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## G = D5×C42⋊C3order 480 = 25·3·5

### Direct product of D5 and C42⋊C3

Aliases: D5×C42⋊C3, (D5×C42)⋊C3, (C4×C20)⋊4C6, C423(C3×D5), C22.3(D5×A4), (C22×D5).3A4, C5⋊(C2×C42⋊C3), (C5×C42⋊C3)⋊5C2, (C2×C10).3(C2×A4), SmallGroup(480,264)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C20 — D5×C42⋊C3
 Chief series C1 — C22 — C2×C10 — C4×C20 — C5×C42⋊C3 — D5×C42⋊C3
 Lower central C4×C20 — D5×C42⋊C3
 Upper central C1

Generators and relations for D5×C42⋊C3
G = < a,b,c,d,e | a5=b2=c4=d4=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, ede-1=c-1d2 >

Subgroups: 432 in 56 conjugacy classes, 12 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, D5, D5, C10, A4, C15, C42, C42, C22×C4, Dic5, C20, D10, C2×C10, C2×A4, C3×D5, C2×C42, C4×D5, C2×Dic5, C2×C20, C22×D5, C42⋊C3, C5×A4, C4×Dic5, C4×C20, C2×C4×D5, C2×C42⋊C3, D5×A4, D5×C42, C5×C42⋊C3, D5×C42⋊C3
Quotients: C1, C2, C3, C6, D5, A4, C2×A4, C3×D5, C42⋊C3, C2×C42⋊C3, D5×A4, D5×C42⋊C3

Smallest permutation representation of D5×C42⋊C3
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 15)(12 14)(16 17)(18 20)(21 25)(22 24)(26 27)(28 30)(31 35)(32 34)(36 37)(38 40)(41 45)(42 44)(46 47)(48 50)(51 55)(52 54)(56 57)(58 60)
(1 21 11 17)(2 22 12 18)(3 23 13 19)(4 24 14 20)(5 25 15 16)(26 36 35 45)(27 37 31 41)(28 38 32 42)(29 39 33 43)(30 40 34 44)
(1 21 11 17)(2 22 12 18)(3 23 13 19)(4 24 14 20)(5 25 15 16)(6 54 60 50)(7 55 56 46)(8 51 57 47)(9 52 58 48)(10 53 59 49)(26 35)(27 31)(28 32)(29 33)(30 34)(36 45)(37 41)(38 42)(39 43)(40 44)
(1 47 27)(2 48 28)(3 49 29)(4 50 30)(5 46 26)(6 44 24)(7 45 25)(8 41 21)(9 42 22)(10 43 23)(11 51 31)(12 52 32)(13 53 33)(14 54 34)(15 55 35)(16 56 36)(17 57 37)(18 58 38)(19 59 39)(20 60 40)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 10A 10B 15A 15B 15C 15D 20A ··· 20H order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 6 10 10 15 15 15 15 20 ··· 20 size 1 3 5 15 16 16 3 3 3 3 15 15 15 15 2 2 80 80 6 6 32 32 32 32 6 ··· 6

32 irreducible representations

 dim 1 1 1 1 2 2 3 3 3 3 6 6 type + + + + + + image C1 C2 C3 C6 D5 C3×D5 A4 C2×A4 C42⋊C3 C2×C42⋊C3 D5×A4 D5×C42⋊C3 kernel D5×C42⋊C3 C5×C42⋊C3 D5×C42 C4×C20 C42⋊C3 C42 C22×D5 C2×C10 D5 C5 C22 C1 # reps 1 1 2 2 2 4 1 1 4 4 2 8

Matrix representation of D5×C42⋊C3 in GL5(𝔽61)

 0 1 0 0 0 60 43 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 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 50 0 0 0 0 50 11 0 0 0 55 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 50 0 0 0 0 56 60 0 0 0 0 0 50
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 59 0 0 0 0 60 1 0 0 0 60 0

G:=sub<GL(5,GF(61))| [0,60,0,0,0,1,43,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,50,50,55,0,0,0,11,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,50,56,0,0,0,0,60,0,0,0,0,0,50],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,59,60,60,0,0,0,1,0] >;

D5×C42⋊C3 in GAP, Magma, Sage, TeX

D_5\times C_4^2\rtimes C_3
% in TeX

G:=Group("D5xC4^2:C3");
// GroupNames label

G:=SmallGroup(480,264);
// by ID

G=gap.SmallGroup(480,264);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-5,-2,2,198,1276,3454,584,3364,5052,8833]);
// Polycyclic

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

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