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G = D5.D15order 300 = 22·3·52

The non-split extension by D5 of D15 acting via D15/C15=C2

Aliases: D5.D15, C5⋊Dic15, C155F5, C151Dic5, C524Dic3, (C5×C15)⋊1C4, C3⋊(D5.D5), C53(C3⋊F5), (C3×D5).1D5, (C5×D5).2S3, (D5×C15).2C2, SmallGroup(300,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — D5.D15
 Chief series C1 — C5 — C52 — C5×C15 — D5×C15 — D5.D15
 Lower central C5×C15 — D5.D15
 Upper central C1

Generators and relations for D5.D15
G = < a,b,c,d | a5=b2=c15=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Smallest permutation representation of D5.D15
On 60 points
Generators in S60
```(1 13 10 7 4)(2 14 11 8 5)(3 15 12 9 6)(16 19 22 25 28)(17 20 23 26 29)(18 21 24 27 30)(31 37 43 34 40)(32 38 44 35 41)(33 39 45 36 42)(46 55 49 58 52)(47 56 50 59 53)(48 57 51 60 54)
(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 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)
(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 53 23 36)(2 52 24 35)(3 51 25 34)(4 50 26 33)(5 49 27 32)(6 48 28 31)(7 47 29 45)(8 46 30 44)(9 60 16 43)(10 59 17 42)(11 58 18 41)(12 57 19 40)(13 56 20 39)(14 55 21 38)(15 54 22 37)```

`G:=sub<Sym(60)| (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30)(31,37,43,34,40)(32,38,44,35,41)(33,39,45,36,42)(46,55,49,58,52)(47,56,50,59,53)(48,57,51,60,54), (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)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)(45,53), (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), (1,53,23,36)(2,52,24,35)(3,51,25,34)(4,50,26,33)(5,49,27,32)(6,48,28,31)(7,47,29,45)(8,46,30,44)(9,60,16,43)(10,59,17,42)(11,58,18,41)(12,57,19,40)(13,56,20,39)(14,55,21,38)(15,54,22,37)>;`

`G:=Group( (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30)(31,37,43,34,40)(32,38,44,35,41)(33,39,45,36,42)(46,55,49,58,52)(47,56,50,59,53)(48,57,51,60,54), (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)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,46)(39,47)(40,48)(41,49)(42,50)(43,51)(44,52)(45,53), (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), (1,53,23,36)(2,52,24,35)(3,51,25,34)(4,50,26,33)(5,49,27,32)(6,48,28,31)(7,47,29,45)(8,46,30,44)(9,60,16,43)(10,59,17,42)(11,58,18,41)(12,57,19,40)(13,56,20,39)(14,55,21,38)(15,54,22,37) );`

`G=PermutationGroup([[(1,13,10,7,4),(2,14,11,8,5),(3,15,12,9,6),(16,19,22,25,28),(17,20,23,26,29),(18,21,24,27,30),(31,37,43,34,40),(32,38,44,35,41),(33,39,45,36,42),(46,55,49,58,52),(47,56,50,59,53),(48,57,51,60,54)], [(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),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,46),(39,47),(40,48),(41,49),(42,50),(43,51),(44,52),(45,53)], [(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)], [(1,53,23,36),(2,52,24,35),(3,51,25,34),(4,50,26,33),(5,49,27,32),(6,48,28,31),(7,47,29,45),(8,46,30,44),(9,60,16,43),(10,59,17,42),(11,58,18,41),(12,57,19,40),(13,56,20,39),(14,55,21,38),(15,54,22,37)]])`

33 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 5C ··· 5G 6 10A 10B 15A 15B 15C 15D 15E ··· 15N 30A 30B 30C 30D order 1 2 3 4 4 5 5 5 ··· 5 6 10 10 15 15 15 15 15 ··· 15 30 30 30 30 size 1 5 2 75 75 2 2 4 ··· 4 10 10 10 2 2 2 2 4 ··· 4 10 10 10 10

33 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + - - + - + image C1 C2 C4 S3 D5 Dic3 Dic5 D15 Dic15 F5 C3⋊F5 D5.D5 D5.D15 kernel D5.D15 D5×C15 C5×C15 C5×D5 C3×D5 C52 C15 D5 C5 C15 C5 C3 C1 # reps 1 1 2 1 2 1 2 4 4 1 2 4 8

Matrix representation of D5.D15 in GL4(𝔽61) generated by

 9 0 0 0 0 34 0 0 0 0 20 0 0 0 0 58
,
 0 34 0 0 9 0 0 0 0 0 0 58 0 0 20 0
,
 42 0 0 0 0 42 0 0 0 0 16 0 0 0 0 16
,
 0 0 16 0 0 0 0 16 0 42 0 0 42 0 0 0
`G:=sub<GL(4,GF(61))| [9,0,0,0,0,34,0,0,0,0,20,0,0,0,0,58],[0,9,0,0,34,0,0,0,0,0,0,20,0,0,58,0],[42,0,0,0,0,42,0,0,0,0,16,0,0,0,0,16],[0,0,0,42,0,0,42,0,16,0,0,0,0,16,0,0] >;`

D5.D15 in GAP, Magma, Sage, TeX

`D_5.D_{15}`
`% in TeX`

`G:=Group("D5.D15");`
`// GroupNames label`

`G:=SmallGroup(300,33);`
`// by ID`

`G=gap.SmallGroup(300,33);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,963,3004,3009]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^2=c^15=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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