direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D15, C60⋊2C2, C20⋊2S3, C12⋊2D5, C4○Dic15, C2.1D30, C10.9D6, C6.9D10, D30.2C2, Dic15⋊5C2, C30.9C22, C5⋊3(C4×S3), C3⋊2(C4×D5), C15⋊7(C2×C4), SmallGroup(120,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C4×D15 |
Generators and relations for C4×D15
G = < a,b,c | a4=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D15
►On 60 points
Generators in S60
(1 54 17 31)(2 55 18 32)(3 56 19 33)(4 57 20 34)(5 58 21 35)(6 59 22 36)(7 60 23 37)(8 46 24 38)(9 47 25 39)(10 48 26 40)(11 49 27 41)(12 50 28 42)(13 51 29 43)(14 52 30 44)(15 53 16 45)
(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 16)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)
G:=sub<Sym(60)| (1,54,17,31)(2,55,18,32)(3,56,19,33)(4,57,20,34)(5,58,21,35)(6,59,22,36)(7,60,23,37)(8,46,24,38)(9,47,25,39)(10,48,26,40)(11,49,27,41)(12,50,28,42)(13,51,29,43)(14,52,30,44)(15,53,16,45), (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), (1,16)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)>;
G:=Group( (1,54,17,31)(2,55,18,32)(3,56,19,33)(4,57,20,34)(5,58,21,35)(6,59,22,36)(7,60,23,37)(8,46,24,38)(9,47,25,39)(10,48,26,40)(11,49,27,41)(12,50,28,42)(13,51,29,43)(14,52,30,44)(15,53,16,45), (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), (1,16)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54) );
G=PermutationGroup([[(1,54,17,31),(2,55,18,32),(3,56,19,33),(4,57,20,34),(5,58,21,35),(6,59,22,36),(7,60,23,37),(8,46,24,38),(9,47,25,39),(10,48,26,40),(11,49,27,41),(12,50,28,42),(13,51,29,43),(14,52,30,44),(15,53,16,45)], [(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)], [(1,16),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,54)]])
C4×D15 is a maximal subgroup of
D15⋊2C8 D30.5C4 C40⋊S3 D20⋊S3 D12⋊D5 D15⋊Q8 D6.D10 C4×S3×D5 C20⋊D6 D60⋊11C2 D4⋊2D15 Q8⋊3D15 C6.D30 C20.6S4
C4×D15 is a maximal quotient of
C40⋊S3 C30.4Q8 D30⋊3C4 C6.D30
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 15 | 15 | 2 | 1 | 1 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D5 | D6 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 |
| kernel | C4×D15 | Dic15 | C60 | D30 | D15 | C20 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C4×D15 ►in GL2(𝔽29) generated by
| 17 | 0 |
| 0 | 17 |
| 28 | 21 |
| 21 | 22 |
| 22 | 23 |
| 8 | 7 |
G:=sub<GL(2,GF(29))| [17,0,0,17],[28,21,21,22],[22,8,23,7] >;
C4×D15 in GAP, Magma, Sage, TeX
C_4\times D_{15} % in TeX
G:=Group("C4xD15"); // GroupNames label
G:=SmallGroup(120,27);
// by ID
G=gap.SmallGroup(120,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,26,323,2404]);
// Polycyclic
G:=Group<a,b,c|a^4=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export