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G = D10⋊S4order 480 = 25·3·5

1st semidirect product of D10 and S4 acting via S4/A4=C2

non-abelian, soluble, monomial

Aliases: D101S4, (C5×A4)⋊2D4, (C2×S4)⋊2D5, (C10×S4)⋊2C2, C2.14(D5×S4), A42(C5⋊D4), C52(A4⋊D4), C10.14(C2×S4), A4⋊Dic54C2, (C2×A4).6D10, C22⋊(C15⋊D4), (C23×D5)⋊1S3, C23.6(S3×D5), (C22×C10).6D6, (C10×A4).6C22, (C2×D5×A4)⋊1C2, (C2×C10)⋊1(C3⋊D4), SmallGroup(480,980)

Series: Derived Chief Lower central Upper central

C1C22C10×A4 — D10⋊S4
C1C22C2×C10C5×A4C10×A4C2×D5×A4 — D10⋊S4
C5×A4C10×A4 — D10⋊S4
C1C2

Generators and relations for D10⋊S4
 G = < a,b,c,d,e,f | a10=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a5b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 1012 in 124 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×3], C22, C22 [×12], C5, S3, C6 [×2], C2×C4 [×3], D4 [×6], C23, C23 [×5], D5 [×2], C10, C10 [×3], Dic3, A4, D6, C2×C6, C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×2], C20, D10, D10 [×6], C2×C10, C2×C10 [×5], C3⋊D4, S4, C2×A4, C2×A4, C5×S3, C3×D5, C30, C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×4], C22×C10, C22×C10, A4⋊C4, C2×S4, C22×A4, Dic15, C5×A4, C6×D5, S3×C10, D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, A4⋊D4, C15⋊D4, C5×S4, D5×A4, C10×A4, C23⋊D10, A4⋊Dic5, C10×S4, C2×D5×A4, D10⋊S4
Quotients: C1, C2 [×3], C22, S3, D4, D5, D6, D10, C3⋊D4, S4, C5⋊D4, C2×S4, S3×D5, A4⋊D4, C15⋊D4, D5×S4, D10⋊S4

Smallest permutation representation of D10⋊S4
On 60 points
Generators in S60
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 40)(8 39)(9 38)(10 37)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 50)(20 49)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 60)(30 59)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)
(1 13 23)(2 14 24)(3 15 25)(4 16 26)(5 17 27)(6 18 28)(7 19 29)(8 20 30)(9 11 21)(10 12 22)(31 41 51)(32 42 52)(33 43 53)(34 44 54)(35 45 55)(36 46 56)(37 47 57)(38 48 58)(39 49 59)(40 50 60)
(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)

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

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

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

34 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C5A5B6A6B6C10A10B10C10D10E10F10G10H10I10J15A15B20A20B20C20D30A30B
order1222222344455666101010101010101010101515202020203030
size113310123081260602284040226666121212121616121212121616

34 irreducible representations

dim111122222223344666
type++++++++++++-++
imageC1C2C2C2S3D4D5D6D10C3⋊D4C5⋊D4S4C2×S4S3×D5C15⋊D4A4⋊D4D5×S4D10⋊S4
kernelD10⋊S4A4⋊Dic5C10×S4C2×D5×A4C23×D5C5×A4C2×S4C22×C10C2×A4C2×C10A4D10C10C23C22C5C2C1
# reps111111212242222144

Matrix representation of D10⋊S4 in GL5(𝔽61)

30000
2341000
00100
00010
00001
,
754000
3354000
00100
00010
00001
,
10000
01000
006000
006001
006010
,
10000
01000
000160
001060
000060
,
10000
01000
00010
00001
00100
,
600000
591000
000600
006000
000060

G:=sub<GL(5,GF(61))| [3,23,0,0,0,0,41,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[7,33,0,0,0,54,54,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[60,59,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60] >;

D10⋊S4 in GAP, Magma, Sage, TeX

D_{10}\rtimes S_4
% in TeX

G:=Group("D10:S4");
// GroupNames label

G:=SmallGroup(480,980);
// by ID

G=gap.SmallGroup(480,980);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,234,3364,5052,1286,2953,2232]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^5*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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