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## G = C5×C24⋊C6order 480 = 25·3·5

### Direct product of C5 and C24⋊C6

Aliases: C5×C24⋊C6, C242C30, C22≀C2⋊C15, C231(C5×A4), C22⋊A42C10, (C22×C10)⋊1A4, (C23×C10)⋊1C6, C22.2(C10×A4), (C5×C22≀C2)⋊C3, (C5×C22⋊A4)⋊1C2, (C2×C10).6(C2×A4), SmallGroup(480,656)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C5×C24⋊C6
 Chief series C1 — C22 — C24 — C23×C10 — C5×C22⋊A4 — C5×C24⋊C6
 Lower central C24 — C5×C24⋊C6
 Upper central C1 — C5

Generators and relations for C5×C24⋊C6
G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f6=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=ec=ce, cd=dc, fcf-1=bcde, fef-1=de=ed, fdf-1=e >

Subgroups: 328 in 70 conjugacy classes, 14 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C5, C6, C2×C4, D4, C23, C23, C10, A4, C15, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×A4, C30, C22≀C2, C2×C20, C5×D4, C22×C10, C22×C10, C22⋊A4, C5×A4, C5×C22⋊C4, D4×C10, C23×C10, C24⋊C6, C10×A4, C5×C22≀C2, C5×C22⋊A4, C5×C24⋊C6
Quotients: C1, C2, C3, C5, C6, C10, A4, C15, C2×A4, C30, C5×A4, C24⋊C6, C10×A4, C5×C24⋊C6

Smallest permutation representation of C5×C24⋊C6
On 40 points
Generators in S40
(1 5 8 9 4)(2 6 7 10 3)(11 29 22 25 39)(12 30 17 26 40)(13 31 18 27 35)(14 32 19 28 36)(15 33 20 23 37)(16 34 21 24 38)
(1 35)(4 27)(5 13)(8 31)(9 18)(11 15)(20 22)(23 25)(29 33)(37 39)
(1 37)(4 23)(5 15)(8 33)(9 20)(11 13)(18 22)(25 27)(29 31)(35 39)
(1 35)(2 38)(3 24)(4 27)(5 13)(6 16)(7 34)(8 31)(9 18)(10 21)(11 15)(12 14)(17 19)(20 22)(23 25)(26 28)(29 33)(30 32)(36 40)(37 39)
(1 37)(2 40)(3 26)(4 23)(5 15)(6 12)(7 30)(8 33)(9 20)(10 17)(11 13)(14 16)(18 22)(19 21)(24 28)(25 27)(29 31)(32 34)(35 39)(36 38)
(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)

G:=sub<Sym(40)| (1,5,8,9,4)(2,6,7,10,3)(11,29,22,25,39)(12,30,17,26,40)(13,31,18,27,35)(14,32,19,28,36)(15,33,20,23,37)(16,34,21,24,38), (1,35)(4,27)(5,13)(8,31)(9,18)(11,15)(20,22)(23,25)(29,33)(37,39), (1,37)(4,23)(5,15)(8,33)(9,20)(11,13)(18,22)(25,27)(29,31)(35,39), (1,35)(2,38)(3,24)(4,27)(5,13)(6,16)(7,34)(8,31)(9,18)(10,21)(11,15)(12,14)(17,19)(20,22)(23,25)(26,28)(29,33)(30,32)(36,40)(37,39), (1,37)(2,40)(3,26)(4,23)(5,15)(6,12)(7,30)(8,33)(9,20)(10,17)(11,13)(14,16)(18,22)(19,21)(24,28)(25,27)(29,31)(32,34)(35,39)(36,38), (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)>;

G:=Group( (1,5,8,9,4)(2,6,7,10,3)(11,29,22,25,39)(12,30,17,26,40)(13,31,18,27,35)(14,32,19,28,36)(15,33,20,23,37)(16,34,21,24,38), (1,35)(4,27)(5,13)(8,31)(9,18)(11,15)(20,22)(23,25)(29,33)(37,39), (1,37)(4,23)(5,15)(8,33)(9,20)(11,13)(18,22)(25,27)(29,31)(35,39), (1,35)(2,38)(3,24)(4,27)(5,13)(6,16)(7,34)(8,31)(9,18)(10,21)(11,15)(12,14)(17,19)(20,22)(23,25)(26,28)(29,33)(30,32)(36,40)(37,39), (1,37)(2,40)(3,26)(4,23)(5,15)(6,12)(7,30)(8,33)(9,20)(10,17)(11,13)(14,16)(18,22)(19,21)(24,28)(25,27)(29,31)(32,34)(35,39)(36,38), (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) );

G=PermutationGroup([[(1,5,8,9,4),(2,6,7,10,3),(11,29,22,25,39),(12,30,17,26,40),(13,31,18,27,35),(14,32,19,28,36),(15,33,20,23,37),(16,34,21,24,38)], [(1,35),(4,27),(5,13),(8,31),(9,18),(11,15),(20,22),(23,25),(29,33),(37,39)], [(1,37),(4,23),(5,15),(8,33),(9,20),(11,13),(18,22),(25,27),(29,31),(35,39)], [(1,35),(2,38),(3,24),(4,27),(5,13),(6,16),(7,34),(8,31),(9,18),(10,21),(11,15),(12,14),(17,19),(20,22),(23,25),(26,28),(29,33),(30,32),(36,40),(37,39)], [(1,37),(2,40),(3,26),(4,23),(5,15),(6,12),(7,30),(8,33),(9,20),(10,17),(11,13),(14,16),(18,22),(19,21),(24,28),(25,27),(29,31),(32,34),(35,39),(36,38)], [(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)]])

50 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 2 2 2 3 3 4 5 5 5 5 6 6 10 10 10 10 10 10 10 10 10 ··· 10 15 ··· 15 20 20 20 20 30 ··· 30 size 1 3 4 6 6 16 16 12 1 1 1 1 16 16 3 3 3 3 4 4 4 4 6 ··· 6 16 ··· 16 12 12 12 12 16 ··· 16

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 6 6 type + + + + + image C1 C2 C3 C5 C6 C10 C15 C30 A4 C2×A4 C5×A4 C10×A4 C24⋊C6 C5×C24⋊C6 kernel C5×C24⋊C6 C5×C22⋊A4 C5×C22≀C2 C24⋊C6 C23×C10 C22⋊A4 C22≀C2 C24 C22×C10 C2×C10 C23 C22 C5 C1 # reps 1 1 2 4 2 4 8 8 1 1 4 4 2 8

Matrix representation of C5×C24⋊C6 in GL6(𝔽61)

 20 0 0 0 0 0 0 20 0 0 0 0 0 0 20 0 0 0 0 0 0 20 0 0 0 0 0 0 20 0 0 0 0 0 0 20
,
 0 1 60 0 0 0 1 0 60 0 0 0 0 0 60 0 0 0 0 0 8 1 0 0 0 0 8 0 1 0 0 0 8 0 0 1
,
 60 0 0 0 0 0 60 0 1 0 0 0 60 1 0 0 0 0 8 0 0 1 0 0 8 0 0 0 1 0 8 0 0 0 0 1
,
 0 1 60 0 0 0 1 0 60 0 0 0 0 0 60 0 0 0 53 53 0 60 60 60 0 0 8 0 0 1 0 0 8 0 1 0
,
 60 0 0 0 0 0 60 0 1 0 0 0 60 1 0 0 0 0 8 0 0 0 1 0 8 0 0 1 0 0 0 53 53 60 60 60
,
 0 0 0 60 0 1 53 53 53 59 60 60 0 0 0 60 1 0 31 31 31 8 0 0 31 32 31 8 0 0 31 31 32 8 0 0

G:=sub<GL(6,GF(61))| [20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20],[0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,8,8,8,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,60,60,8,8,8,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,53,0,0,1,0,0,53,0,0,60,60,60,0,8,8,0,0,0,60,0,0,0,0,0,60,0,1,0,0,0,60,1,0],[60,60,60,8,8,0,0,0,1,0,0,53,0,1,0,0,0,53,0,0,0,0,1,60,0,0,0,1,0,60,0,0,0,0,0,60],[0,53,0,31,31,31,0,53,0,31,32,31,0,53,0,31,31,32,60,59,60,8,8,8,0,60,1,0,0,0,1,60,0,0,0,0] >;

C5×C24⋊C6 in GAP, Magma, Sage, TeX

C_5\times C_2^4\rtimes C_6
% in TeX

G:=Group("C5xC2^4:C6");
// GroupNames label

G:=SmallGroup(480,656);
// by ID

G=gap.SmallGroup(480,656);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,4203,850,10504,1586,5052,8833]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=f^6=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^-1=e*c=c*e,c*d=d*c,f*c*f^-1=b*c*d*e,f*e*f^-1=d*e=e*d,f*d*f^-1=e>;
// generators/relations

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