C5xS3xD4 - GroupNames

G = C₅×S₃×D₄ order 240 = 2⁴·3·5

Direct product of C₅, S₃ and D₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₅×S₃×D₄, C₂₀⋊₆D₆, D₁₂⋊₃C₁₀, C₆₀⋊₇C₂², C₃₀.₅₂C₂³, C₁₂⋊(C₂×C₁₀), C₃⋊₂(D₄×C₁₀), C₄⋊₁(S₃×C₁₀), (C₂×C₁₀)⋊₇D₆, C₁₅⋊₁₅(C₂×D₄), (S₃×C₂₀)⋊₆C₂, (C₄×S₃)⋊₁C₁₀, D₆⋊₂(C₂×C₁₀), (D₄×C₁₅)⋊₈C₂, (C₃×D₄)⋊₂C₁₀, (C₅×D₁₂)⋊₉C₂, C₃⋊D₄⋊₁C₁₀, (C₂×C₃₀)⋊₇C₂², C₂²⋊₂(S₃×C₁₀), (C₂²×S₃)⋊₂C₁₀, Dic₃⋊₁(C₂×C₁₀), C₆.₅(C₂²×C₁₀), (S₃×C₁₀)⋊₁₀C₂², C₁₀.₄₂(C₂²×S₃), (C₅×Dic₃)⋊₈C₂², (C₂×C₆)⋊(C₂×C₁₀), (S₃×C₂×C₁₀)⋊₆C₂, C₂.₆(S₃×C₂×C₁₀), (C₅×C₃⋊D₄)⋊₅C₂, SmallGroup(240,169)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₅×S₃×D₄

Chief series

C₁ — C₃ — C₆ — C₃₀ — S₃×C₁₀ — S₃×C₂×C₁₀ — C₅×S₃×D₄

Lower central

C₃ — C₆ — C₅×S₃×D₄

Upper central

C₁ — C₁₀ — C₅×D₄

Generators and relations for C₅×S₃×D₄
G = < a,b,c,d,e | a⁵=b³=c²=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 240 in 108 conjugacy classes, 50 normal (26 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, S₃, C₆, C₆, C₂×C₄, D₄, D₄, C₂³, C₁₀, C₁₀, Dic₃, C₁₂, D₆, D₆, D₆, C₂×C₆, C₁₅, C₂×D₄, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₄×S₃, D₁₂, C₃⋊D₄, C₃×D₄, C₂²×S₃, C₅×S₃, C₅×S₃, C₃₀, C₃₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₂²×C₁₀, S₃×D₄, C₅×Dic₃, C₆₀, S₃×C₁₀, S₃×C₁₀, S₃×C₁₀, C₂×C₃₀, D₄×C₁₀, S₃×C₂₀, C₅×D₁₂, C₅×C₃⋊D₄, D₄×C₁₅, S₃×C₂×C₁₀, C₅×S₃×D₄
Quotients: C₁, C₂, C₂², C₅, S₃, D₄, C₂³, C₁₀, D₆, C₂×D₄, C₂×C₁₀, C₂²×S₃, C₅×S₃, C₅×D₄, C₂²×C₁₀, S₃×D₄, S₃×C₁₀, D₄×C₁₀, S₃×C₂×C₁₀, C₅×S₃×D₄

Smallest permutation representation of C₅×S₃×D₄
►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)

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

(6 13)(7 14)(8 15)(9 11)(10 12)(21 51)(22 52)(23 53)(24 54)(25 55)(31 37)(32 38)(33 39)(34 40)(35 36)(41 47)(42 48)(43 49)(44 50)(45 46)

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

(6 21)(7 22)(8 23)(9 24)(10 25)(11 54)(12 55)(13 51)(14 52)(15 53)(16 59)(17 60)(18 56)(19 57)(20 58)

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

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

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

C₅×S₃×D₄ is a maximal subgroup of D₂₀⋊₁₀D₆ D₁₂.₉D₁₀ D₂₀⋊₁₃D₆ D₁₂⋊₁₄D₁₀

75 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	5A	5B	5C	5D	6A	6B	6C	10A	10B	10C	10D	10E	···	10L	10M	···	10T	10U	···	10AB	12	15A	15B	15C	15D	20A	20B	20C	20D	20E	20F	20G	20H	30A	30B	30C	30D	30E	···	30L	60A	60B	60C	60D
order	1	2	2	2	2	2	2	2	3	4	4	5	5	5	5	6	6	6	10	10	10	10	10	···	10	10	···	10	10	···	10	12	15	15	15	15	20	20	20	20	20	20	20	20	30	30	30	30	30	···	30	60	60	60	60
size	1	1	2	2	3	3	6	6	2	2	6	1	1	1	1	2	4	4	1	1	1	1	2	···	2	3	···	3	6	···	6	4	2	2	2	2	2	2	2	2	6	6	6	6	2	2	2	2	4	···	4	4	4	4	4

75 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	S₃	D₄	D₆	D₆	C₅×S₃	C₅×D₄	S₃×C₁₀	S₃×C₁₀	S₃×D₄	C₅×S₃×D₄
kernel	C₅×S₃×D₄	S₃×C₂₀	C₅×D₁₂	C₅×C₃⋊D₄	D₄×C₁₅	S₃×C₂×C₁₀	S₃×D₄	C₄×S₃	D₁₂	C₃⋊D₄	C₃×D₄	C₂²×S₃	C₅×D₄	C₅×S₃	C₂₀	C₂×C₁₀	D₄	S₃	C₄	C₂²	C₅	C₁
# reps	1	1	1	2	1	2	4	4	4	8	4	8	1	2	1	2	4	8	4	8	1	4

Matrix representation of C₅×S₃×D₄ ►in GL₄(𝔽₆₁) generated by

20	0	0	0
0	20	0	0
0	0	34	0
0	0	0	34

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

0	1	0	0
1	0	0	0
0	0	1	0
0	0	0	1

60	0	0	0
0	60	0	0
0	0	1	2
0	0	60	60

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

G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,34,0,0,0,0,34],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,2,60],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C₅×S₃×D₄ in GAP, Magma, Sage, TeX

C_5\times S_3\times D_4

% in TeX

G:=Group("C5xS3xD4");

// GroupNames label

G:=SmallGroup(240,169);

// by ID

G=gap.SmallGroup(240,169);

# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,404,5765]);

// Polycyclic

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

// generators/relations

G = C5×S3×D4 order 240 = 24·3·5

Direct product of C5, S3 and D4

G = C₅×S₃×D₄ order 240 = 2⁴·3·5

Direct product of C₅, S₃ and D₄