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## G = C5×C4.3S4order 480 = 25·3·5

### Direct product of C5 and C4.3S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C5×C4.3S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C5×SL2(𝔽3) — C5×GL2(𝔽3) — C5×C4.3S4
 Lower central SL2(𝔽3) — C5×C4.3S4
 Upper central C1 — C10 — C20

Generators and relations for C5×C4.3S4
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, fbf=b-1, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 346 in 84 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, C10, C10, C12, D6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, C2×C10, SL2(𝔽3), D12, C5×S3, C30, C8⋊C22, C40, C2×C20, C5×D4, C5×Q8, C22×C10, GL2(𝔽3), C4.A4, C60, S3×C10, C5×M4(2), C5×D8, C5×SD16, D4×C10, C5×C4○D4, C4.3S4, C5×SL2(𝔽3), C5×D12, C5×C8⋊C22, C5×GL2(𝔽3), C5×C4.A4, C5×C4.3S4
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, S4, C5×S3, C2×S4, S3×C10, C4.3S4, C5×S4, C10×S4, C5×C4.3S4

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 type + + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 S3×C10 S4 C2×S4 C5×S4 C10×S4 C4.3S4 C5×C4.3S4 kernel C5×C4.3S4 C5×GL2(𝔽3) C5×C4.A4 C4.3S4 GL2(𝔽3) C4.A4 C5×C4○D4 C5×Q8 C4○D4 Q8 C20 C10 C4 C2 C5 C1 # reps 1 2 1 4 8 4 1 1 4 4 2 2 8 8 3 12

Matrix representation of C5×C4.3S4 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 91 0 0 0 0 91
,
 99 0 198 198 198 99 0 198 0 198 99 198 43 43 43 185
,
 0 1 0 0 240 0 0 0 1 1 1 2 0 240 240 240
,
 240 240 240 239 0 0 240 0 0 1 0 0 1 0 1 1
,
 1 0 0 0 0 0 240 0 1 1 1 2 240 240 0 240
,
 1 0 0 0 0 0 1 0 0 1 0 0 240 240 240 240
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[99,198,0,43,0,99,198,43,198,0,99,43,198,198,198,185],[0,240,1,0,1,0,1,240,0,0,1,240,0,0,2,240],[240,0,0,1,240,0,1,0,240,240,0,1,239,0,0,1],[1,0,1,240,0,0,1,240,0,240,1,0,0,0,2,240],[1,0,0,240,0,0,1,240,0,1,0,240,0,0,0,240] >;

C5×C4.3S4 in GAP, Magma, Sage, TeX

C_5\times C_4._3S_4
% in TeX

G:=Group("C5xC4.3S4");
// GroupNames label

G:=SmallGroup(480,1021);
// by ID

G=gap.SmallGroup(480,1021);
# by ID

G:=PCGroup([7,-2,-2,-5,-3,-2,2,-2,3389,1688,1123,4204,655,172,2525,404,285,124]);
// 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,f*b*f=b^-1,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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