C2xC6xF5 - GroupNames

G = C₂xC₆xF₅ order 240 = 2⁴·3·5

Direct product of C₂xC₆ and F₅

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

Aliases: C₂xC₆xF₅, D₁₀:₃C₁₂, C₁₀:(C₂xC₁₂), D₅:(C₂xC₁₂), C₅:(C₂²xC₁₂), C₃₀:₃(C₂xC₄), (C₂xC₃₀):₄C₄, (C₆xD₅):₇C₄, (C₂xC₁₀):₄C₁₂, C₁₅:₄(C₂²xC₄), D₅.(C₂²xC₆), D₁₀.₇(C₂xC₆), (C₃xD₅).₃C₂³, (C₂²xD₅).₄C₆, (C₆xD₅).₂₆C₂², (D₅xC₂xC₆).₇C₂, (C₃xD₅):₅(C₂xC₄), SmallGroup(240,200)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅ — C₂xC₆xF₅

Chief series

C₁ — C₅ — D₅ — C₃xD₅ — C₃xF₅ — C₆xF₅ — C₂xC₆xF₅

Lower central

C₅ — C₂xC₆xF₅

Upper central

C₁ — C₂xC₆

Generators and relations for C₂xC₆xF₅
G = < a,b,c,d | a²=b⁶=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Subgroups: 284 in 108 conjugacy classes, 64 normal (14 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₆, C₂xC₄, C₂³, D₅, D₅, C₁₀, C₁₂, C₂xC₆, C₂xC₆, C₁₅, C₂²xC₄, F₅, D₁₀, C₂xC₁₀, C₂xC₁₂, C₂²xC₆, C₃xD₅, C₃xD₅, C₃₀, C₂xF₅, C₂²xD₅, C₂²xC₁₂, C₃xF₅, C₆xD₅, C₂xC₃₀, C₂²xF₅, C₆xF₅, D₅xC₂xC₆, C₂xC₆xF₅
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂xC₄, C₂³, C₁₂, C₂xC₆, C₂²xC₄, F₅, C₂xC₁₂, C₂²xC₆, C₂xF₅, C₂²xC₁₂, C₃xF₅, C₂²xF₅, C₆xF₅, C₂xC₆xF₅

Smallest permutation representation of C₂xC₆xF₅
►On 60 points

Generators in S₆₀

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

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

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

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

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

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

C₂xC₆xF₅ is a maximal subgroup of D₁₀.₂₀D₁₂

60 conjugacy classes

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

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	4	4	4	4
type	+	+	+								+	+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	F₅	C₂xF₅	C₃xF₅	C₆xF₅
kernel	C₂xC₆xF₅	C₆xF₅	D₅xC₂xC₆	C₂²xF₅	C₆xD₅	C₂xC₃₀	C₂xF₅	C₂²xD₅	D₁₀	C₂xC₁₀	C₂xC₆	C₆	C₂²	C₂
# reps	1	6	1	2	6	2	12	2	12	4	1	3	2	6

Matrix representation of C₂xC₆xF₅ ►in GL₆(F₆₁)

1	0	0	0	0	0
0	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

13	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

50	0	0	0	0	0
0	50	0	0	0	0
0	0	60	0	0	0
0	0	0	0	0	60
0	0	0	60	0	0
0	0	1	1	1	1

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[13,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[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],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,1,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,60,0,1] >;

C₂xC₆xF₅ in GAP, Magma, Sage, TeX

C_2\times C_6\times F_5

% in TeX

G:=Group("C2xC6xF5");

// GroupNames label

G:=SmallGroup(240,200);

// by ID

G=gap.SmallGroup(240,200);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-5,144,3461,317]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^5=d^4=1,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^3>;

// generators/relations

G = C2xC6xF5 order 240 = 24·3·5

Direct product of C2xC6 and F5

G = C₂xC₆xF₅ order 240 = 2⁴·3·5

Direct product of C₂xC₆ and F₅