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## G = C5×C4⋊S4order 480 = 25·3·5

### Direct product of C5 and C4⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — C5×C4⋊S4
 Chief series C1 — C22 — A4 — C2×A4 — C10×A4 — C10×S4 — C5×C4⋊S4
 Lower central A4 — C2×A4 — C5×C4⋊S4
 Upper central C1 — C10 — C20

Generators and relations for C5×C4⋊S4
G = < a,b,c,d,e,f | a5=b4=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 436 in 112 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, C10, C10, C12, A4, D6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, D12, S4, C2×A4, C5×S3, C30, C4⋊D4, C2×C20, C5×D4, C22×C10, C22×C10, C4×A4, C2×S4, C60, C5×A4, S3×C10, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C4⋊S4, C5×D12, C5×S4, C10×A4, C5×C4⋊D4, A4×C20, C10×S4, C5×C4⋊S4
Quotients: C1, C2, C22, C5, S3, D4, C10, D6, C2×C10, D12, S4, C5×S3, C5×D4, C2×S4, S3×C10, C4⋊S4, C5×D12, C5×S4, C10×S4, C5×C4⋊S4

Smallest permutation representation of C5×C4⋊S4
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 17 45 6)(2 18 41 7)(3 19 42 8)(4 20 43 9)(5 16 44 10)(11 34 57 27)(12 35 58 28)(13 31 59 29)(14 32 60 30)(15 33 56 26)(21 37 51 49)(22 38 52 50)(23 39 53 46)(24 40 54 47)(25 36 55 48)
(1 45)(2 41)(3 42)(4 43)(5 44)(6 17)(7 18)(8 19)(9 20)(10 16)(21 51)(22 52)(23 53)(24 54)(25 55)(36 48)(37 49)(38 50)(39 46)(40 47)
(11 57)(12 58)(13 59)(14 60)(15 56)(21 51)(22 52)(23 53)(24 54)(25 55)(26 33)(27 34)(28 35)(29 31)(30 32)(36 48)(37 49)(38 50)(39 46)(40 47)
(1 47 33)(2 48 34)(3 49 35)(4 50 31)(5 46 32)(6 54 15)(7 55 11)(8 51 12)(9 52 13)(10 53 14)(16 23 60)(17 24 56)(18 25 57)(19 21 58)(20 22 59)(26 45 40)(27 41 36)(28 42 37)(29 43 38)(30 44 39)
(6 17)(7 18)(8 19)(9 20)(10 16)(11 25)(12 21)(13 22)(14 23)(15 24)(26 40)(27 36)(28 37)(29 38)(30 39)(31 50)(32 46)(33 47)(34 48)(35 49)(51 58)(52 59)(53 60)(54 56)(55 57)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 5C 5D 6 10A 10B 10C 10D 10E ··· 10L 10M ··· 10T 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 20E 20F 20G 20H 20I ··· 20P 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 5 5 6 10 10 10 10 10 ··· 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 20 20 20 20 20 ··· 20 30 30 30 30 60 ··· 60 size 1 1 3 3 12 12 8 2 6 12 12 1 1 1 1 8 1 1 1 1 3 ··· 3 12 ··· 12 8 8 8 8 8 8 2 2 2 2 6 6 6 6 12 ··· 12 8 8 8 8 8 ··· 8

70 irreducible representations

 dim 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 C5 C10 C10 S3 D4 D6 D12 C5×S3 C5×D4 S3×C10 C5×D12 S4 C2×S4 C5×S4 C10×S4 C4⋊S4 C5×C4⋊S4 kernel C5×C4⋊S4 A4×C20 C10×S4 C4⋊S4 C4×A4 C2×S4 C22×C20 C5×A4 C22×C10 C2×C10 C22×C4 A4 C23 C22 C20 C10 C4 C2 C5 C1 # reps 1 1 2 4 4 8 1 1 1 2 4 4 4 8 2 2 8 8 1 4

Matrix representation of C5×C4⋊S4 in GL5(𝔽61)

 20 0 0 0 0 0 20 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 60 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 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 0 1 60 0 0 1 0 60 0 0 0 0 60
,
 30 57 0 0 0 4 30 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 31 57 0 0 0 57 30 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(61))| [20,0,0,0,0,0,20,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,60,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[30,4,0,0,0,57,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[31,57,0,0,0,57,30,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C5×C4⋊S4 in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes S_4
% in TeX

G:=Group("C5xC4:S4");
// GroupNames label

G:=SmallGroup(480,1015);
// by ID

G=gap.SmallGroup(480,1015);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,309,148,2804,10085,285,5886,475]);
// Polycyclic

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

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