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## G = C2×S3×D15order 360 = 23·32·5

### Direct product of C2, S3 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C2×S3×D15
 Chief series C1 — C5 — C15 — C3×C15 — C3×D15 — S3×D15 — C2×S3×D15
 Lower central C3×C15 — C2×S3×D15
 Upper central C1 — C2

Generators and relations for C2×S3×D15
G = < a,b,c,d,e | a2=b3=c2=d15=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: 1092 in 138 conjugacy classes, 42 normal (24 characteristic)
C1, C2, C2, C3, C3, C22, C5, S3, S3, C6, C6, C23, C32, D5, C10, C10, D6, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3⋊S3, C3×C6, D10, C2×C10, C22×S3, C5×S3, C3×D5, D15, D15, C30, C30, S32, S3×C6, S3×C6, C2×C3⋊S3, C22×D5, C3×C15, S3×D5, C6×D5, S3×C10, D30, D30, C2×C30, C2×S32, S3×C15, C3×D15, C3⋊D15, C3×C30, C2×S3×D5, C22×D15, S3×D15, S3×C30, C6×D15, C2×C3⋊D15, C2×S3×D15
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22×S3, D15, S32, C22×D5, S3×D5, D30, C2×S32, C2×S3×D5, C22×D15, S3×D15, C2×S3×D15

Smallest permutation representation of C2×S3×D15
On 60 points
Generators in S60
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 16)(13 17)(14 18)(15 19)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 51 56)(47 52 57)(48 53 58)(49 54 59)(50 55 60)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 31)(13 32)(14 33)(15 34)(16 57)(17 58)(18 59)(19 60)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)
(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 34)(2 33)(3 32)(4 31)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 49)(17 48)(18 47)(19 46)(20 60)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 5A 5B 6A 6B 6C 6D 6E 6F 6G 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 5 5 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 15 15 45 45 2 2 4 2 2 2 2 4 6 6 30 30 2 2 6 6 6 6 2 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D5 D6 D6 D6 D10 D10 D15 D30 D30 S32 S3×D5 C2×S32 C2×S3×D5 S3×D15 C2×S3×D15 kernel C2×S3×D15 S3×D15 S3×C30 C6×D15 C2×C3⋊D15 S3×C10 D30 S3×C6 C5×S3 D15 C30 C3×S3 C3×C6 D6 S3 C6 C10 C6 C5 C3 C2 C1 # reps 1 4 1 1 1 1 1 2 2 2 2 4 2 4 8 4 1 2 1 2 4 4

Matrix representation of C2×S3×D15 in GL6(𝔽31)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 1 0 0 0 0 30 0
,
 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 13 18 0 0 0 0 13 30 0 0 0 0 0 0 0 30 0 0 0 0 1 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 30 0 0 0 0 30 0 0 0 0 0 0 0 30 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[13,13,0,0,0,0,18,30,0,0,0,0,0,0,0,1,0,0,0,0,30,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,30,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×S3×D15 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_{15}
% in TeX

G:=Group("C2xS3xD15");
// GroupNames label

G:=SmallGroup(360,154);
// by ID

G=gap.SmallGroup(360,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,201,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^15=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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