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## G = D4×D15order 240 = 24·3·5

### Direct product of D4 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — D4×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — D4×D15
 Lower central C15 — C30 — D4×D15
 Upper central C1 — C2 — D4

Generators and relations for D4×D15
G = < a,b,c,d | a4=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 640 in 108 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×2], C22 [×7], C5, S3 [×4], C6, C6 [×2], C2×C4, D4, D4 [×3], C23 [×2], D5 [×4], C10, C10 [×2], Dic3, C12, D6 [×7], C2×C6 [×2], C15, C2×D4, Dic5, C20, D10 [×7], C2×C10 [×2], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], D15 [×2], D15 [×2], C30, C30 [×2], C4×D5, D20, C5⋊D4 [×2], C5×D4, C22×D5 [×2], S3×D4, Dic15, C60, D30, D30 [×2], D30 [×4], C2×C30 [×2], D4×D5, C4×D15, D60, C157D4 [×2], D4×C15, C22×D15 [×2], D4×D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C22×D5, S3×D4, D30 [×3], D4×D5, C22×D15, D4×D15

Smallest permutation representation of D4×D15
On 60 points
Generators in S60
(1 50 16 31)(2 51 17 32)(3 52 18 33)(4 53 19 34)(5 54 20 35)(6 55 21 36)(7 56 22 37)(8 57 23 38)(9 58 24 39)(10 59 25 40)(11 60 26 41)(12 46 27 42)(13 47 28 43)(14 48 29 44)(15 49 30 45)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 46)(28 47)(29 48)(30 49)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(46 53)(47 52)(48 51)(49 50)(54 60)(55 59)(56 58)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 12 15A 15B 15C 15D 20A 20B 30A 30B 30C 30D 30E ··· 30L 60A 60B 60C 60D order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 10 10 10 10 10 12 15 15 15 15 20 20 30 30 30 30 30 ··· 30 60 60 60 60 size 1 1 2 2 15 15 30 30 2 2 30 2 2 2 4 4 2 2 4 4 4 4 4 2 2 2 2 4 4 2 2 2 2 4 ··· 4 4 4 4 4

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 D15 D30 D30 S3×D4 D4×D5 D4×D15 kernel D4×D15 C4×D15 D60 C15⋊7D4 D4×C15 C22×D15 C5×D4 D15 C3×D4 C20 C2×C10 C12 C2×C6 D4 C4 C22 C5 C3 C1 # reps 1 1 1 2 1 2 1 2 2 1 2 2 4 4 4 8 1 2 4

Matrix representation of D4×D15 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 0 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 1 0
,
 31 14 0 0 47 37 0 0 0 0 1 0 0 0 0 1
,
 28 25 0 0 37 33 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0],[31,47,0,0,14,37,0,0,0,0,1,0,0,0,0,1],[28,37,0,0,25,33,0,0,0,0,1,0,0,0,0,1] >;

D4×D15 in GAP, Magma, Sage, TeX

D_4\times D_{15}
% in TeX

G:=Group("D4xD15");
// GroupNames label

G:=SmallGroup(240,179);
// by ID

G=gap.SmallGroup(240,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,116,964,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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