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## G = A4×C5⋊D4order 480 = 25·3·5

### Direct product of A4 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — A4×C5⋊D4
 Chief series C1 — C5 — C2×C10 — C22×C10 — C10×A4 — C2×D5×A4 — A4×C5⋊D4
 Lower central C2×C10 — C22×C10 — A4×C5⋊D4
 Upper central C1 — C2 — C22

Generators and relations for A4×C5⋊D4
G = < a,b,c,d,e,f | a2=b2=c3=d5=e4=f2=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 880 in 132 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, A4, C2×C6, C15, C22×C4, C2×D4, C24, C24, Dic5, Dic5, D10, D10, C2×C10, C2×C10, C3×D4, C2×A4, C2×A4, C3×D5, C30, C22×D4, C2×Dic5, C5⋊D4, C5⋊D4, C22×D5, C22×C10, C22×C10, C4×A4, C22×A4, C22×A4, C3×Dic5, C5×A4, C6×D5, C2×C30, C22×Dic5, C2×C5⋊D4, C23×D5, C23×C10, D4×A4, C3×C5⋊D4, D5×A4, C10×A4, C10×A4, C22×C5⋊D4, A4×Dic5, C2×D5×A4, A4×C2×C10, A4×C5⋊D4
Quotients: C1, C2, C3, C22, C6, D4, D5, A4, C2×C6, D10, C3×D4, C2×A4, C3×D5, C5⋊D4, C22×A4, C6×D5, D4×A4, C3×C5⋊D4, D5×A4, C2×D5×A4, A4×C5⋊D4

Smallest permutation representation of A4×C5⋊D4
On 60 points
Generators in S60
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(51 56)(52 57)(53 58)(54 59)(55 60)
(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 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(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 39 6 34)(2 38 7 33)(3 37 8 32)(4 36 9 31)(5 40 10 35)(11 49 16 44)(12 48 17 43)(13 47 18 42)(14 46 19 41)(15 50 20 45)(21 59 26 54)(22 58 27 53)(23 57 28 52)(24 56 29 51)(25 60 30 55)
(1 34)(2 33)(3 32)(4 31)(5 35)(6 39)(7 38)(8 37)(9 36)(10 40)(11 44)(12 43)(13 42)(14 41)(15 45)(16 49)(17 48)(18 47)(19 46)(20 50)(21 54)(22 53)(23 52)(24 51)(25 55)(26 59)(27 58)(28 57)(29 56)(30 60)

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

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

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

52 conjugacy classes

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

52 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 D10 C3×D4 C3×D5 C5⋊D4 C6×D5 C3×C5⋊D4 A4 C2×A4 C2×A4 C2×A4 D4×A4 D5×A4 C2×D5×A4 A4×C5⋊D4 kernel A4×C5⋊D4 A4×Dic5 C2×D5×A4 A4×C2×C10 C22×C5⋊D4 C22×Dic5 C23×D5 C23×C10 C5×A4 C22×A4 C2×A4 C2×C10 C24 A4 C23 C22 C5⋊D4 Dic5 D10 C2×C10 C5 C22 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 4 4 4 8 1 1 1 1 1 2 2 4

Matrix representation of A4×C5⋊D4 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 60 0 0 0 1 60 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 60 0 1 0 0 60 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 52 18 0 0 0 18 52 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 21 25 0 0 0 36 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 36 40 0 0 0 21 25 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[52,18,0,0,0,18,52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[21,36,0,0,0,25,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,21,0,0,0,40,25,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60] >;

A4×C5⋊D4 in GAP, Magma, Sage, TeX

A_4\times C_5\rtimes D_4
% in TeX

G:=Group("A4xC5:D4");
// GroupNames label

G:=SmallGroup(480,1045);
// by ID

G=gap.SmallGroup(480,1045);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,197,648,271,18822]);
// Polycyclic

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

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