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## G = C5×A4⋊D4order 480 = 25·3·5

### Direct product of C5 and A4⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C5×A4⋊D4
 Chief series C1 — C22 — A4 — C2×A4 — C10×A4 — C10×S4 — C5×A4⋊D4
 Lower central A4 — C2×A4 — C5×A4⋊D4
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×A4⋊D4
G = < a,b,c,d,e,f | a5=b2=c2=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=ebe-1=fbf=bc=cb, dcd-1=b, ce=ec, cf=fc, ede-1=fdf=d-1, fef=e-1 >

Subgroups: 468 in 124 conjugacy classes, 24 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, C10, C10, Dic3, A4, D6, C2×C6, C15, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C3⋊D4, S4, C2×A4, C2×A4, C5×S3, C30, C22≀C2, C2×C20, C5×D4, C22×C10, C22×C10, A4⋊C4, C2×S4, C22×A4, C5×Dic3, C5×A4, S3×C10, C2×C30, C5×C22⋊C4, D4×C10, C23×C10, A4⋊D4, C5×C3⋊D4, C5×S4, C10×A4, C10×A4, C5×C22≀C2, C5×A4⋊C4, C10×S4, A4×C2×C10, C5×A4⋊D4
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, C2×C10, C3⋊D4, S4, C5×S3, C5×D4, C2×S4, S3×C10, A4⋊D4, C5×C3⋊D4, C5×S4, C10×S4, C5×A4⋊D4

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

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 10Q 10R 10S 10T 10U 10V 10W 10X 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L order 1 2 2 2 2 2 2 3 4 4 4 5 5 5 5 6 6 6 10 10 10 10 10 10 10 10 10 ··· 10 10 10 10 10 10 10 10 10 15 15 15 15 20 ··· 20 30 ··· 30 size 1 1 2 3 3 6 12 8 12 12 12 1 1 1 1 8 8 8 1 1 1 1 2 2 2 2 3 ··· 3 6 6 6 6 12 12 12 12 8 8 8 8 12 ··· 12 8 ··· 8

70 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 type + + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 D6 C3⋊D4 C5×S3 C5×D4 S3×C10 C5×C3⋊D4 S4 C2×S4 C5×S4 C10×S4 A4⋊D4 C5×A4⋊D4 kernel C5×A4⋊D4 C5×A4⋊C4 C10×S4 A4×C2×C10 A4⋊D4 A4⋊C4 C2×S4 C22×A4 C23×C10 C5×A4 C22×C10 C2×C10 C24 A4 C23 C22 C2×C10 C10 C22 C2 C5 C1 # reps 1 1 1 1 4 4 4 4 1 1 1 2 4 4 4 8 2 2 8 8 1 4

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

 20 0 0 0 0 0 20 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 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 1 0 0 0 0 0 1 0 0 1 0 0
,
 1 1 0 0 0 59 60 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1
,
 1 1 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [20,0,0,0,0,0,20,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,60,0,0,1,0,60,0,0,0,0,60],[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],[1,59,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C5×A4⋊D4 in GAP, Magma, Sage, TeX

C_5\times A_4\rtimes D_4
% in TeX

G:=Group("C5xA4:D4");
// GroupNames label

G:=SmallGroup(480,1023);
// by ID

G=gap.SmallGroup(480,1023);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,309,2804,10085,285,5886,475]);
// Polycyclic

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