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## G = D5×A4⋊C4order 480 = 25·3·5

### Direct product of D5 and A4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — D5×A4⋊C4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C10×A4 — C2×D5×A4 — D5×A4⋊C4
 Lower central C5×A4 — D5×A4⋊C4
 Upper central C1 — C2

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, C3, C4, C22, C22, C5, C6, C2×C4, C23, C23, D5, D5, C10, C10, Dic3, A4, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×A4, C2×A4, C3×D5, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, A4⋊C4, A4⋊C4, C22×A4, C5×Dic3, Dic15, C5×A4, C6×D5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, C2×A4⋊C4, D5×Dic3, D5×A4, C10×A4, D5×C22⋊C4, C5×A4⋊C4, A4⋊Dic5, C2×D5×A4, D5×A4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, D10, C2×Dic3, S4, C4×D5, A4⋊C4, 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 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)
(16 25)(17 21)(18 22)(19 23)(20 24)(26 35)(27 31)(28 32)(29 33)(30 34)(36 45)(37 41)(38 42)(39 43)(40 44)(46 55)(47 51)(48 52)(49 53)(50 54)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 60)(7 56)(8 57)(9 58)(10 59)(26 35)(27 31)(28 32)(29 33)(30 34)(36 45)(37 41)(38 42)(39 43)(40 44)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 54 44)(7 55 45)(8 51 41)(9 52 42)(10 53 43)(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 41 11 37)(2 42 12 38)(3 43 13 39)(4 44 14 40)(5 45 15 36)(6 34 60 30)(7 35 56 26)(8 31 57 27)(9 32 58 28)(10 33 59 29)(16 55 25 46)(17 51 21 47)(18 52 22 48)(19 53 23 49)(20 54 24 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)(11,15)(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), (16,25)(17,21)(18,22)(19,23)(20,24)(26,35)(27,31)(28,32)(29,33)(30,34)(36,45)(37,41)(38,42)(39,43)(40,44)(46,55)(47,51)(48,52)(49,53)(50,54), (1,11)(2,12)(3,13)(4,14)(5,15)(6,60)(7,56)(8,57)(9,58)(10,59)(26,35)(27,31)(28,32)(29,33)(30,34)(36,45)(37,41)(38,42)(39,43)(40,44), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,54,44)(7,55,45)(8,51,41)(9,52,42)(10,53,43)(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), (1,41,11,37)(2,42,12,38)(3,43,13,39)(4,44,14,40)(5,45,15,36)(6,34,60,30)(7,35,56,26)(8,31,57,27)(9,32,58,28)(10,33,59,29)(16,55,25,46)(17,51,21,47)(18,52,22,48)(19,53,23,49)(20,54,24,50)>;

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 15A 15B 20A ··· 20H 30A 30B order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 10 10 10 10 10 10 15 15 20 ··· 20 30 30 size 1 1 3 3 5 5 15 15 8 6 6 6 6 30 30 30 30 2 2 8 40 40 2 2 6 6 6 6 16 16 12 ··· 12 16 16

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 6 6 type + + + + + + - + + + + + - + image C1 C2 C2 C2 C4 S3 D5 Dic3 D6 D10 C4×D5 S4 A4⋊C4 C2×S4 S3×D5 D5×Dic3 D5×S4 D5×A4⋊C4 kernel D5×A4⋊C4 C5×A4⋊C4 A4⋊Dic5 C2×D5×A4 D5×A4 C23×D5 A4⋊C4 C22×D5 C22×C10 C2×A4 A4 D10 D5 C10 C23 C22 C2 C1 # reps 1 1 1 1 4 1 2 2 1 2 4 2 4 2 2 2 4 4

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

 43 1 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 18 0 0 0 0 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 0 0 0 0 60 0 0 0 1 60 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 1 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 1 60 2 0 0 0 0 1
,
 11 0 0 0 0 0 11 0 0 0 0 0 60 1 59 0 0 0 1 0 0 0 0 0 1

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

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