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G = D4xD15order 240 = 24·3·5

Direct product of D4 and D15

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

Aliases: D4xD15, C4:1D30, C20:3D6, D60:3C2, C12:3D10, C60:1C22, C22:2D30, D30:6C22, C30.32C23, Dic15:3C22, C5:4(S3xD4), C3:4(D4xD5), (C5xD4):2S3, (C3xD4):2D5, (C2xC10):6D6, (C2xC6):3D10, C15:13(C2xD4), (D4xC15):2C2, (C4xD15):1C2, C15:7D4:1C2, (C2xC30):1C22, (C22xD15):2C2, C6.32(C22xD5), C2.6(C22xD15), C10.32(C22xS3), SmallGroup(240,179)

Series: Derived Chief Lower central Upper central

C1C30 — D4xD15
C1C5C15C30D30C22xD15 — D4xD15
C15C30 — D4xD15
C1C2D4

Generators and relations for D4xD15
 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, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2xC4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2xC6, C15, C2xD4, Dic5, C20, D10, C2xC10, C4xS3, D12, C3:D4, C3xD4, C22xS3, D15, D15, C30, C30, C4xD5, D20, C5:D4, C5xD4, C22xD5, S3xD4, Dic15, C60, D30, D30, D30, C2xC30, D4xD5, C4xD15, D60, C15:7D4, D4xC15, C22xD15, D4xD15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, C22xS3, D15, C22xD5, S3xD4, D30, D4xD5, C22xD15, D4xD15

Smallest permutation representation of D4xD15
On 60 points
Generators in S60
(1 58 16 43)(2 59 17 44)(3 60 18 45)(4 46 19 31)(5 47 20 32)(6 48 21 33)(7 49 22 34)(8 50 23 35)(9 51 24 36)(10 52 25 37)(11 53 26 38)(12 54 27 39)(13 55 28 40)(14 56 29 41)(15 57 30 42)
(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 58)(17 59)(18 60)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)
(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 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)

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

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

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

D4xD15 is a maximal subgroup of
D15:D8  Dic10:D6  Dic6:D10  D12:5D10  D8:D15  Q8:3D30  S3xD4xD5  D30.C23  D20:14D6  D12:14D10  D4:6D30  D4:8D30
D4xD15 is a maximal quotient of
C22:2Dic30  Dic15:19D4  D30:16D4  D30.28D4  D30:9D4  C23.11D30  C4:Dic30  D60:11C4  D30.29D4  C4:D60  D30:5Q8  D8:D15  D8:3D15  Q8:3D30  SD16:D15  D4.5D30  Q16:D15  D120:8C2  D30:17D4  C60:2D4  Dic15:12D4  C60:3D4

45 conjugacy classes

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

45 irreducible representations

dim1111112222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D5D6D6D10D10D15D30D30S3xD4D4xD5D4xD15
kernelD4xD15C4xD15D60C15:7D4D4xC15C22xD15C5xD4D15C3xD4C20C2xC10C12C2xC6D4C4C22C5C3C1
# reps1112121221224448124

Matrix representation of D4xD15 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] >;

D4xD15 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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x
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Z
F
o
wr
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