direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×S3×D5, C15⋊C24, C30⋊C23, D15⋊C23, D30⋊11C22, (C2×C6)⋊6D10, (C2×C10)⋊9D6, (C5×S3)⋊C23, C5⋊1(S3×C23), (C3×D5)⋊C23, C3⋊1(C23×D5), C6⋊1(C22×D5), (C2×C30)⋊5C22, C10⋊1(C22×S3), (C6×D5)⋊8C22, (S3×C10)⋊8C22, (C22×D15)⋊7C2, (D5×C2×C6)⋊4C2, (S3×C2×C10)⋊4C2, SmallGroup(240,202)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C22×S3×D5 |
Generators and relations for C22×S3×D5
G = < a,b,c,d,e,f | a2=b2=c3=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1200 in 268 conjugacy classes, 104 normal (14 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, S3, C6, C6, C23, D5, D5, C10, C10, D6, D6, C2×C6, C2×C6, C15, C24, D10, D10, C2×C10, C2×C10, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C22×D5, C22×D5, C22×C10, S3×C23, S3×D5, C6×D5, S3×C10, D30, C2×C30, C23×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22×S3×D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, C22×D5, S3×C23, S3×D5, C23×D5, C2×S3×D5, C22×S3×D5
►On 60 points
(1 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(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 34)(2 35)(3 31)(4 32)(5 33)(6 41)(7 42)(8 43)(9 44)(10 45)(11 36)(12 37)(13 38)(14 39)(15 40)(16 46)(17 47)(18 48)(19 49)(20 50)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)
(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 33)(2 32)(3 31)(4 35)(5 34)(6 36)(7 40)(8 39)(9 38)(10 37)(11 41)(12 45)(13 44)(14 43)(15 42)(16 46)(17 50)(18 49)(19 48)(20 47)(21 51)(22 55)(23 54)(24 53)(25 52)(26 56)(27 60)(28 59)(29 58)(30 57)
G:=sub<Sym(60)| (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,41)(7,42)(8,43)(9,44)(10,45)(11,36)(12,37)(13,38)(14,39)(15,40)(16,46)(17,47)(18,48)(19,49)(20,50)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55), (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), (1,33)(2,32)(3,31)(4,35)(5,34)(6,36)(7,40)(8,39)(9,38)(10,37)(11,41)(12,45)(13,44)(14,43)(15,42)(16,46)(17,50)(18,49)(19,48)(20,47)(21,51)(22,55)(23,54)(24,53)(25,52)(26,56)(27,60)(28,59)(29,58)(30,57)>;
G:=Group( (1,34)(2,35)(3,31)(4,32)(5,33)(6,36)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(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), (1,34)(2,35)(3,31)(4,32)(5,33)(6,41)(7,42)(8,43)(9,44)(10,45)(11,36)(12,37)(13,38)(14,39)(15,40)(16,46)(17,47)(18,48)(19,49)(20,50)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55), (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), (1,33)(2,32)(3,31)(4,35)(5,34)(6,36)(7,40)(8,39)(9,38)(10,37)(11,41)(12,45)(13,44)(14,43)(15,42)(16,46)(17,50)(18,49)(19,48)(20,47)(21,51)(22,55)(23,54)(24,53)(25,52)(26,56)(27,60)(28,59)(29,58)(30,57) );
G=PermutationGroup([[(1,34),(2,35),(3,31),(4,32),(5,33),(6,36),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(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)], [(1,34),(2,35),(3,31),(4,32),(5,33),(6,41),(7,42),(8,43),(9,44),(10,45),(11,36),(12,37),(13,38),(14,39),(15,40),(16,46),(17,47),(18,48),(19,49),(20,50),(21,56),(22,57),(23,58),(24,59),(25,60),(26,51),(27,52),(28,53),(29,54),(30,55)], [(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)], [(1,33),(2,32),(3,31),(4,35),(5,34),(6,36),(7,40),(8,39),(9,38),(10,37),(11,41),(12,45),(13,44),(14,43),(15,42),(16,46),(17,50),(18,49),(19,48),(20,47),(21,51),(22,55),(23,54),(24,53),(25,52),(26,56),(27,60),(28,59),(29,58),(30,57)]])
C22×S3×D5 is a maximal subgroup of
D30.27D4 D6⋊4D20 D30⋊4D4 D30⋊5D4 D30⋊8D4
C22×S3×D5 is a maximal quotient of
D20.38D6 D20.39D6 C30.C24 D20⋊24D6 D20⋊25D6 D20⋊26D6 D20⋊29D6 C15⋊2- 1+4 D30.C23 D20⋊13D6 D20⋊14D6 D12⋊14D10 D20.29D6 C30.33C24 D12.29D10 D20⋊16D6 D20⋊17D6 C15⋊2+ 1+4
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 15A | 15B | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D10 | D10 | S3×D5 | C2×S3×D5 |
| kernel | C22×S3×D5 | C2×S3×D5 | D5×C2×C6 | S3×C2×C10 | C22×D15 | C22×D5 | C22×S3 | D10 | C2×C10 | D6 | C2×C6 | C22 | C2 |
| # reps | 1 | 12 | 1 | 1 | 1 | 1 | 2 | 6 | 1 | 12 | 2 | 2 | 6 |
Matrix representation of C22×S3×D5 ►in GL4(𝔽31) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 29 | 26 |
| 0 | 0 | 13 | 1 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 30 |
| 30 | 1 | 0 | 0 |
| 17 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 14 | 30 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,29,13,0,0,26,1],[30,0,0,0,0,30,0,0,0,0,1,18,0,0,0,30],[30,17,0,0,1,13,0,0,0,0,1,0,0,0,0,1],[1,14,0,0,0,30,0,0,0,0,1,0,0,0,0,1] >;
C22×S3×D5 in GAP, Magma, Sage, TeX
C_2^2\times S_3\times D_5
% in TeX
G:=Group("C2^2xS3xD5"); // GroupNames label
G:=SmallGroup(240,202);
// by ID
G=gap.SmallGroup(240,202);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,490,6917]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^2=e^5=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations