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## G = C10×C22⋊A4order 480 = 25·3·5

### Direct product of C10 and C22⋊A4

Aliases: C10×C22⋊A4, C254C15, C246C30, C22⋊(C10×A4), C233(C5×A4), (C24×C10)⋊2C3, (C22×C10)⋊3A4, (C23×C10)⋊8C6, (C2×C10)⋊2(C2×A4), SmallGroup(480,1209)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C10×C22⋊A4
G = < a,b,c,d,e,f | a10=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 896 in 296 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C23, C23, C10, C10, A4, C15, C24, C24, C2×C10, C2×C10, C2×A4, C30, C25, C22×C10, C22×C10, C22⋊A4, C5×A4, C23×C10, C23×C10, C2×C22⋊A4, C10×A4, C24×C10, C5×C22⋊A4, C10×C22⋊A4
Quotients: C1, C2, C3, C5, C6, C10, A4, C15, C2×A4, C30, C22⋊A4, C5×A4, C2×C22⋊A4, C10×A4, C5×C22⋊A4, C10×C22⋊A4

Smallest permutation representation of C10×C22⋊A4
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 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 51)(18 52)(19 53)(20 54)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(31 47)(32 48)(33 49)(34 50)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 41)
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 60)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)
(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), (11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,41), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,60)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59), (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), (11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(31,47)(32,48)(33,49)(34,50)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(31,42)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,41), (1,30)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,60)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59), (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)], [(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,51),(18,52),(19,53),(20,54),(31,47),(32,48),(33,49),(34,50),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,21),(8,22),(9,23),(10,24),(31,47),(32,48),(33,49),(34,50),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,30),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(31,42),(32,43),(33,44),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,41)], [(1,30),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,60),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59)], [(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 ··· 2K 3A 3B 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 10E ··· 10AR 15A ··· 15H 30A ··· 30H order 1 2 2 ··· 2 3 3 5 5 5 5 6 6 10 10 10 10 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 3 ··· 3 16 16 1 1 1 1 16 16 1 1 1 1 3 ··· 3 16 ··· 16 16 ··· 16

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C5 C6 C10 C15 C30 A4 C2×A4 C5×A4 C10×A4 kernel C10×C22⋊A4 C5×C22⋊A4 C24×C10 C2×C22⋊A4 C23×C10 C22⋊A4 C25 C24 C22×C10 C2×C10 C23 C22 # reps 1 1 2 4 2 4 8 8 5 5 20 20

Matrix representation of C10×C22⋊A4 in GL6(𝔽31)

 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 5 30 0 0 0 0 25 0 30 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 30 0 30
,
 30 0 0 0 0 0 0 30 0 0 0 0 6 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 1 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 30 0 0 0 0 1 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 30 0 0 0 0 0 0 1 0 0 0 0 0 30 30
,
 5 29 0 0 0 0 0 26 1 0 0 0 0 6 0 0 0 0 0 0 0 0 1 0 0 0 0 30 30 29 0 0 0 0 0 1

G:=sub<GL(6,GF(31))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,5,25,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,30,0,0,0,0,30,0,0,0,0,0,0,30],[30,0,6,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,1,0,0,0,0,30,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,1,0,0,0,0,30,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,1,30,0,0,0,0,0,30],[5,0,0,0,0,0,29,26,6,0,0,0,0,1,0,0,0,0,0,0,0,0,30,0,0,0,0,1,30,0,0,0,0,0,29,1] >;

C10×C22⋊A4 in GAP, Magma, Sage, TeX

C_{10}\times C_2^2\rtimes A_4
% in TeX

G:=Group("C10xC2^2:A4");
// GroupNames label

G:=SmallGroup(480,1209);
// by ID

G=gap.SmallGroup(480,1209);
# by ID

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

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

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