C3xD5.D5 - GroupNames

G = C₃×D₅.D₅ order 300 = 2²·3·5²

Direct product of C₃ and D₅.D₅

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

Aliases: C₃×D₅.D₅, C₁₅⋊₆F₅, C₅²⋊₄C₁₂, C₁₅⋊₂Dic₅, (C₅×C₁₅)⋊₆C₄, D₅.(C₃×D₅), C₅⋊(C₃×Dic₅), C₅⋊₃(C₃×F₅), (C₃×D₅).₂D₅, (C₅×D₅).₂C₆, (D₅×C₁₅).₃C₂, SmallGroup(300,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₃×D₅.D₅

Chief series

C₁ — C₅ — C₅² — C₅×D₅ — D₅×C₁₅ — C₃×D₅.D₅

Lower central

C₅² — C₃×D₅.D₅

Upper central

C₁ — C₃

Generators and relations for C₃×D₅.D₅
G = < a,b,c,d,e | a³=b⁵=c²=d⁵=1, e²=b^-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, ebe^-1=b², cd=dc, ece^-1=bc, ede^-1=d^-1 >

C₁

₅C₂

C₃

C₅

₄C₅

₂₅C₄

₅C₆

D₅

₅C₁₀

C₁₅

₄C₁₅

C₅²

₂₅C₁₂

₅F₅

₅Dic₅

C₃×D₅

₅C₃₀

C₅×D₅

C₅×C₁₅

₅C₃×F₅

₅C₃×Dic₅

D₅.D₅

D₅×C₁₅

C₃×D₅.D₅

Smallest permutation representation of C₃×D₅.D₅
►On 60 points

Generators in S₆₀

(1 24 14)(2 25 15)(3 21 11)(4 22 12)(5 23 13)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)

(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)

(1 10)(2 9)(3 8)(4 7)(5 6)(11 18)(12 17)(13 16)(14 20)(15 19)(21 28)(22 27)(23 26)(24 30)(25 29)(31 38)(32 37)(33 36)(34 40)(35 39)(41 48)(42 47)(43 46)(44 50)(45 49)(51 58)(52 57)(53 56)(54 60)(55 59)

(1 5 4 3 2)(6 7 8 9 10)(11 15 14 13 12)(16 17 18 19 20)(21 25 24 23 22)(26 27 28 29 30)(31 33 35 32 34)(36 39 37 40 38)(41 43 45 42 44)(46 49 47 50 48)(51 53 55 52 54)(56 59 57 60 58)

(1 39 6 31)(2 37 10 33)(3 40 9 35)(4 38 8 32)(5 36 7 34)(11 50 19 45)(12 48 18 42)(13 46 17 44)(14 49 16 41)(15 47 20 43)(21 60 29 55)(22 58 28 52)(23 56 27 54)(24 59 26 51)(25 57 30 53)

G:=sub<Sym(60)| (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,18)(12,17)(13,16)(14,20)(15,19)(21,28)(22,27)(23,26)(24,30)(25,29)(31,38)(32,37)(33,36)(34,40)(35,39)(41,48)(42,47)(43,46)(44,50)(45,49)(51,58)(52,57)(53,56)(54,60)(55,59), (1,5,4,3,2)(6,7,8,9,10)(11,15,14,13,12)(16,17,18,19,20)(21,25,24,23,22)(26,27,28,29,30)(31,33,35,32,34)(36,39,37,40,38)(41,43,45,42,44)(46,49,47,50,48)(51,53,55,52,54)(56,59,57,60,58), (1,39,6,31)(2,37,10,33)(3,40,9,35)(4,38,8,32)(5,36,7,34)(11,50,19,45)(12,48,18,42)(13,46,17,44)(14,49,16,41)(15,47,20,43)(21,60,29,55)(22,58,28,52)(23,56,27,54)(24,59,26,51)(25,57,30,53)>;

G:=Group( (1,24,14)(2,25,15)(3,21,11)(4,22,12)(5,23,13)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,18)(12,17)(13,16)(14,20)(15,19)(21,28)(22,27)(23,26)(24,30)(25,29)(31,38)(32,37)(33,36)(34,40)(35,39)(41,48)(42,47)(43,46)(44,50)(45,49)(51,58)(52,57)(53,56)(54,60)(55,59), (1,5,4,3,2)(6,7,8,9,10)(11,15,14,13,12)(16,17,18,19,20)(21,25,24,23,22)(26,27,28,29,30)(31,33,35,32,34)(36,39,37,40,38)(41,43,45,42,44)(46,49,47,50,48)(51,53,55,52,54)(56,59,57,60,58), (1,39,6,31)(2,37,10,33)(3,40,9,35)(4,38,8,32)(5,36,7,34)(11,50,19,45)(12,48,18,42)(13,46,17,44)(14,49,16,41)(15,47,20,43)(21,60,29,55)(22,58,28,52)(23,56,27,54)(24,59,26,51)(25,57,30,53) );

G=PermutationGroup([[(1,24,14),(2,25,15),(3,21,11),(4,22,12),(5,23,13),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,18),(12,17),(13,16),(14,20),(15,19),(21,28),(22,27),(23,26),(24,30),(25,29),(31,38),(32,37),(33,36),(34,40),(35,39),(41,48),(42,47),(43,46),(44,50),(45,49),(51,58),(52,57),(53,56),(54,60),(55,59)], [(1,5,4,3,2),(6,7,8,9,10),(11,15,14,13,12),(16,17,18,19,20),(21,25,24,23,22),(26,27,28,29,30),(31,33,35,32,34),(36,39,37,40,38),(41,43,45,42,44),(46,49,47,50,48),(51,53,55,52,54),(56,59,57,60,58)], [(1,39,6,31),(2,37,10,33),(3,40,9,35),(4,38,8,32),(5,36,7,34),(11,50,19,45),(12,48,18,42),(13,46,17,44),(14,49,16,41),(15,47,20,43),(21,60,29,55),(22,58,28,52),(23,56,27,54),(24,59,26,51),(25,57,30,53)]])

39 conjugacy classes

class	1	2	3A	3B	4A	4B	5A	5B	5C	···	5G	6A	6B	10A	10B	12A	12B	12C	12D	15A	15B	15C	15D	15E	···	15N	30A	30B	30C	30D
order	1	2	3	3	4	4	5	5	5	···	5	6	6	10	10	12	12	12	12	15	15	15	15	15	···	15	30	30	30	30
size	1	5	1	1	25	25	2	2	4	···	4	5	5	10	10	25	25	25	25	2	2	2	2	4	···	4	10	10	10	10

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+					+	-			+
image	C₁	C₂	C₃	C₄	C₆	C₁₂	D₅	Dic₅	C₃×D₅	C₃×Dic₅	F₅	C₃×F₅	D₅.D₅	C₃×D₅.D₅
kernel	C₃×D₅.D₅	D₅×C₁₅	D₅.D₅	C₅×C₁₅	C₅×D₅	C₅²	C₃×D₅	C₁₅	D₅	C₅	C₁₅	C₅	C₃	C₁
# reps	1	1	2	2	2	4	2	2	4	4	1	2	4	8

Matrix representation of C₃×D₅.D₅ ►in GL₆(𝔽₆₁)

47	0	0	0	0	0
0	47	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	9	0	0	0
0	0	0	34	0	0
0	0	0	0	58	0
0	0	0	0	0	20

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	34	0	0
0	0	9	0	0	0
0	0	0	0	0	20
0	0	0	0	58	0

43	60	0	0	0	0
1	0	0	0	0	0
0	0	34	0	0	0
0	0	0	34	0	0
0	0	0	0	9	0
0	0	0	0	0	9

18	1	0	0	0	0
43	43	0	0	0	0
0	0	0	0	9	0
0	0	0	0	0	9
0	0	0	34	0	0
0	0	34	0	0	0

G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,34,0,0,0,0,0,0,58,0,0,0,0,0,0,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,34,0,0,0,0,0,0,0,0,58,0,0,0,0,20,0],[43,1,0,0,0,0,60,0,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[18,43,0,0,0,0,1,43,0,0,0,0,0,0,0,0,0,34,0,0,0,0,34,0,0,0,9,0,0,0,0,0,0,9,0,0] >;

C₃×D₅.D₅ in GAP, Magma, Sage, TeX

C_3\times D_5.D_5

% in TeX

G:=Group("C3xD5.D5");

// GroupNames label

G:=SmallGroup(300,29);

// by ID

G=gap.SmallGroup(300,29);

# by ID

G:=PCGroup([5,-2,-3,-2,-5,-5,30,963,4504,1014]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₃×D₅.D₅ in TeX

G = C3×D5.D5 order 300 = 22·3·52

Direct product of C3 and D5.D5

G = C₃×D₅.D₅ order 300 = 2²·3·5²

Direct product of C₃ and D₅.D₅