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## G = C5×C23.3A4order 480 = 25·3·5

### Direct product of C5 and C23.3A4

Aliases: C5×C23.3A4, C10.(C42⋊C3), C2.C42⋊C15, C23.3(C5×A4), (C22×C10).3A4, (C2×C10).SL2(𝔽3), C22.(C5×SL2(𝔽3)), C2.(C5×C42⋊C3), (C5×C2.C42)⋊C3, SmallGroup(480,74)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2.C42 — C5×C23.3A4
 Chief series C1 — C2 — C23 — C2.C42 — C5×C2.C42 — C5×C23.3A4
 Lower central C2.C42 — C5×C23.3A4
 Upper central C1 — C10

Generators and relations for C5×C23.3A4
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=g3=1, e2=gbg-1=bcd, f2=gcg-1=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef-1=de=ed, df=fd, dg=gd, geg-1=bef, gfg-1=cde >

Smallest permutation representation of C5×C23.3A4
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 21)(2 22)(3 23)(4 24)(5 25)(26 31)(27 32)(28 33)(29 34)(30 35)
(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 56)(7 57)(8 58)(9 59)(10 60)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 56)(7 57)(8 58)(9 59)(10 60)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46 41 51)(37 47 42 52)(38 48 43 53)(39 49 44 54)(40 50 45 55)
(1 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 11)(7 12)(8 13)(9 14)(10 15)(16 56)(17 57)(18 58)(19 59)(20 60)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 56 36)(2 57 37)(3 58 38)(4 59 39)(5 60 40)(6 41 21)(7 42 22)(8 43 23)(9 44 24)(10 45 25)(11 46 26)(12 47 27)(13 48 28)(14 49 29)(15 50 30)(16 51 31)(17 52 32)(18 53 33)(19 54 34)(20 55 35)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E ··· 10L 15A ··· 15H 20A ··· 20P 30A ··· 30H order 1 2 2 2 3 3 4 4 4 4 5 5 5 5 6 6 10 10 10 10 10 ··· 10 15 ··· 15 20 ··· 20 30 ··· 30 size 1 1 3 3 16 16 6 6 6 6 1 1 1 1 16 16 1 1 1 1 3 ··· 3 16 ··· 16 6 ··· 6 16 ··· 16

60 irreducible representations

 dim 1 1 1 1 2 2 2 3 3 3 3 6 6 type + - + + image C1 C3 C5 C15 SL2(𝔽3) SL2(𝔽3) C5×SL2(𝔽3) A4 C42⋊C3 C5×A4 C5×C42⋊C3 C23.3A4 C5×C23.3A4 kernel C5×C23.3A4 C5×C2.C42 C23.3A4 C2.C42 C2×C10 C2×C10 C22 C22×C10 C10 C23 C2 C5 C1 # reps 1 2 4 8 1 2 12 1 4 4 16 1 4

Matrix representation of C5×C23.3A4 in GL5(𝔽61)

 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 48 0 60
,
 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 13 47 1
,
 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 48 47 0 0 0 47 13 0 0 0 0 0 50 0 0 0 0 0 60 0 0 0 0 9 50
,
 47 13 0 0 0 13 14 0 0 0 0 0 1 0 0 0 0 0 50 0 0 0 4 29 11
,
 1 0 0 0 0 47 13 0 0 0 0 0 0 13 0 0 0 14 60 35 0 0 0 0 1

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,48,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,13,0,0,0,60,47,0,0,0,0,1],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[48,47,0,0,0,47,13,0,0,0,0,0,50,0,0,0,0,0,60,9,0,0,0,0,50],[47,13,0,0,0,13,14,0,0,0,0,0,1,0,4,0,0,0,50,29,0,0,0,0,11],[1,47,0,0,0,0,13,0,0,0,0,0,0,14,0,0,0,13,60,0,0,0,0,35,1] >;

C5×C23.3A4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._3A_4
% in TeX

G:=Group("C5xC2^3.3A4");
// GroupNames label

G:=SmallGroup(480,74);
// by ID

G=gap.SmallGroup(480,74);
# by ID

G:=PCGroup([7,-3,-5,-2,2,-2,2,-2,632,268,4623,521,80,12604,17645]);
// Polycyclic

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

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