direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D15, C10⋊2D6, C6⋊2D10, C15⋊2C23, C30⋊2C22, (C2×C6)⋊3D5, (C2×C10)⋊5S3, (C2×C30)⋊3C2, C5⋊2(C22×S3), C3⋊2(C22×D5), SmallGroup(120,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C22×D15 |
Generators and relations for C22×D15
G = < a,b,c,d | a2=b2=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 284 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, D5, C10, D6, C2×C6, C15, D10, C2×C10, C22×S3, D15, C30, C22×D5, D30, C2×C30, C22×D15
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22×S3, D15, C22×D5, D30, C22×D15
►On 60 points
Generators in S60
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 31)(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 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 60)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 35)(17 34)(18 33)(19 32)(20 31)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)
G:=sub<Sym(60)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,31)(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), (1,20)(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)(31,60)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,35)(17,34)(18,33)(19,32)(20,31)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,31)(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), (1,20)(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)(31,60)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,35)(17,34)(18,33)(19,32)(20,31)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36) );
G=PermutationGroup([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,31),(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)], [(1,20),(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),(31,60),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,35),(17,34),(18,33),(19,32),(20,31),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36)]])
C22×D15 is a maximal subgroup of
D30⋊4C4 D30⋊3C4 D10⋊D6 C22×S3×D5
C22×D15 is a maximal quotient of D60⋊11C2 D4⋊2D15 Q8⋊3D15
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D30 |
| kernel | C22×D15 | D30 | C2×C30 | C2×C10 | C2×C6 | C10 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 1 | 2 | 3 | 6 | 4 | 12 |
Matrix representation of C22×D15 ►in GL3(𝔽31) generated by
| 30 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 30 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 12 | 4 |
| 0 | 27 | 22 |
| 30 | 0 | 0 |
| 0 | 19 | 27 |
| 0 | 28 | 12 |
G:=sub<GL(3,GF(31))| [30,0,0,0,30,0,0,0,30],[30,0,0,0,1,0,0,0,1],[1,0,0,0,12,27,0,4,22],[30,0,0,0,19,28,0,27,12] >;
C22×D15 in GAP, Magma, Sage, TeX
C_2^2\times D_{15} % in TeX
G:=Group("C2^2xD15"); // GroupNames label
G:=SmallGroup(120,46);
// by ID
G=gap.SmallGroup(120,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,323,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations