F5xC2xC10 - GroupNames

G = F₅xC₂xC₁₀ order 400 = 2⁴·5²

Direct product of C₂xC₁₀ and F₅

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

Aliases: F₅xC₂xC₁₀, D₁₀:₃C₂₀, C₁₀²:₅C₄, C₁₀:(C₂xC₂₀), D₅:(C₂xC₂₀), C₅:(C₂²xC₂₀), (C₂xC₁₀):₂C₂₀, (D₅xC₁₀):₁₀C₄, D₅.(C₂²xC₁₀), C₅²:₄(C₂²xC₄), D₁₀.₇(C₂xC₁₀), (C₅xD₅).₂C₂³, (C₂²xD₅).₃C₁₀, (D₅xC₁₀).₂₃C₂², (C₅xC₁₀):₃(C₂xC₄), (C₅xD₅):₄(C₂xC₄), (D₅xC₂xC₁₀).₆C₂, SmallGroup(400,214)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — F₅xC₂xC₁₀

Chief series

C₁ — C₅ — D₅ — C₅xD₅ — C₅xF₅ — C₁₀xF₅ — F₅xC₂xC₁₀

Lower central

C₅ — F₅xC₂xC₁₀

Upper central

C₁ — C₂xC₁₀

Generators and relations for F₅xC₂xC₁₀
G = < a,b,c,d | a²=b¹⁰=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Subgroups: 304 in 113 conjugacy classes, 64 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂xC₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²xC₄, C₂₀, F₅, D₁₀, C₂xC₁₀, C₂xC₁₀, C₅², C₂xC₂₀, C₂xF₅, C₂²xD₅, C₂²xC₁₀, C₅xD₅, C₅xD₅, C₅xC₁₀, C₂²xC₂₀, C₂²xF₅, C₅xF₅, D₅xC₁₀, C₁₀², C₁₀xF₅, D₅xC₂xC₁₀, F₅xC₂xC₁₀
Quotients: C₁, C₂, C₄, C₂², C₅, C₂xC₄, C₂³, C₁₀, C₂²xC₄, C₂₀, F₅, C₂xC₁₀, C₂xC₂₀, C₂xF₅, C₂²xC₁₀, C₂²xC₂₀, C₂²xF₅, C₅xF₅, C₁₀xF₅, F₅xC₂xC₁₀

Smallest permutation representation of F₅xC₂xC₁₀
►On 80 points

Generators in S₈₀

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

(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)

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

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

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

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

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

100 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4H	5A	5B	5C	5D	5E	···	5I	10A	···	10L	10M	···	10AA	10AB	···	10AQ	20A	···	20AF
order	1	2	2	2	2	2	2	2	4	···	4	5	5	5	5	5	···	5	10	···	10	10	···	10	10	···	10	20	···	20
size	1	1	1	1	5	5	5	5	5	···	5	1	1	1	1	4	···	4	1	···	1	4	···	4	5	···	5	5	···	5

100 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	F₅	C₂xF₅	C₅xF₅	C₁₀xF₅
kernel	F₅xC₂xC₁₀	C₁₀xF₅	D₅xC₂xC₁₀	D₅xC₁₀	C₁₀²	C₂²xF₅	C₂xF₅	C₂²xD₅	D₁₀	C₂xC₁₀	C₂xC₁₀	C₁₀	C₂²	C₂
# reps	1	6	1	6	2	4	24	4	24	8	1	3	4	12

Matrix representation of F₅xC₂xC₁₀ ►in GL₅(F₄₁)

40	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

31	0	0	0	0
0	4	0	0	0
0	0	4	0	0
0	0	0	4	0
0	0	0	0	4

1	0	0	0	0
0	16	0	0	0
0	17	18	0	0
0	10	0	10	0
0	6	0	0	37

32	0	0	0	0
0	1	0	24	0
0	0	0	1	40
0	0	40	40	0
0	0	0	40	0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[31,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,16,17,10,6,0,0,18,0,0,0,0,0,10,0,0,0,0,0,37],[32,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,24,1,40,40,0,0,40,0,0] >;

F₅xC₂xC₁₀ in GAP, Magma, Sage, TeX

F_5\times C_2\times C_{10}

% in TeX

G:=Group("F5xC2xC10");

// GroupNames label

G:=SmallGroup(400,214);

// by ID

G=gap.SmallGroup(400,214);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,5765,317]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;

// generators/relations

G = F5xC2xC10 order 400 = 24·52

Direct product of C2xC10 and F5

G = F₅xC₂xC₁₀ order 400 = 2⁴·5²

Direct product of C₂xC₁₀ and F₅