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## G = C10×C42⋊C3order 480 = 25·3·5

### Direct product of C10 and C42⋊C3

Aliases: C10×C42⋊C3, C423C30, (C2×C42)⋊C15, (C4×C20)⋊6C6, C23.4(C5×A4), (C22×C10).4A4, C22.1(C10×A4), (C2×C4×C20)⋊C3, (C2×C10).5(C2×A4), SmallGroup(480,654)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C10×C42⋊C3
 Chief series C1 — C22 — C42 — C4×C20 — C5×C42⋊C3 — C10×C42⋊C3
 Lower central C42 — C10×C42⋊C3
 Upper central C1 — C10

Generators and relations for C10×C42⋊C3
G = < a,b,c,d | a10=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

Subgroups: 192 in 56 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C10, C10, A4, C15, C42, C42, C22×C4, C20, C2×C10, C2×C10, C2×A4, C30, C2×C42, C2×C20, C22×C10, C42⋊C3, C5×A4, C4×C20, C4×C20, C22×C20, C2×C42⋊C3, C10×A4, C2×C4×C20, C5×C42⋊C3, C10×C42⋊C3
Quotients: C1, C2, C3, C5, C6, C10, A4, C15, C2×A4, C30, C42⋊C3, C5×A4, C2×C42⋊C3, C10×A4, C5×C42⋊C3, C10×C42⋊C3

Smallest permutation representation of C10×C42⋊C3
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 6)(2 7)(3 8)(4 9)(5 10)(11 55 16 60)(12 56 17 51)(13 57 18 52)(14 58 19 53)(15 59 20 54)(21 26)(22 27)(23 28)(24 29)(25 30)(31 42 36 47)(32 43 37 48)(33 44 38 49)(34 45 39 50)(35 46 40 41)
(1 30 6 25)(2 21 7 26)(3 22 8 27)(4 23 9 28)(5 24 10 29)(31 47 36 42)(32 48 37 43)(33 49 38 44)(34 50 39 45)(35 41 40 46)
(1 59 39)(2 60 40)(3 51 31)(4 52 32)(5 53 33)(6 54 34)(7 55 35)(8 56 36)(9 57 37)(10 58 38)(11 41 21)(12 42 22)(13 43 23)(14 44 24)(15 45 25)(16 46 26)(17 47 27)(18 48 28)(19 49 29)(20 50 30)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 A4 C2×A4 C42⋊C3 C5×A4 C2×C42⋊C3 C10×A4 C5×C42⋊C3 C10×C42⋊C3 kernel C10×C42⋊C3 C5×C42⋊C3 C2×C4×C20 C2×C42⋊C3 C4×C20 C42⋊C3 C2×C42 C42 C22×C10 C2×C10 C10 C23 C5 C22 C2 C1 # reps 1 1 2 4 2 4 8 8 1 1 4 4 4 4 16 16

Matrix representation of C10×C42⋊C3 in GL4(𝔽61) generated by

 27 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 60 0 0 0 52 50 0 0 4 0 50
,
 1 0 0 0 0 50 0 0 0 32 11 0 0 44 0 1
,
 47 0 0 0 0 47 59 0 0 0 14 1 0 0 48 0
G:=sub<GL(4,GF(61))| [27,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,52,4,0,0,50,0,0,0,0,50],[1,0,0,0,0,50,32,44,0,0,11,0,0,0,0,1],[47,0,0,0,0,47,0,0,0,59,14,48,0,0,1,0] >;

C10×C42⋊C3 in GAP, Magma, Sage, TeX

C_{10}\times C_4^2\rtimes C_3
% in TeX

G:=Group("C10xC4^2:C3");
// GroupNames label

G:=SmallGroup(480,654);
// by ID

G=gap.SmallGroup(480,654);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,850,360,5786,102,5052,8833]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations

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