direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×S3×D15, C30⋊3D6, C6⋊1D30, C10⋊1S32, C6⋊1(S3×D5), (C5×S3)⋊2D6, (S3×C6)⋊5D5, (C3×C6)⋊1D10, (S3×C30)⋊6C2, (S3×C10)⋊3S3, (C3×S3)⋊2D10, (C3×C15)⋊4C23, (C6×D15)⋊10C2, (C3×C30)⋊3C22, C15⋊4(C22×S3), C3⋊D15⋊3C22, C3⋊1(C22×D15), C32⋊2(C22×D5), (S3×C15)⋊2C22, (C3×D15)⋊3C22, C5⋊2(C2×S32), C3⋊3(C2×S3×D5), (C2×C3⋊D15)⋊8C2, SmallGroup(360,154)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — C2×S3×D15 |
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
►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