non-abelian, soluble, monomial
Aliases: C24:2D15, C23.5D30, (C2xC10):4S4, (C5xA4):7D4, C22:2(C5:S4), A4:3(C5:D4), C5:3(A4:D4), C10.26(C2xS4), A4:Dic5:1C2, (C23xC10):4S3, C22:(C15:7D4), (C22xA4):2D5, (C2xA4).12D10, (C22xC10).17D6, (C10xA4).12C22, (C2xC5:S4):2C2, (A4xC2xC10):2C2, C2.11(C2xC5:S4), (C2xC10):4(C3:D4), SmallGroup(480,1034)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:2D15
G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, faf=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fdf=cd=dc, cf=fc, ede-1=c, fef=e-1 >
Subgroups: 1036 in 124 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2xC4, D4, C23, C23, D5, C10, C10, Dic3, A4, D6, C2xC6, C15, C22:C4, C2xD4, C24, Dic5, D10, C2xC10, C2xC10, C3:D4, S4, C2xA4, C2xA4, D15, C30, C22wrC2, C2xDic5, C5:D4, C22xD5, C22xC10, C22xC10, A4:C4, C2xS4, C22xA4, Dic15, C5xA4, D30, C2xC30, C23.D5, C2xC5:D4, C23xC10, A4:D4, C15:7D4, C5:S4, C10xA4, C10xA4, C24:2D5, A4:Dic5, C2xC5:S4, A4xC2xC10, C24:2D15
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, C3:D4, S4, D15, C5:D4, C2xS4, D30, A4:D4, C15:7D4, C5:S4, C2xC5:S4, C24:2D15
►On 60 points
Generators in S60
(16 43)(17 44)(18 45)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 43)(17 44)(18 45)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)
(2 57)(3 58)(5 60)(6 46)(8 48)(9 49)(11 51)(12 52)(14 54)(15 55)(17 44)(18 45)(20 32)(21 33)(23 35)(24 36)(26 38)(27 39)(29 41)(30 42)
(1 56)(3 58)(4 59)(6 46)(7 47)(9 49)(10 50)(12 52)(13 53)(15 55)(16 43)(18 45)(19 31)(21 33)(22 34)(24 36)(25 37)(27 39)(28 40)(30 42)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 50)(17 49)(18 48)(19 47)(20 46)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)
G:=sub<Sym(60)| (16,43)(17,44)(18,45)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42), (1,56)(2,57)(3,58)(4,59)(5,60)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,43)(17,44)(18,45)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42), (2,57)(3,58)(5,60)(6,46)(8,48)(9,49)(11,51)(12,52)(14,54)(15,55)(17,44)(18,45)(20,32)(21,33)(23,35)(24,36)(26,38)(27,39)(29,41)(30,42), (1,56)(3,58)(4,59)(6,46)(7,47)(9,49)(10,50)(12,52)(13,53)(15,55)(16,43)(18,45)(19,31)(21,33)(22,34)(24,36)(25,37)(27,39)(28,40)(30,42), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,50)(17,49)(18,48)(19,47)(20,46)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)>;
G:=Group( (16,43)(17,44)(18,45)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42), (1,56)(2,57)(3,58)(4,59)(5,60)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,43)(17,44)(18,45)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42), (2,57)(3,58)(5,60)(6,46)(8,48)(9,49)(11,51)(12,52)(14,54)(15,55)(17,44)(18,45)(20,32)(21,33)(23,35)(24,36)(26,38)(27,39)(29,41)(30,42), (1,56)(3,58)(4,59)(6,46)(7,47)(9,49)(10,50)(12,52)(13,53)(15,55)(16,43)(18,45)(19,31)(21,33)(22,34)(24,36)(25,37)(27,39)(28,40)(30,42), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,50)(17,49)(18,48)(19,47)(20,46)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51) );
G=PermutationGroup([[(16,43),(17,44),(18,45),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,43),(17,44),(18,45),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42)], [(2,57),(3,58),(5,60),(6,46),(8,48),(9,49),(11,51),(12,52),(14,54),(15,55),(17,44),(18,45),(20,32),(21,33),(23,35),(24,36),(26,38),(27,39),(29,41),(30,42)], [(1,56),(3,58),(4,59),(6,46),(7,47),(9,49),(10,50),(12,52),(13,53),(15,55),(16,43),(18,45),(19,31),(21,33),(22,34),(24,36),(25,37),(27,39),(28,40),(30,42)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,50),(17,49),(18,48),(19,47),(20,46),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 60 | 8 | 60 | 60 | 60 | 2 | 2 | 8 | 8 | 8 | 2 | ··· | 2 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3:D4 | D15 | C5:D4 | D30 | C15:7D4 | S4 | C2xS4 | A4:D4 | C5:S4 | C2xC5:S4 | C24:2D15 |
| kernel | C24:2D15 | A4:Dic5 | C2xC5:S4 | A4xC2xC10 | C23xC10 | C5xA4 | C22xA4 | C22xC10 | C2xA4 | C2xC10 | C24 | A4 | C23 | C22 | C2xC10 | C10 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of C24:2D15 ►in GL5(F61)
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 60 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 60 | 0 |
| 39 | 19 | 0 | 0 | 0 |
| 19 | 39 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 22 | 19 | 0 | 0 | 0 |
| 42 | 39 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(61))| [0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[39,19,0,0,0,19,39,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[22,42,0,0,0,19,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C24:2D15 in GAP, Magma, Sage, TeX
C_2^4\rtimes_2D_{15} % in TeX
G:=Group("C2^4:2D15"); // GroupNames label
G:=SmallGroup(480,1034);
// by ID
G=gap.SmallGroup(480,1034);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,451,3364,10085,1286,5886,2232]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=f^2=1,f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*d*f=c*d=d*c,c*f=f*c,e*d*e^-1=c,f*e*f=e^-1>;
// generators/relations