Copied to
clipboard

G = D5×A4⋊C4order 480 = 25·3·5

Direct product of D5 and A4⋊C4

direct product, non-abelian, soluble, monomial

Aliases: D5×A4⋊C4, D10.7S4, A43(C4×D5), (D5×A4)⋊1C4, C2.3(D5×S4), C22⋊(D5×Dic3), C10.13(C2×S4), A4⋊Dic53C2, (C2×A4).5D10, C23.5(S3×D5), (C23×D5).1S3, (C22×C10).5D6, (C10×A4).5C22, (C22×D5)⋊1Dic3, C52(C2×A4⋊C4), (C5×A4⋊C4)⋊2C2, (C2×D5×A4).1C2, (C5×A4)⋊6(C2×C4), (C2×C10)⋊2(C2×Dic3), SmallGroup(480,979)

Series: Derived Chief Lower central Upper central

C1C22C5×A4 — D5×A4⋊C4
C1C22C2×C10C5×A4C10×A4C2×D5×A4 — D5×A4⋊C4
C5×A4 — D5×A4⋊C4
C1C2

Generators and relations for D5×A4⋊C4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e3=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >

Subgroups: 920 in 126 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22, C22 [×11], C5, C6 [×3], C2×C4 [×8], C23, C23 [×6], D5 [×2], D5 [×2], C10, C10 [×2], Dic3 [×2], A4, C2×C6, C15, C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×2], C20 [×2], D10, D10 [×8], C2×C10, C2×C10 [×2], C2×Dic3, C2×A4, C2×A4 [×2], C3×D5 [×2], C30, C2×C22⋊C4, C4×D5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×4], C22×C10, A4⋊C4, A4⋊C4, C22×A4, C5×Dic3, Dic15, C5×A4, C6×D5, D10⋊C4 [×2], C23.D5, C5×C22⋊C4, C2×C4×D5 [×2], C23×D5, C2×A4⋊C4, D5×Dic3, D5×A4 [×2], C10×A4, D5×C22⋊C4, C5×A4⋊C4, A4⋊Dic5, C2×D5×A4, D5×A4⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D5, Dic3 [×2], D6, D10, C2×Dic3, S4, C4×D5, A4⋊C4 [×2], C2×S4, S3×D5, C2×A4⋊C4, D5×Dic3, D5×S4, D5×A4⋊C4

Smallest permutation representation of D5×A4⋊C4
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 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)(31 34)(32 33)(36 37)(38 40)(41 44)(42 43)(46 47)(48 50)(51 54)(52 53)(56 57)(58 60)
(16 22)(17 23)(18 24)(19 25)(20 21)(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)(46 52)(47 53)(48 54)(49 55)(50 51)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 60)(7 56)(8 57)(9 58)(10 59)(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 51 41)(7 52 42)(8 53 43)(9 54 44)(10 55 45)(11 21 31)(12 22 32)(13 23 33)(14 24 34)(15 25 35)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(1 43 13 37)(2 44 14 38)(3 45 15 39)(4 41 11 40)(5 42 12 36)(6 31 60 30)(7 32 56 26)(8 33 57 27)(9 34 58 28)(10 35 59 29)(16 52 22 46)(17 53 23 47)(18 54 24 48)(19 55 25 49)(20 51 21 50)

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

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

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

40 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A10B10C10D10E10F15A15B20A···20H30A30B
order1222222234444444455666101010101010151520···203030
size113355151586666303030302284040226666161612···121616

40 irreducible representations

dim111112222223334466
type++++++-+++++-+
imageC1C2C2C2C4S3D5Dic3D6D10C4×D5S4A4⋊C4C2×S4S3×D5D5×Dic3D5×S4D5×A4⋊C4
kernelD5×A4⋊C4C5×A4⋊C4A4⋊Dic5C2×D5×A4D5×A4C23×D5A4⋊C4C22×D5C22×C10C2×A4A4D10D5C10C23C22C2C1
# reps111141221242422244

Matrix representation of D5×A4⋊C4 in GL5(𝔽61)

431000
600000
00100
00010
00001
,
6018000
01000
00100
00010
00001
,
10000
01000
006000
000600
001601
,
10000
01000
006000
00010
000160
,
10000
01000
000600
001602
00001
,
110000
011000
0060159
00010
00001

G:=sub<GL(5,GF(61))| [43,60,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,18,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,1,0,0,0,60,60,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,1,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,60,60,0,0,0,0,2,1],[11,0,0,0,0,0,11,0,0,0,0,0,60,0,0,0,0,1,1,0,0,0,59,0,1] >;

D5×A4⋊C4 in GAP, Magma, Sage, TeX

D_5\times A_4\rtimes C_4
% in TeX

G:=Group("D5xA4:C4");
// GroupNames label

G:=SmallGroup(480,979);
// by ID

G=gap.SmallGroup(480,979);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,36,234,3364,5052,1286,2953,2232]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^3=f^4=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,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽