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G = D5.D15order 300 = 22·3·52

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

metabelian, supersoluble, monomial, A-group

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

C1C5×C15 — D5.D15
C1C5C52C5×C15D5×C15 — D5.D15
C5×C15 — D5.D15
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 >

5C2
4C5
75C4
5C6
5C10
4C15
25Dic3
15F5
15Dic5
5C30
5C3⋊F5
5Dic15
3D5.D5

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

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

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

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

33 conjugacy classes

class 1  2  3 4A4B5A5B5C···5G 6 10A10B15A15B15C15D15E···15N30A30B30C30D
order12344555···5610101515151515···1530303030
size1527575224···410101022224···410101010

33 irreducible representations

dim1112222224444
type++++--+-+
imageC1C2C4S3D5Dic3Dic5D15Dic15F5C3⋊F5D5.D5D5.D15
kernelD5.D15D5×C15C5×C15C5×D5C3×D5C52C15D5C5C15C5C3C1
# reps1121212441248

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

9000
03400
00200
00058
,
03400
9000
00058
00200
,
42000
04200
00160
00016
,
00160
00016
04200
42000
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

Export

Subgroup lattice of D5.D15 in TeX

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