C10xC2^2:A4 - GroupNames

G = C₁₀xC₂²:A₄ order 480 = 2⁵·3·5

Direct product of C₁₀ and C₂²:A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: C₁₀xC₂²:A₄, C₂⁵:₄C₁₅, C₂⁴:₆C₃₀, C₂²:(C₁₀xA₄), C₂³:₃(C₅xA₄), (C₂⁴xC₁₀):₂C₃, (C₂²xC₁₀):₃A₄, (C₂³xC₁₀):₈C₆, (C₂xC₁₀):₂(C₂xA₄), SmallGroup(480,1209)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₁₀xC₂²:A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂³xC₁₀ — C₅xC₂²:A₄ — C₁₀xC₂²:A₄

Lower central

C₂⁴ — C₁₀xC₂²:A₄

Upper central

C₁ — C₁₀

Generators and relations for C₁₀xC₂²: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, C₂xC₁₀, C₂xC₁₀, C₂xA₄, C₃₀, C₂⁵, C₂²xC₁₀, C₂²xC₁₀, C₂²:A₄, C₅xA₄, C₂³xC₁₀, C₂³xC₁₀, C₂xC₂²:A₄, C₁₀xA₄, C₂⁴xC₁₀, C₅xC₂²:A₄, C₁₀xC₂²:A₄
Quotients: C₁, C₂, C₃, C₅, C₆, C₁₀, A₄, C₁₅, C₂xA₄, C₃₀, C₂²:A₄, C₅xA₄, C₂xC₂²:A₄, C₁₀xA₄, C₅xC₂²:A₄, C₁₀xC₂²:A₄

Smallest permutation representation of C₁₀xC₂²: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	C₂xA₄	C₅xA₄	C₁₀xA₄
kernel	C₁₀xC₂²:A₄	C₅xC₂²:A₄	C₂⁴xC₁₀	C₂xC₂²:A₄	C₂³xC₁₀	C₂²:A₄	C₂⁵	C₂⁴	C₂²xC₁₀	C₂xC₁₀	C₂³	C₂²
# reps	1	1	2	4	2	4	8	8	5	5	20	20

Matrix representation of C₁₀xC₂²: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] >;

C₁₀xC₂²: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

G = C10xC22:A4 order 480 = 25·3·5

Direct product of C10 and C22:A4

G = C₁₀xC₂²:A₄ order 480 = 2⁵·3·5

Direct product of C₁₀ and C₂²:A₄