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## G = C10×C32⋊C4order 360 = 23·32·5

### Direct product of C10 and C32⋊C4

Aliases: C10×C32⋊C4, (C3×C6)⋊C20, C3⋊S32C20, (C3×C30)⋊5C4, C321(C2×C20), (C5×C3⋊S3)⋊7C4, (C3×C15)⋊11(C2×C4), (C2×C3⋊S3).2C10, (C10×C3⋊S3).4C2, C3⋊S3.3(C2×C10), (C5×C3⋊S3).7C22, SmallGroup(360,148)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C10×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C5×C3⋊S3 — C5×C32⋊C4 — C10×C32⋊C4
 Lower central C32 — C10×C32⋊C4
 Upper central C1 — C10

Generators and relations for C10×C32⋊C4
G = < a,b,c,d | a10=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Smallest permutation representation of C10×C32⋊C4
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)
(11 57 26)(12 58 27)(13 59 28)(14 60 29)(15 51 30)(16 52 21)(17 53 22)(18 54 23)(19 55 24)(20 56 25)
(1 49 39)(2 50 40)(3 41 31)(4 42 32)(5 43 33)(6 44 34)(7 45 35)(8 46 36)(9 47 37)(10 48 38)(11 57 26)(12 58 27)(13 59 28)(14 60 29)(15 51 30)(16 52 21)(17 53 22)(18 54 23)(19 55 24)(20 56 25)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 45 26 35)(12 46 27 36)(13 47 28 37)(14 48 29 38)(15 49 30 39)(16 50 21 40)(17 41 22 31)(18 42 23 32)(19 43 24 33)(20 44 25 34)

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

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

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

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 9 9 4 4 9 9 9 9 1 1 1 1 4 4 1 1 1 1 9 ··· 9 4 ··· 4 9 ··· 9 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 C32⋊C4 C2×C32⋊C4 C5×C32⋊C4 C10×C32⋊C4 kernel C10×C32⋊C4 C5×C32⋊C4 C10×C3⋊S3 C5×C3⋊S3 C3×C30 C2×C32⋊C4 C32⋊C4 C2×C3⋊S3 C3⋊S3 C3×C6 C10 C5 C2 C1 # reps 1 2 1 2 2 4 8 4 8 8 2 2 8 8

Matrix representation of C10×C32⋊C4 in GL4(𝔽61) generated by

 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 60
,
 60 60 0 0 1 0 0 0 0 0 0 1 0 0 60 60
,
 0 0 60 0 0 0 0 60 60 0 0 0 1 1 0 0
G:=sub<GL(4,GF(61))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,60],[60,1,0,0,60,0,0,0,0,0,0,60,0,0,1,60],[0,0,60,1,0,0,0,1,60,0,0,0,0,60,0,0] >;

C10×C32⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C10xC3^2:C4");
// GroupNames label

G:=SmallGroup(360,148);
// by ID

G=gap.SmallGroup(360,148);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,3,120,8404,142,11525,455]);
// Polycyclic

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

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