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G = D4×D15order 240 = 24·3·5

Direct product of D4 and D15

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4×D15, C41D30, C203D6, D603C2, C123D10, C601C22, C222D30, D306C22, C30.32C23, Dic153C22, C54(S3×D4), C34(D4×D5), (C5×D4)⋊2S3, (C3×D4)⋊2D5, (C2×C10)⋊6D6, (C2×C6)⋊3D10, C1513(C2×D4), (D4×C15)⋊2C2, (C4×D15)⋊1C2, C157D41C2, (C2×C30)⋊1C22, (C22×D15)⋊2C2, C6.32(C22×D5), C2.6(C22×D15), C10.32(C22×S3), SmallGroup(240,179)

Series: Derived Chief Lower central Upper central

C1C30 — D4×D15
C1C5C15C30D30C22×D15 — D4×D15
C15C30 — D4×D15
C1C2D4

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)])

D4×D15 is a maximal subgroup of
D15⋊D8  Dic10⋊D6  Dic6⋊D10  D125D10  D8⋊D15  Q83D30  S3×D4×D5  D30.C23  D2014D6  D1214D10  D46D30  D48D30
D4×D15 is a maximal quotient of
C222Dic30  Dic1519D4  D3016D4  D30.28D4  D309D4  C23.11D30  C4⋊Dic30  D6011C4  D30.29D4  C4⋊D60  D305Q8  D8⋊D15  D83D15  Q83D30  SD16⋊D15  D4.5D30  Q16⋊D15  D1208C2  D3017D4  C602D4  Dic1512D4  C603D4

45 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A10B10C10D10E10F 12 15A15B15C15D20A20B30A30B30C30D30E···30L60A60B60C60D
order1222222234455666101010101010121515151520203030303030···3060606060
size112215153030223022244224444422224422224···44444

45 irreducible representations

dim1111112222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6D10D10D15D30D30S3×D4D4×D5D4×D15
kernelD4×D15C4×D15D60C157D4D4×C15C22×D15C5×D4D15C3×D4C20C2×C10C12C2×C6D4C4C22C5C3C1
# reps1112121221224448124

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

60000
06000
0001
00600
,
60000
06000
0001
0010
,
311400
473700
0010
0001
,
282500
373300
0010
0001
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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