Dic5xC2xC10 - GroupNames

G = Dic₅xC₂xC₁₀ order 400 = 2⁴·5²

Direct product of C₂xC₁₀ and Dic₅

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

Aliases: Dic₅xC₂xC₁₀, C₁₀²:₁₂C₄, C₁₀².₃₀C₂², (C₂xC₁₀):₅C₂₀, C₁₀:₃(C₂xC₂₀), C₅:₃(C₂²xC₂₀), C₂³.₂(C₅xD₅), (C₂xC₁₀).₄₈D₁₀, (C₂xC₁₀²).₄C₂, C₅²:₁₂(C₂²xC₄), (C₂²xC₁₀).₈D₅, (C₅xC₁₀).₂₇C₂³, C₁₀.₉(C₂²xC₁₀), (C₂²xC₁₀).₅C₁₀, C₁₀.₄₈(C₂²xD₅), C₂².₁₁(D₅xC₁₀), C₂.₂(D₅xC₂xC₁₀), (C₅xC₁₀):₁₁(C₂xC₄), (C₂xC₁₀).₁₄(C₂xC₁₀), SmallGroup(400,189)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — Dic₅xC₂xC₁₀

Chief series

C₁ — C₅ — C₁₀ — C₅xC₁₀ — C₅xDic₅ — C₁₀xDic₅ — Dic₅xC₂xC₁₀

Lower central

C₅ — Dic₅xC₂xC₁₀

Upper central

C₁ — C₂²xC₁₀

Generators and relations for Dic₅xC₂xC₁₀
G = < a,b,c,d | a²=b¹⁰=c¹⁰=1, d²=c⁵, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 260 in 140 conjugacy classes, 86 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₂², C₅, C₅, C₂xC₄, C₂³, C₁₀, C₁₀, C₁₀, C₂²xC₄, Dic₅, C₂₀, C₂xC₁₀, C₂xC₁₀, C₅², C₂xDic₅, C₂xC₂₀, C₂²xC₁₀, C₂²xC₁₀, C₅xC₁₀, C₅xC₁₀, C₂²xDic₅, C₂²xC₂₀, C₅xDic₅, C₁₀², C₁₀xDic₅, C₂xC₁₀², Dic₅xC₂xC₁₀
Quotients: C₁, C₂, C₄, C₂², C₅, C₂xC₄, C₂³, D₅, C₁₀, C₂²xC₄, Dic₅, C₂₀, D₁₀, C₂xC₁₀, C₂xDic₅, C₂xC₂₀, C₂²xD₅, C₂²xC₁₀, C₅xD₅, C₂²xDic₅, C₂²xC₂₀, C₅xDic₅, D₅xC₁₀, C₁₀xDic₅, D₅xC₂xC₁₀, Dic₅xC₂xC₁₀

Smallest permutation representation of Dic₅xC₂xC₁₀
►On 80 points

Generators in S₈₀

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

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

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

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

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

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

160 conjugacy classes

class	1	2A	···	2G	4A	···	4H	5A	5B	5C	5D	5E	···	5N	10A	···	10AB	10AC	···	10CT	20A	···	20AF
order	1	2	···	2	4	···	4	5	5	5	5	5	···	5	10	···	10	10	···	10	20	···	20
size	1	1	···	1	5	···	5	1	1	1	1	2	···	2	1	···	1	2	···	2	5	···	5

160 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+						+	-	+
image	C₁	C₂	C₂	C₄	C₅	C₁₀	C₁₀	C₂₀	D₅	Dic₅	D₁₀	C₅xD₅	C₅xDic₅	D₅xC₁₀
kernel	Dic₅xC₂xC₁₀	C₁₀xDic₅	C₂xC₁₀²	C₁₀²	C₂²xDic₅	C₂xDic₅	C₂²xC₁₀	C₂xC₁₀	C₂²xC₁₀	C₂xC₁₀	C₂xC₁₀	C₂³	C₂²	C₂²
# reps	1	6	1	8	4	24	4	32	2	8	6	8	32	24

Matrix representation of Dic₅xC₂xC₁₀ ►in GL₄(F₄₁) generated by

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

31	0	0	0
0	31	0	0
0	0	37	0
0	0	0	37

40	0	0	0
0	1	0	0
0	0	10	0
0	0	38	37

9	0	0	0
0	40	0	0
0	0	1	32
0	0	0	40

G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[31,0,0,0,0,31,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,1,0,0,0,0,10,38,0,0,0,37],[9,0,0,0,0,40,0,0,0,0,1,0,0,0,32,40] >;

Dic₅xC₂xC₁₀ in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_2\times C_{10}

% in TeX

G:=Group("Dic5xC2xC10");

// GroupNames label

G:=SmallGroup(400,189);

// by ID

G=gap.SmallGroup(400,189);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,11525]);

// Polycyclic

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

// generators/relations

G = Dic5xC2xC10 order 400 = 24·52

Direct product of C2xC10 and Dic5

G = Dic₅xC₂xC₁₀ order 400 = 2⁴·5²

Direct product of C₂xC₁₀ and Dic₅