C2xC3^2:Dic5 - GroupNames

G = C₂×C₃²⋊Dic₅ order 360 = 2³·3²·5

Direct product of C₂ and C₃²⋊Dic₅

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

Aliases: C₂×C₃²⋊Dic₅, (C₃×C₆)⋊Dic₅, (C₃×C₃₀)⋊₆C₄, C₃⋊S₃⋊₂Dic₅, C₃⋊S₃.₅D₁₀, C₁₀⋊₂(C₃²⋊C₄), C₃²⋊₁(C₂×Dic₅), C₅⋊₄(C₂×C₃²⋊C₄), (C₅×C₃⋊S₃)⋊₈C₄, (C₃×C₁₅)⋊₁₂(C₂×C₄), (C₂×C₃⋊S₃).₂D₅, (C₁₀×C₃⋊S₃).₅C₂, (C₅×C₃⋊S₃).₈C₂², SmallGroup(360,149)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₅ — C₂×C₃²⋊Dic₅

Chief series

C₁ — C₅ — C₃×C₁₅ — C₅×C₃⋊S₃ — C₃²⋊Dic₅ — C₂×C₃²⋊Dic₅

Lower central

C₃×C₁₅ — C₂×C₃²⋊Dic₅

Upper central

C₁ — C₂

Generators and relations for C₂×C₃²⋊Dic₅
G = < a,b,c,d,e | a²=b³=c³=d¹⁰=1, e²=d⁵, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=b^-1, ebe^-1=bc^-1, dcd^-1=c^-1, ece^-1=b^-1c^-1, ede^-1=d^-1 >

C₁

C₂

₉C₂

₂C₃

C₅

₉C₂²

₄₅C₄

₂C₆

₆S₃

C₃²

C₁₀

₉C₁₀

₂C₁₅

₄₅C₂×C₄

₆D₆

C₃×C₆

C₃⋊S₃

₉Dic₅

₉C₂×C₁₀

₉Dic₅

₂C₃₀

₆C₅×S₃

C₃×C₁₅

C₂×C₃⋊S₃

₅C₃²⋊C₄

₉C₂×Dic₅

₆S₃×C₁₀

C₅×C₃⋊S₃

C₃×C₃₀

C₅×C₃⋊S₃

₅C₂×C₃²⋊C₄

C₃²⋊Dic₅

C₁₀×C₃⋊S₃

C₂×C₃²⋊Dic₅

Smallest permutation representation of C₂×C₃²⋊Dic₅
►On 60 points

Generators in S₆₀

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

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

(1 35 40)(2 31 36)(3 37 32)(4 33 38)(5 39 34)(6 43 48)(7 49 44)(8 45 50)(9 41 46)(10 47 42)

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+			+	-	+	-	+	+
image	C₁	C₂	C₂	C₄	C₄	D₅	Dic₅	D₁₀	Dic₅	C₃²⋊C₄	C₂×C₃²⋊C₄	C₃²⋊Dic₅	C₂×C₃²⋊Dic₅
kernel	C₂×C₃²⋊Dic₅	C₃²⋊Dic₅	C₁₀×C₃⋊S₃	C₅×C₃⋊S₃	C₃×C₃₀	C₂×C₃⋊S₃	C₃⋊S₃	C₃⋊S₃	C₃×C₆	C₁₀	C₅	C₂	C₁
# reps	1	2	1	2	2	2	2	2	2	2	2	8	8

Matrix representation of C₂×C₃²⋊Dic₅ ►in GL₆(𝔽₆₁)

60	0	0	0	0	0
0	60	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	0	1	0	0
0	0	60	60	0	0
0	0	0	0	60	60
0	0	0	0	1	0

1	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	0	1
0	0	0	0	60	60

0	1	0	0	0	0
60	18	0	0	0	0
0	0	58	0	0	0
0	0	3	3	0	0
0	0	0	0	20	0
0	0	0	0	41	41

0	11	0	0	0	0
11	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	60	60	0	0

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,60,1,0,0,0,0,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,60],[0,60,0,0,0,0,1,18,0,0,0,0,0,0,58,3,0,0,0,0,0,3,0,0,0,0,0,0,20,41,0,0,0,0,0,41],[0,11,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂×C₃²⋊Dic₅ in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes {\rm Dic}_5

% in TeX

G:=Group("C2xC3^2:Dic5");

// GroupNames label

G:=SmallGroup(360,149);

// by ID

G=gap.SmallGroup(360,149);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,963,111,964,376,10373]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂×C₃²⋊Dic₅ in TeX

G = C2×C32⋊Dic5 order 360 = 23·32·5

Direct product of C2 and C32⋊Dic5

G = C₂×C₃²⋊Dic₅ order 360 = 2³·3²·5

Direct product of C₂ and C₃²⋊Dic₅