metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊7D4, D30⋊2C2, C2.5D30, C6.12D10, C10.12D6, C22⋊2D15, Dic15⋊1C2, C30.12C22, (C2×C6)⋊2D5, (C2×C10)⋊4S3, (C2×C30)⋊2C2, C3⋊3(C5⋊D4), C5⋊3(C3⋊D4), SmallGroup(120,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊7D4
G = < a,b,c | a15=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C15⋊7D4
►On 60 points
(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 32 17 54)(2 31 18 53)(3 45 19 52)(4 44 20 51)(5 43 21 50)(6 42 22 49)(7 41 23 48)(8 40 24 47)(9 39 25 46)(10 38 26 60)(11 37 27 59)(12 36 28 58)(13 35 29 57)(14 34 30 56)(15 33 16 55)
(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(16 18)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)
G:=sub<Sym(60)| (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), (1,32,17,54)(2,31,18,53)(3,45,19,52)(4,44,20,51)(5,43,21,50)(6,42,22,49)(7,41,23,48)(8,40,24,47)(9,39,25,46)(10,38,26,60)(11,37,27,59)(12,36,28,58)(13,35,29,57)(14,34,30,56)(15,33,16,55), (2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(16,18)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56)>;
G:=Group( (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), (1,32,17,54)(2,31,18,53)(3,45,19,52)(4,44,20,51)(5,43,21,50)(6,42,22,49)(7,41,23,48)(8,40,24,47)(9,39,25,46)(10,38,26,60)(11,37,27,59)(12,36,28,58)(13,35,29,57)(14,34,30,56)(15,33,16,55), (2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(16,18)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56) );
G=PermutationGroup([[(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)], [(1,32,17,54),(2,31,18,53),(3,45,19,52),(4,44,20,51),(5,43,21,50),(6,42,22,49),(7,41,23,48),(8,40,24,47),(9,39,25,46),(10,38,26,60),(11,37,27,59),(12,36,28,58),(13,35,29,57),(14,34,30,56),(15,33,16,55)], [(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(16,18),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,60),(42,59),(43,58),(44,57),(45,56)]])
C15⋊7D4 is a maximal subgroup of
Dic5.D6 Dic3.D10 D5×C3⋊D4 S3×C5⋊D4 D60⋊11C2 D4×D15 D4⋊2D15 C45⋊7D4 C22⋊D45 D6⋊D15 D6⋊2D15 C62⋊D5 A4⋊D15 Q8.D30 C24⋊2D15
C15⋊7D4 is a maximal quotient of
C30.4Q8 D30⋊3C4 D4⋊D15 D4.D15 Q8⋊2D15 C15⋊7Q16 C30.38D4 C45⋊7D4 D6⋊D15 D6⋊2D15 C62⋊D5 C24⋊2D15
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 30 | 2 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 |
| kernel | C15⋊7D4 | Dic15 | D30 | C2×C30 | C2×C10 | C15 | C2×C6 | C10 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C15⋊7D4 ►in GL2(𝔽31) generated by
| 2 | 3 |
| 3 | 5 |
| 5 | 15 |
| 21 | 26 |
| 1 | 1 |
| 0 | 30 |
G:=sub<GL(2,GF(31))| [2,3,3,5],[5,21,15,26],[1,0,1,30] >;
C15⋊7D4 in GAP, Magma, Sage, TeX
C_{15}\rtimes_7D_4 % in TeX
G:=Group("C15:7D4"); // GroupNames label
G:=SmallGroup(120,30);
// by ID
G=gap.SmallGroup(120,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-5,61,323,2404]);
// Polycyclic
G:=Group<a,b,c|a^15=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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