S3×C6×D5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C3×S3×D5 — S3×C6×D5
 Lower central C15 — S3×C6×D5
 Upper central C1 — C6

Generators and relations for S3×C6×D5
G = < a,b,c,d,e | a6=b3=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 564 in 138 conjugacy classes, 56 normal (28 characteristic)
C1, C2, C2, C3, C3, C22, C5, S3, S3, C6, C6, C23, C32, D5, D5, C10, C10, D6, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3×C6, C3×C6, D10, D10, C2×C10, C22×S3, C22×C6, C5×S3, C3×D5, C3×D5, D15, C30, C30, S3×C6, S3×C6, C62, C22×D5, C3×C15, S3×D5, C6×D5, C6×D5, S3×C10, D30, C2×C30, S3×C2×C6, C32×D5, S3×C15, C3×D15, C3×C30, C2×S3×D5, D5×C2×C6, C3×S3×D5, D5×C3×C6, S3×C30, C6×D15, S3×C6×D5
Quotients: C1, C2, C3, C22, S3, C6, C23, D5, D6, C2×C6, C3×S3, D10, C22×S3, C22×C6, C3×D5, S3×C6, C22×D5, S3×D5, C6×D5, S3×C2×C6, C2×S3×D5, D5×C2×C6, C3×S3×D5, S3×C6×D5

Smallest permutation representation of S3×C6×D5
On 60 points
Generators in S60
(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)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(7 52)(8 53)(9 54)(10 49)(11 50)(12 51)(13 48)(14 43)(15 44)(16 45)(17 46)(18 47)(19 33)(20 34)(21 35)(22 36)(23 31)(24 32)(25 39)(26 40)(27 41)(28 42)(29 37)(30 38)
(1 39 46 53 31)(2 40 47 54 32)(3 41 48 49 33)(4 42 43 50 34)(5 37 44 51 35)(6 38 45 52 36)(7 22 57 30 16)(8 23 58 25 17)(9 24 59 26 18)(10 19 60 27 13)(11 20 55 28 14)(12 21 56 29 15)
(1 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 16)(14 17)(15 18)(19 57)(20 58)(21 59)(22 60)(23 55)(24 56)(37 54)(38 49)(39 50)(40 51)(41 52)(42 53)(43 46)(44 47)(45 48)

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

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), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,59,57)(56,60,58), (1,58)(2,59)(3,60)(4,55)(5,56)(6,57)(7,52)(8,53)(9,54)(10,49)(11,50)(12,51)(13,48)(14,43)(15,44)(16,45)(17,46)(18,47)(19,33)(20,34)(21,35)(22,36)(23,31)(24,32)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38), (1,39,46,53,31)(2,40,47,54,32)(3,41,48,49,33)(4,42,43,50,34)(5,37,44,51,35)(6,38,45,52,36)(7,22,57,30,16)(8,23,58,25,17)(9,24,59,26,18)(10,19,60,27,13)(11,20,55,28,14)(12,21,56,29,15), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,16)(14,17)(15,18)(19,57)(20,58)(21,59)(22,60)(23,55)(24,56)(37,54)(38,49)(39,50)(40,51)(41,52)(42,53)(43,46)(44,47)(45,48) );

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)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,59,57),(56,60,58)], [(1,58),(2,59),(3,60),(4,55),(5,56),(6,57),(7,52),(8,53),(9,54),(10,49),(11,50),(12,51),(13,48),(14,43),(15,44),(16,45),(17,46),(18,47),(19,33),(20,34),(21,35),(22,36),(23,31),(24,32),(25,39),(26,40),(27,41),(28,42),(29,37),(30,38)], [(1,39,46,53,31),(2,40,47,54,32),(3,41,48,49,33),(4,42,43,50,34),(5,37,44,51,35),(6,38,45,52,36),(7,22,57,30,16),(8,23,58,25,17),(9,24,59,26,18),(10,19,60,27,13),(11,20,55,28,14),(12,21,56,29,15)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,16),(14,17),(15,18),(19,57),(20,58),(21,59),(22,60),(23,55),(24,56),(37,54),(38,49),(39,50),(40,51),(41,52),(42,53),(43,46),(44,47),(45,48)]])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N ··· 6S 6T 6U 6V 6W 10A 10B 10C 10D 10E 10F 15A 15B 15C 15D 15E ··· 15J 30A 30B 30C 30D 30E ··· 30J 30K ··· 30R order 1 2 2 2 2 2 2 2 3 3 3 3 3 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 10 10 10 10 10 10 15 15 15 15 15 ··· 15 30 30 30 30 30 ··· 30 30 ··· 30 size 1 1 3 3 5 5 15 15 1 1 2 2 2 2 2 1 1 2 2 2 3 3 3 3 5 5 5 5 10 ··· 10 15 15 15 15 2 2 6 6 6 6 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D5 D6 D6 C3×S3 D10 D10 C3×D5 S3×C6 S3×C6 C6×D5 C6×D5 S3×D5 C2×S3×D5 C3×S3×D5 S3×C6×D5 kernel S3×C6×D5 C3×S3×D5 D5×C3×C6 S3×C30 C6×D15 C2×S3×D5 S3×D5 C6×D5 S3×C10 D30 C6×D5 S3×C6 C3×D5 C30 D10 C3×S3 C3×C6 D6 D5 C10 S3 C6 C6 C3 C2 C1 # reps 1 4 1 1 1 2 8 2 2 2 1 2 2 1 2 4 2 4 4 2 8 4 2 2 4 4

Matrix representation of S3×C6×D5 in GL4(𝔽31) generated by

 1 0 0 0 0 1 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 0 5 0 0 0 25 25
,
 1 0 0 0 0 1 0 0 0 0 30 24 0 0 0 1
,
 0 1 0 0 30 12 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 30 0 0 0 0 30
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,5,25,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,30,0,0,0,24,1],[0,30,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,30,0,0,0,0,30] >;

S3×C6×D5 in GAP, Magma, Sage, TeX

S_3\times C_6\times D_5
% in TeX

G:=Group("S3xC6xD5");
// GroupNames label

G:=SmallGroup(360,151);
// by ID

G=gap.SmallGroup(360,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,730,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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