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## G = C5×C42⋊S3order 480 = 25·3·5

### Direct product of C5 and C42⋊S3

Aliases: C5×C42⋊S3, C42⋊(C5×S3), (C4×C20)⋊2S3, C22.(C5×S4), C42⋊C32C10, (C2×C10).1S4, (C5×C42⋊C3)⋊6C2, SmallGroup(480,254)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C3 — C5×C42⋊S3
 Chief series C1 — C22 — C42 — C42⋊C3 — C5×C42⋊C3 — C5×C42⋊S3
 Lower central C42⋊C3 — C5×C42⋊S3
 Upper central C1 — C5

Generators and relations for C5×C42⋊S3
G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=c, dcd-1=b-1c-1, ece=b, ede=d-1 >

Smallest permutation representation of C5×C42⋊S3
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 58 36 20)(2 59 37 16)(3 60 38 17)(4 56 39 18)(5 57 40 19)(6 12)(7 13)(8 14)(9 15)(10 11)(21 46 29 55)(22 47 30 51)(23 48 26 52)(24 49 27 53)(25 50 28 54)(31 44)(32 45)(33 41)(34 42)(35 43)
(1 58 36 20)(2 59 37 16)(3 60 38 17)(4 56 39 18)(5 57 40 19)(6 45 12 32)(7 41 13 33)(8 42 14 34)(9 43 15 35)(10 44 11 31)(21 29)(22 30)(23 26)(24 27)(25 28)(46 55)(47 51)(48 52)(49 53)(50 54)
(1 23 32)(2 24 33)(3 25 34)(4 21 35)(5 22 31)(6 58 48)(7 59 49)(8 60 50)(9 56 46)(10 57 47)(11 19 51)(12 20 52)(13 16 53)(14 17 54)(15 18 55)(26 45 36)(27 41 37)(28 42 38)(29 43 39)(30 44 40)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 52)(7 53)(8 54)(9 55)(10 51)(11 47)(12 48)(13 49)(14 50)(15 46)(16 59)(17 60)(18 56)(19 57)(20 58)(21 43)(22 44)(23 45)(24 41)(25 42)(26 32)(27 33)(28 34)(29 35)(30 31)

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

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

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

50 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 40A ··· 40H order 1 2 2 3 4 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 15 15 15 15 20 ··· 20 20 20 20 20 20 20 20 20 40 ··· 40 size 1 3 12 32 3 3 6 12 1 1 1 1 12 12 3 3 3 3 12 12 12 12 32 32 32 32 3 ··· 3 6 6 6 6 12 12 12 12 12 ··· 12

50 irreducible representations

 dim 1 1 1 1 2 2 3 3 3 3 6 6 type + + + + + image C1 C2 C5 C10 S3 C5×S3 S4 C42⋊S3 C5×S4 C5×C42⋊S3 C42⋊S3 C5×C42⋊S3 kernel C5×C42⋊S3 C5×C42⋊C3 C42⋊S3 C42⋊C3 C4×C20 C42 C2×C10 C5 C22 C1 C5 C1 # reps 1 1 4 4 1 4 2 4 8 16 1 4

Matrix representation of C5×C42⋊S3 in GL3(𝔽241) generated by

 98 0 0 0 98 0 0 0 98
,
 177 0 0 0 240 0 0 0 177
,
 177 0 0 0 177 0 0 0 240
,
 0 1 0 0 0 1 1 0 0
,
 240 0 0 0 0 240 0 240 0
G:=sub<GL(3,GF(241))| [98,0,0,0,98,0,0,0,98],[177,0,0,0,240,0,0,0,177],[177,0,0,0,177,0,0,0,240],[0,0,1,1,0,0,0,1,0],[240,0,0,0,0,240,0,240,0] >;

C5×C42⋊S3 in GAP, Magma, Sage, TeX

C_5\times C_4^2\rtimes S_3
% in TeX

G:=Group("C5xC4^2:S3");
// GroupNames label

G:=SmallGroup(480,254);
// by ID

G=gap.SmallGroup(480,254);
# by ID

G:=PCGroup([7,-2,-5,-3,-2,2,-2,2,422,1683,185,360,1054,1173,102,15125,1027,1784]);
// Polycyclic

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

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