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## G = C5×C4.6S4order 480 = 25·3·5

### Direct product of C5 and C4.6S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C5×C4.6S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C5×C4.6S4
 Lower central SL2(𝔽3) — C5×C4.6S4
 Upper central C1 — C20

Generators and relations for C5×C4.6S4
G = < a,b,c,d,e,f | a5=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 258 in 78 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, C10, C10, Dic3, C12, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C20, C20, C2×C10, SL2(𝔽3), C4×S3, C5×S3, C30, C4○D8, C40, C2×C20, C5×D4, C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C4.A4, C5×Dic3, C60, S3×C10, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4○D4, C4.6S4, C5×SL2(𝔽3), S3×C20, C5×C4○D8, C5×CSU2(𝔽3), C5×GL2(𝔽3), C5×C4.A4, C5×C4.6S4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, C4.6S4, C5×S4, C10×S4, C5×C4.6S4

Smallest permutation representation of C5×C4.6S4
On 80 points
Generators in S80
(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)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 65 12 7)(2 61 13 8)(3 62 14 9)(4 63 15 10)(5 64 11 6)(16 55 57 43)(17 51 58 44)(18 52 59 45)(19 53 60 41)(20 54 56 42)(21 68 79 39)(22 69 80 40)(23 70 76 36)(24 66 77 37)(25 67 78 38)(26 35 50 71)(27 31 46 72)(28 32 47 73)(29 33 48 74)(30 34 49 75)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 76 64 23)(7 77 65 24)(8 78 61 25)(9 79 62 21)(10 80 63 22)(16 33 57 74)(17 34 58 75)(18 35 59 71)(19 31 60 72)(20 32 56 73)(26 52 50 45)(27 53 46 41)(28 54 47 42)(29 55 48 43)(30 51 49 44)
(1 47 12 28)(2 48 13 29)(3 49 14 30)(4 50 15 26)(5 46 11 27)(6 31 64 72)(7 32 65 73)(8 33 61 74)(9 34 62 75)(10 35 63 71)(16 25 57 78)(17 21 58 79)(18 22 59 80)(19 23 60 76)(20 24 56 77)(36 53 70 41)(37 54 66 42)(38 55 67 43)(39 51 68 44)(40 52 69 45)
(16 25 74)(17 21 75)(18 22 71)(19 23 72)(20 24 73)(26 52 69)(27 53 70)(28 54 66)(29 55 67)(30 51 68)(31 60 76)(32 56 77)(33 57 78)(34 58 79)(35 59 80)(36 46 41)(37 47 42)(38 48 43)(39 49 44)(40 50 45)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 64)(7 65)(8 61)(9 62)(10 63)(16 25)(17 21)(18 22)(19 23)(20 24)(36 41)(37 42)(38 43)(39 44)(40 45)(51 68)(52 69)(53 70)(54 66)(55 67)(56 77)(57 78)(58 79)(59 80)(60 76)

G:=sub<Sym(80)| (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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,65,12,7)(2,61,13,8)(3,62,14,9)(4,63,15,10)(5,64,11,6)(16,55,57,43)(17,51,58,44)(18,52,59,45)(19,53,60,41)(20,54,56,42)(21,68,79,39)(22,69,80,40)(23,70,76,36)(24,66,77,37)(25,67,78,38)(26,35,50,71)(27,31,46,72)(28,32,47,73)(29,33,48,74)(30,34,49,75), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,76,64,23)(7,77,65,24)(8,78,61,25)(9,79,62,21)(10,80,63,22)(16,33,57,74)(17,34,58,75)(18,35,59,71)(19,31,60,72)(20,32,56,73)(26,52,50,45)(27,53,46,41)(28,54,47,42)(29,55,48,43)(30,51,49,44), (1,47,12,28)(2,48,13,29)(3,49,14,30)(4,50,15,26)(5,46,11,27)(6,31,64,72)(7,32,65,73)(8,33,61,74)(9,34,62,75)(10,35,63,71)(16,25,57,78)(17,21,58,79)(18,22,59,80)(19,23,60,76)(20,24,56,77)(36,53,70,41)(37,54,66,42)(38,55,67,43)(39,51,68,44)(40,52,69,45), (16,25,74)(17,21,75)(18,22,71)(19,23,72)(20,24,73)(26,52,69)(27,53,70)(28,54,66)(29,55,67)(30,51,68)(31,60,76)(32,56,77)(33,57,78)(34,58,79)(35,59,80)(36,46,41)(37,47,42)(38,48,43)(39,49,44)(40,50,45), (1,12)(2,13)(3,14)(4,15)(5,11)(6,64)(7,65)(8,61)(9,62)(10,63)(16,25)(17,21)(18,22)(19,23)(20,24)(36,41)(37,42)(38,43)(39,44)(40,45)(51,68)(52,69)(53,70)(54,66)(55,67)(56,77)(57,78)(58,79)(59,80)(60,76)>;

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)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,65,12,7)(2,61,13,8)(3,62,14,9)(4,63,15,10)(5,64,11,6)(16,55,57,43)(17,51,58,44)(18,52,59,45)(19,53,60,41)(20,54,56,42)(21,68,79,39)(22,69,80,40)(23,70,76,36)(24,66,77,37)(25,67,78,38)(26,35,50,71)(27,31,46,72)(28,32,47,73)(29,33,48,74)(30,34,49,75), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,76,64,23)(7,77,65,24)(8,78,61,25)(9,79,62,21)(10,80,63,22)(16,33,57,74)(17,34,58,75)(18,35,59,71)(19,31,60,72)(20,32,56,73)(26,52,50,45)(27,53,46,41)(28,54,47,42)(29,55,48,43)(30,51,49,44), (1,47,12,28)(2,48,13,29)(3,49,14,30)(4,50,15,26)(5,46,11,27)(6,31,64,72)(7,32,65,73)(8,33,61,74)(9,34,62,75)(10,35,63,71)(16,25,57,78)(17,21,58,79)(18,22,59,80)(19,23,60,76)(20,24,56,77)(36,53,70,41)(37,54,66,42)(38,55,67,43)(39,51,68,44)(40,52,69,45), (16,25,74)(17,21,75)(18,22,71)(19,23,72)(20,24,73)(26,52,69)(27,53,70)(28,54,66)(29,55,67)(30,51,68)(31,60,76)(32,56,77)(33,57,78)(34,58,79)(35,59,80)(36,46,41)(37,47,42)(38,48,43)(39,49,44)(40,50,45), (1,12)(2,13)(3,14)(4,15)(5,11)(6,64)(7,65)(8,61)(9,62)(10,63)(16,25)(17,21)(18,22)(19,23)(20,24)(36,41)(37,42)(38,43)(39,44)(40,45)(51,68)(52,69)(53,70)(54,66)(55,67)(56,77)(57,78)(58,79)(59,80)(60,76) );

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),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,65,12,7),(2,61,13,8),(3,62,14,9),(4,63,15,10),(5,64,11,6),(16,55,57,43),(17,51,58,44),(18,52,59,45),(19,53,60,41),(20,54,56,42),(21,68,79,39),(22,69,80,40),(23,70,76,36),(24,66,77,37),(25,67,78,38),(26,35,50,71),(27,31,46,72),(28,32,47,73),(29,33,48,74),(30,34,49,75)], [(1,37,12,66),(2,38,13,67),(3,39,14,68),(4,40,15,69),(5,36,11,70),(6,76,64,23),(7,77,65,24),(8,78,61,25),(9,79,62,21),(10,80,63,22),(16,33,57,74),(17,34,58,75),(18,35,59,71),(19,31,60,72),(20,32,56,73),(26,52,50,45),(27,53,46,41),(28,54,47,42),(29,55,48,43),(30,51,49,44)], [(1,47,12,28),(2,48,13,29),(3,49,14,30),(4,50,15,26),(5,46,11,27),(6,31,64,72),(7,32,65,73),(8,33,61,74),(9,34,62,75),(10,35,63,71),(16,25,57,78),(17,21,58,79),(18,22,59,80),(19,23,60,76),(20,24,56,77),(36,53,70,41),(37,54,66,42),(38,55,67,43),(39,51,68,44),(40,52,69,45)], [(16,25,74),(17,21,75),(18,22,71),(19,23,72),(20,24,73),(26,52,69),(27,53,70),(28,54,66),(29,55,67),(30,51,68),(31,60,76),(32,56,77),(33,57,78),(34,58,79),(35,59,80),(36,46,41),(37,47,42),(38,48,43),(39,49,44),(40,50,45)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,64),(7,65),(8,61),(9,62),(10,63),(16,25),(17,21),(18,22),(19,23),(20,24),(36,41),(37,42),(38,43),(39,44),(40,45),(51,68),(52,69),(53,70),(54,66),(55,67),(56,77),(57,78),(58,79),(59,80),(60,76)]])

80 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 5C 5D 6 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 15A 15B 15C 15D 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 30A 30B 30C 30D 40A ··· 40P 60A ··· 60H order 1 2 2 2 3 4 4 4 4 5 5 5 5 6 8 8 8 8 10 10 10 10 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 40 ··· 40 60 ··· 60 size 1 1 6 12 8 1 1 6 12 1 1 1 1 8 6 6 6 6 1 1 1 1 6 6 6 6 12 12 12 12 8 8 8 8 8 8 1 ··· 1 6 6 6 6 12 12 12 12 8 8 8 8 6 ··· 6 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 type + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D6 C5×S3 S3×C10 C4.6S4 C5×C4.6S4 S4 C2×S4 C5×S4 C10×S4 C4.6S4 C5×C4.6S4 kernel C5×C4.6S4 C5×CSU2(𝔽3) C5×GL2(𝔽3) C5×C4.A4 C4.6S4 CSU2(𝔽3) GL2(𝔽3) C4.A4 C5×C4○D4 C5×Q8 C4○D4 Q8 C5 C1 C20 C10 C4 C2 C5 C1 # reps 1 1 1 1 4 4 4 4 1 1 4 4 4 16 2 2 8 8 2 8

Matrix representation of C5×C4.6S4 in GL2(𝔽41) generated by

 18 0 0 18
,
 9 0 0 9
,
 0 15 30 0
,
 32 0 0 9
,
 4 7 38 36
,
 0 4 31 0
G:=sub<GL(2,GF(41))| [18,0,0,18],[9,0,0,9],[0,30,15,0],[32,0,0,9],[4,38,7,36],[0,31,4,0] >;

C5×C4.6S4 in GAP, Magma, Sage, TeX

C_5\times C_4._6S_4
% in TeX

G:=Group("C5xC4.6S4");
// GroupNames label

G:=SmallGroup(480,1020);
// by ID

G=gap.SmallGroup(480,1020);
# by ID

G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,1688,1123,4204,655,172,2525,404,285,124]);
// Polycyclic

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

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