C3xD4xF5 - GroupNames

G = C₃×D₄×F₅ order 480 = 2⁵·3·5

Direct product of C₃, D₄ and F₅

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

Aliases: C₃×D₄×F₅, D₂₀⋊₃C₁₂, C₅⋊(D₄×C₁₂), C₂₀⋊(C₂×C₁₂), C₅⋊D₄⋊C₁₂, C₄⋊F₅⋊₂C₆, C₄⋊₁(C₆×F₅), C₆₀⋊₆(C₂×C₄), D₁₀⋊(C₂×C₁₂), C₁₅⋊₂₄(C₄×D₄), (C₄×F₅)⋊₃C₆, C₁₂⋊₆(C₂×F₅), (C₅×D₄)⋊₃C₁₂, (C₃×D₂₀)⋊₆C₄, (D₄×C₁₅)⋊₆C₄, Dic₅⋊(C₂×C₁₂), (C₁₂×F₅)⋊₈C₂, (D₄×D₅).₃C₆, D₅.₂(C₆×D₄), C₂²⋊F₅⋊₃C₆, C₂²⋊₂(C₆×F₅), (C₂²×F₅)⋊₂C₆, C₆.₅₂(C₂²×F₅), C₃₀.₉₀(C₂²×C₄), C₁₀.₈(C₂²×C₁₂), (C₆×D₅).₇₀C₂³, (C₆×F₅).₁₇C₂², (D₅×C₁₂).₈₇C₂², D₁₀.₁₁(C₂²×C₆), (C₂×C₆×F₅)⋊₄C₂, C₂.₉(C₂×C₆×F₅), (C₃×C₄⋊F₅)⋊₆C₂, (C₂×C₆)⋊₄(C₂×F₅), (C₃×D₄×D₅).₆C₂, (C₂×C₃₀)⋊₆(C₂×C₄), (C₃×C₅⋊D₄)⋊₂C₄, (C₂×C₁₀)⋊₂(C₂×C₁₂), (C₆×D₅)⋊₁₆(C₂×C₄), D₅.₂(C₃×C₄○D₄), (C₃×C₂²⋊F₅)⋊₇C₂, (C₂×F₅).₃(C₂×C₆), (C₃×D₅).₁₂(C₂×D₄), (C₄×D₅).₁₂(C₂×C₆), (D₅×C₂×C₆).₉₁C₂², (C₃×Dic₅)⋊₁₂(C₂×C₄), (C₃×D₅).₁₃(C₄○D₄), (C₂²×D₅).₁₈(C₂×C₆), SmallGroup(480,1054)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₃×D₄×F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₆×D₅ — C₆×F₅ — C₂×C₆×F₅ — C₃×D₄×F₅

Lower central

C₅ — C₁₀ — C₃×D₄×F₅

Upper central

C₁ — C₆ — C₃×D₄

Generators and relations for C₃×D₄×F₅
G = < a,b,c,d,e | a³=b⁴=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 664 in 188 conjugacy classes, 76 normal (40 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, C₆, C₆, C₂×C₄, D₄, D₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, C₂₀, F₅, F₅, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂²×C₆, C₃×D₅, C₃×D₅, C₃₀, C₃₀, C₄×D₄, C₄×D₅, D₂₀, C₅⋊D₄, C₅×D₄, C₂×F₅, C₂×F₅, C₂×F₅, C₂²×D₅, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂²×C₁₂, C₆×D₄, C₃×Dic₅, C₆₀, C₃×F₅, C₃×F₅, C₆×D₅, C₆×D₅, C₆×D₅, C₂×C₃₀, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₄×D₅, C₂²×F₅, D₄×C₁₂, D₅×C₁₂, C₃×D₂₀, C₃×C₅⋊D₄, D₄×C₁₅, C₆×F₅, C₆×F₅, C₆×F₅, D₅×C₂×C₆, D₄×F₅, C₁₂×F₅, C₃×C₄⋊F₅, C₃×C₂²⋊F₅, C₃×D₄×D₅, C₂×C₆×F₅, C₃×D₄×F₅
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₂³, C₁₂, C₂×C₆, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₄×D₄, C₂×F₅, C₂²×C₁₂, C₆×D₄, C₃×C₄○D₄, C₃×F₅, C₂²×F₅, D₄×C₁₂, C₆×F₅, D₄×F₅, C₂×C₆×F₅, C₃×D₄×F₅

Smallest permutation representation of C₃×D₄×F₅
►On 60 points

Generators in S₆₀

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

(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 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)

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

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

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

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

75 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	4A	4B	4C	4D	4E	4F	···	4L	5	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	6L	6M	6N	10A	10B	10C	12A	12B	12C	···	12J	12K	···	12X	15A	15B	20	30A	30B	30C	30D	30E	30F	60A	60B
order	1	2	2	2	2	2	2	2	3	3	4	4	4	4	4	4	···	4	5	6	6	6	6	6	6	6	6	6	6	6	6	6	6	10	10	10	12	12	12	···	12	12	···	12	15	15	20	30	30	30	30	30	30	60	60
size	1	1	2	2	5	5	10	10	1	1	2	5	5	5	5	10	···	10	4	1	1	2	2	2	2	5	5	5	5	10	10	10	10	4	8	8	2	2	5	···	5	10	···	10	4	4	8	4	4	8	8	8	8	8	8

75 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	4	4	8	8
type	+	+	+	+	+	+													+				+	+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₄	C₄	C₄	C₆	C₆	C₆	C₆	C₆	C₁₂	C₁₂	C₁₂	D₄	C₄○D₄	C₃×D₄	C₃×C₄○D₄	F₅	C₂×F₅	C₂×F₅	C₃×F₅	C₆×F₅	C₆×F₅	D₄×F₅	C₃×D₄×F₅
kernel	C₃×D₄×F₅	C₁₂×F₅	C₃×C₄⋊F₅	C₃×C₂²⋊F₅	C₃×D₄×D₅	C₂×C₆×F₅	D₄×F₅	C₃×D₂₀	C₃×C₅⋊D₄	D₄×C₁₅	C₄×F₅	C₄⋊F₅	C₂²⋊F₅	D₄×D₅	C₂²×F₅	D₂₀	C₅⋊D₄	C₅×D₄	C₃×F₅	C₃×D₅	F₅	D₅	C₃×D₄	C₁₂	C₂×C₆	D₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	2	1	2	2	2	4	2	2	2	4	2	4	4	8	4	2	2	4	4	1	1	2	2	2	4	1	2

Matrix representation of C₃×D₄×F₅ ►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

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

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

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

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],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;

C₃×D₄×F₅ in GAP, Magma, Sage, TeX

C_3\times D_4\times F_5

% in TeX

G:=Group("C3xD4xF5");

// GroupNames label

G:=SmallGroup(480,1054);

// by ID

G=gap.SmallGroup(480,1054);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,555,9414,818]);

// Polycyclic

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

// generators/relations

G = C3×D4×F5 order 480 = 25·3·5

Direct product of C3, D4 and F5

G = C₃×D₄×F₅ order 480 = 2⁵·3·5

Direct product of C₃, D₄ and F₅