C3xC2^2:F5 - GroupNames

G = C₃×C₂²⋊F₅ order 240 = 2⁴·3·5

Direct product of C₃ and C₂²⋊F₅

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

Aliases: C₃×C₂²⋊F₅, D₁₀⋊₂C₁₂, (C₂×F₅)⋊C₆, (C₂×C₆)⋊₁F₅, (C₂×C₃₀)⋊₃C₄, (C₆×D₅)⋊₆C₄, (C₆×F₅)⋊₃C₂, (C₂×C₁₀)⋊₃C₁₂, C₂.₇(C₆×F₅), (C₃×D₅).₆D₄, D₅.₂(C₃×D₄), C₆.₂₁(C₂×F₅), C₂²⋊₂(C₃×F₅), C₁₅⋊₃(C₂²⋊C₄), C₃₀.₂₁(C₂×C₄), C₁₀.₇(C₂×C₁₂), D₁₀.₆(C₂×C₆), (C₂²×D₅).₃C₆, (C₆×D₅).₂₅C₂², C₅⋊(C₃×C₂²⋊C₄), (D₅×C₂×C₆).₆C₂, SmallGroup(240,117)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — C₃×C₂²⋊F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₆×D₅ — C₆×F₅ — C₃×C₂²⋊F₅

Lower central

C₅ — C₁₀ — C₃×C₂²⋊F₅

Upper central

C₁ — C₆ — C₂×C₆

Generators and relations for C₃×C₂²⋊F₅
G = < a,b,c,d,e | a³=b²=c²=d⁵=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d³ >

Subgroups: 236 in 68 conjugacy classes, 28 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₆, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₁₂, C₂×C₆, C₂×C₆, C₁₅, C₂²⋊C₄, F₅, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₂, C₂²×C₆, C₃×D₅, C₃×D₅, C₃₀, C₃₀, C₂×F₅, C₂²×D₅, C₃×C₂²⋊C₄, C₃×F₅, C₆×D₅, C₆×D₅, C₂×C₃₀, C₂²⋊F₅, C₆×F₅, D₅×C₂×C₆, C₃×C₂²⋊F₅
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₁₂, C₂×C₆, C₂²⋊C₄, F₅, C₂×C₁₂, C₃×D₄, C₂×F₅, C₃×C₂²⋊C₄, C₃×F₅, C₂²⋊F₅, C₆×F₅, C₃×C₂²⋊F₅

Smallest permutation representation of C₃×C₂²⋊F₅
►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 34)(2 35)(3 31)(4 32)(5 33)(6 36)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)

(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(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)

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

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

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

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

C₃×C₂²⋊F₅ is a maximal subgroup of D₁₀.D₁₂ D₁₀.₄D₁₂ C₂²⋊F₅.S₃ C₃⋊D₄⋊F₅ C₃×D₄×F₅

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2	4	4	4	4	4	4
type	+	+	+								+		+	+		+
image	C₁	C₂	C₂	C₃	C₄	C₄	C₆	C₆	C₁₂	C₁₂	D₄	C₃×D₄	F₅	C₂×F₅	C₃×F₅	C₂²⋊F₅	C₆×F₅	C₃×C₂²⋊F₅
kernel	C₃×C₂²⋊F₅	C₆×F₅	D₅×C₂×C₆	C₂²⋊F₅	C₆×D₅	C₂×C₃₀	C₂×F₅	C₂²×D₅	D₁₀	C₂×C₁₀	C₃×D₅	D₅	C₂×C₆	C₆	C₂²	C₃	C₂	C₁
# reps	1	2	1	2	2	2	4	2	4	4	2	4	1	1	2	2	2	4

Matrix representation of C₃×C₂²⋊F₅ ►in GL₆(𝔽₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	0
0	0	0	0	0	13

1	59	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

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

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

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[60,60,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C₃×C₂²⋊F₅ in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes F_5

% in TeX

G:=Group("C3xC2^2:F5");

// GroupNames label

G:=SmallGroup(240,117);

// by ID

G=gap.SmallGroup(240,117);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,313,3461,599]);

// Polycyclic

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

// generators/relations

G = C3×C22⋊F5 order 240 = 24·3·5

Direct product of C3 and C22⋊F5

G = C₃×C₂²⋊F₅ order 240 = 2⁴·3·5

Direct product of C₃ and C₂²⋊F₅