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## G = C10×C3.A4order 360 = 23·32·5

### Direct product of C10 and C3.A4

Aliases: C10×C3.A4, C23⋊C45, C22⋊C90, C30.2A4, C3.(C10×A4), (C2×C6).C30, (C22×C10)⋊C9, C6.2(C5×A4), (C2×C10)⋊2C18, C15.2(C2×A4), (C2×C30).2C6, (C22×C6).C15, (C22×C30).C3, SmallGroup(360,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×C3.A4
 Chief series C1 — C22 — C2×C6 — C2×C30 — C5×C3.A4 — C10×C3.A4
 Lower central C22 — C10×C3.A4
 Upper central C1 — C30

Generators and relations for C10×C3.A4
G = < a,b,c,d,e | a10=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C10×C3.A4
On 90 points
Generators in S90
(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)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)
(1 44 34)(2 45 35)(3 46 36)(4 47 37)(5 48 38)(6 49 39)(7 50 40)(8 41 31)(9 42 32)(10 43 33)(11 81 30)(12 82 21)(13 83 22)(14 84 23)(15 85 24)(16 86 25)(17 87 26)(18 88 27)(19 89 28)(20 90 29)(51 61 78)(52 62 79)(53 63 80)(54 64 71)(55 65 72)(56 66 73)(57 67 74)(58 68 75)(59 69 76)(60 70 77)
(1 6)(2 7)(3 8)(4 9)(5 10)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(81 86)(82 87)(83 88)(84 89)(85 90)
(1 17 74 44 87 57 34 26 67)(2 18 75 45 88 58 35 27 68)(3 19 76 46 89 59 36 28 69)(4 20 77 47 90 60 37 29 70)(5 11 78 48 81 51 38 30 61)(6 12 79 49 82 52 39 21 62)(7 13 80 50 83 53 40 22 63)(8 14 71 41 84 54 31 23 64)(9 15 72 42 85 55 32 24 65)(10 16 73 43 86 56 33 25 66)

G:=sub<Sym(90)| (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90), (1,44,34)(2,45,35)(3,46,36)(4,47,37)(5,48,38)(6,49,39)(7,50,40)(8,41,31)(9,42,32)(10,43,33)(11,81,30)(12,82,21)(13,83,22)(14,84,23)(15,85,24)(16,86,25)(17,87,26)(18,88,27)(19,89,28)(20,90,29)(51,61,78)(52,62,79)(53,63,80)(54,64,71)(55,65,72)(56,66,73)(57,67,74)(58,68,75)(59,69,76)(60,70,77), (1,6)(2,7)(3,8)(4,9)(5,10)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(81,86)(82,87)(83,88)(84,89)(85,90), (1,17,74,44,87,57,34,26,67)(2,18,75,45,88,58,35,27,68)(3,19,76,46,89,59,36,28,69)(4,20,77,47,90,60,37,29,70)(5,11,78,48,81,51,38,30,61)(6,12,79,49,82,52,39,21,62)(7,13,80,50,83,53,40,22,63)(8,14,71,41,84,54,31,23,64)(9,15,72,42,85,55,32,24,65)(10,16,73,43,86,56,33,25,66)>;

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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90), (1,44,34)(2,45,35)(3,46,36)(4,47,37)(5,48,38)(6,49,39)(7,50,40)(8,41,31)(9,42,32)(10,43,33)(11,81,30)(12,82,21)(13,83,22)(14,84,23)(15,85,24)(16,86,25)(17,87,26)(18,88,27)(19,89,28)(20,90,29)(51,61,78)(52,62,79)(53,63,80)(54,64,71)(55,65,72)(56,66,73)(57,67,74)(58,68,75)(59,69,76)(60,70,77), (1,6)(2,7)(3,8)(4,9)(5,10)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (1,6)(2,7)(3,8)(4,9)(5,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(81,86)(82,87)(83,88)(84,89)(85,90), (1,17,74,44,87,57,34,26,67)(2,18,75,45,88,58,35,27,68)(3,19,76,46,89,59,36,28,69)(4,20,77,47,90,60,37,29,70)(5,11,78,48,81,51,38,30,61)(6,12,79,49,82,52,39,21,62)(7,13,80,50,83,53,40,22,63)(8,14,71,41,84,54,31,23,64)(9,15,72,42,85,55,32,24,65)(10,16,73,43,86,56,33,25,66) );

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),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90)], [(1,44,34),(2,45,35),(3,46,36),(4,47,37),(5,48,38),(6,49,39),(7,50,40),(8,41,31),(9,42,32),(10,43,33),(11,81,30),(12,82,21),(13,83,22),(14,84,23),(15,85,24),(16,86,25),(17,87,26),(18,88,27),(19,89,28),(20,90,29),(51,61,78),(52,62,79),(53,63,80),(54,64,71),(55,65,72),(56,66,73),(57,67,74),(58,68,75),(59,69,76),(60,70,77)], [(1,6),(2,7),(3,8),(4,9),(5,10),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(81,86),(82,87),(83,88),(84,89),(85,90)], [(1,17,74,44,87,57,34,26,67),(2,18,75,45,88,58,35,27,68),(3,19,76,46,89,59,36,28,69),(4,20,77,47,90,60,37,29,70),(5,11,78,48,81,51,38,30,61),(6,12,79,49,82,52,39,21,62),(7,13,80,50,83,53,40,22,63),(8,14,71,41,84,54,31,23,64),(9,15,72,42,85,55,32,24,65),(10,16,73,43,86,56,33,25,66)]])

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 9A ··· 9F 10A 10B 10C 10D 10E ··· 10L 15A ··· 15H 18A ··· 18F 30A ··· 30H 30I ··· 30X 45A ··· 45X 90A ··· 90X order 1 2 2 2 3 3 5 5 5 5 6 6 6 6 6 6 9 ··· 9 10 10 10 10 10 ··· 10 15 ··· 15 18 ··· 18 30 ··· 30 30 ··· 30 45 ··· 45 90 ··· 90 size 1 1 3 3 1 1 1 1 1 1 1 1 3 3 3 3 4 ··· 4 1 1 1 1 3 ··· 3 1 ··· 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 type + + + + image C1 C2 C3 C5 C6 C9 C10 C15 C18 C30 C45 C90 A4 C2×A4 C3.A4 C5×A4 C2×C3.A4 C10×A4 C5×C3.A4 C10×C3.A4 kernel C10×C3.A4 C5×C3.A4 C22×C30 C2×C3.A4 C2×C30 C22×C10 C3.A4 C22×C6 C2×C10 C2×C6 C23 C22 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 4 2 6 4 8 6 8 24 24 1 1 2 4 2 4 8 8

Matrix representation of C10×C3.A4 in GL4(𝔽181) generated by

 122 0 0 0 0 180 0 0 0 0 180 0 0 0 0 180
,
 1 0 0 0 0 48 0 0 0 0 48 0 0 0 0 48
,
 1 0 0 0 0 180 0 0 0 0 180 0 0 177 43 1
,
 1 0 0 0 0 180 0 0 0 0 1 0 0 0 138 180
,
 1 0 0 0 0 0 1 0 0 4 138 179 0 71 112 43
G:=sub<GL(4,GF(181))| [122,0,0,0,0,180,0,0,0,0,180,0,0,0,0,180],[1,0,0,0,0,48,0,0,0,0,48,0,0,0,0,48],[1,0,0,0,0,180,0,177,0,0,180,43,0,0,0,1],[1,0,0,0,0,180,0,0,0,0,1,138,0,0,0,180],[1,0,0,0,0,0,4,71,0,1,138,112,0,0,179,43] >;

C10×C3.A4 in GAP, Magma, Sage, TeX

C_{10}\times C_3.A_4
% in TeX

G:=Group("C10xC3.A4");
// GroupNames label

G:=SmallGroup(360,46);
// by ID

G=gap.SmallGroup(360,46);
# by ID

G:=PCGroup([6,-2,-3,-5,-3,-2,2,187,2710,4871]);
// Polycyclic

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

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