direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×S3×D5, C20⋊5D6, C12⋊5D10, C60⋊5C22, Dic5⋊5D6, D6.9D10, Dic3⋊5D10, D10.17D6, C30.11C23, Dic15⋊5C22, D30.12C22, (S3×C20)⋊5C2, D15⋊2(C2×C4), (C4×D15)⋊9C2, (D5×C12)⋊5C2, C15⋊2(C22×C4), D30.C2⋊6C2, (D5×Dic3)⋊6C2, (S3×Dic5)⋊6C2, C6.11(C22×D5), (S3×C10).9C22, C10.11(C22×S3), (C5×Dic3)⋊3C22, (C3×Dic5)⋊3C22, (C6×D5).13C22, C5⋊2(S3×C2×C4), C3⋊1(C2×C4×D5), C2.1(C2×S3×D5), (C2×S3×D5).3C2, (C5×S3)⋊2(C2×C4), (C3×D5)⋊2(C2×C4), SmallGroup(240,135)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C4×S3×D5 |
Generators and relations for C4×S3×D5
G = < a,b,c,d,e | a4=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: 448 in 108 conjugacy classes, 46 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, S3×C2×C4, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, S3×C10, D30, C2×C4×D5, D5×Dic3, S3×Dic5, D30.C2, D5×C12, S3×C20, C4×D15, C2×S3×D5, C4×S3×D5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, C2×S3×D5, C4×S3×D5
►On 60 points
(1 49 19 34)(2 50 20 35)(3 46 16 31)(4 47 17 32)(5 48 18 33)(6 51 21 36)(7 52 22 37)(8 53 23 38)(9 54 24 39)(10 55 25 40)(11 56 26 41)(12 57 27 42)(13 58 28 43)(14 59 29 44)(15 60 30 45)
(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 19)(2 20)(3 16)(4 17)(5 18)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)(31 46)(32 47)(33 48)(34 49)(35 50)(36 56)(37 57)(38 58)(39 59)(40 60)(41 51)(42 52)(43 53)(44 54)(45 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 18)(2 17)(3 16)(4 20)(5 19)(6 21)(7 25)(8 24)(9 23)(10 22)(11 26)(12 30)(13 29)(14 28)(15 27)(31 46)(32 50)(33 49)(34 48)(35 47)(36 51)(37 55)(38 54)(39 53)(40 52)(41 56)(42 60)(43 59)(44 58)(45 57)
G:=sub<Sym(60)| (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45), (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), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25)(31,46)(32,47)(33,48)(34,49)(35,50)(36,56)(37,57)(38,58)(39,59)(40,60)(41,51)(42,52)(43,53)(44,54)(45,55), (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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,21)(7,25)(8,24)(9,23)(10,22)(11,26)(12,30)(13,29)(14,28)(15,27)(31,46)(32,50)(33,49)(34,48)(35,47)(36,51)(37,55)(38,54)(39,53)(40,52)(41,56)(42,60)(43,59)(44,58)(45,57)>;
G:=Group( (1,49,19,34)(2,50,20,35)(3,46,16,31)(4,47,17,32)(5,48,18,33)(6,51,21,36)(7,52,22,37)(8,53,23,38)(9,54,24,39)(10,55,25,40)(11,56,26,41)(12,57,27,42)(13,58,28,43)(14,59,29,44)(15,60,30,45), (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), (1,19)(2,20)(3,16)(4,17)(5,18)(6,26)(7,27)(8,28)(9,29)(10,30)(11,21)(12,22)(13,23)(14,24)(15,25)(31,46)(32,47)(33,48)(34,49)(35,50)(36,56)(37,57)(38,58)(39,59)(40,60)(41,51)(42,52)(43,53)(44,54)(45,55), (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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,21)(7,25)(8,24)(9,23)(10,22)(11,26)(12,30)(13,29)(14,28)(15,27)(31,46)(32,50)(33,49)(34,48)(35,47)(36,51)(37,55)(38,54)(39,53)(40,52)(41,56)(42,60)(43,59)(44,58)(45,57) );
G=PermutationGroup([[(1,49,19,34),(2,50,20,35),(3,46,16,31),(4,47,17,32),(5,48,18,33),(6,51,21,36),(7,52,22,37),(8,53,23,38),(9,54,24,39),(10,55,25,40),(11,56,26,41),(12,57,27,42),(13,58,28,43),(14,59,29,44),(15,60,30,45)], [(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)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,26),(7,27),(8,28),(9,29),(10,30),(11,21),(12,22),(13,23),(14,24),(15,25),(31,46),(32,47),(33,48),(34,49),(35,50),(36,56),(37,57),(38,58),(39,59),(40,60),(41,51),(42,52),(43,53),(44,54),(45,55)], [(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)], [(1,18),(2,17),(3,16),(4,20),(5,19),(6,21),(7,25),(8,24),(9,23),(10,22),(11,26),(12,30),(13,29),(14,28),(15,27),(31,46),(32,50),(33,49),(34,48),(35,47),(36,51),(37,55),(38,54),(39,53),(40,52),(41,56),(42,60),(43,59),(44,58),(45,57)]])
C4×S3×D5 is a maximal subgroup of
C40⋊D6 C4⋊F5⋊3S3 (C4×S3)⋊F5 D15⋊M4(2) C5⋊C8⋊D6 D20⋊24D6 D30.C23 D20⋊16D6
C4×S3×D5 is a maximal quotient of
C40⋊D6 C40.54D6 C40.34D6 C40.55D6 C40.35D6 Dic5⋊5Dic6 Dic3⋊5Dic10 Dic15⋊5Q8 (D5×Dic3)⋊C4 D10.19(C4×S3) Dic3⋊4D20 Dic15⋊13D4 D6.(C4×D5) (S3×Dic5)⋊C4 D30.C2⋊C4 D30.23(C2×C4) D30.Q8 Dic5⋊4D12 Dic15⋊14D4 D6⋊(C4×D5) C15⋊17(C4×D4) Dic15⋊9D4 C15⋊20(C4×D4) C15⋊22(C4×D4) D30.27D4
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 3 | 3 | 5 | 5 | 15 | 15 | 2 | 1 | 1 | 3 | 3 | 5 | 5 | 15 | 15 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 10 | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4×S3 | C4×D5 | S3×D5 | C2×S3×D5 | C4×S3×D5 |
| kernel | C4×S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 | D5×C12 | S3×C20 | C4×D15 | C2×S3×D5 | S3×D5 | C4×D5 | C4×S3 | Dic5 | C20 | D10 | Dic3 | C12 | D6 | D5 | S3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of C4×S3×D5 ►in GL4(𝔽61) generated by
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 60 |
| 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 60 | 43 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,0,1],[0,60,0,0,1,43,0,0,0,0,1,0,0,0,0,1],[60,18,0,0,0,1,0,0,0,0,60,0,0,0,0,60] >;
C4×S3×D5 in GAP, Magma, Sage, TeX
C_4\times S_3\times D_5
% in TeX
G:=Group("C4xS3xD5"); // GroupNames label
G:=SmallGroup(240,135);
// by ID
G=gap.SmallGroup(240,135);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=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