C4xC3:F5 - GroupNames

G = C₄×C₃⋊F₅ order 240 = 2⁴·3·5

Direct product of C₄ and C₃⋊F₅

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

Aliases: C₄×C₃⋊F₅, C₆₀⋊₂C₄, C₁₂⋊₂F₅, C₁₅⋊₂C₄², C₂₀⋊₂Dic₃, D₁₀.₁₂D₆, Dic₅⋊₂Dic₃, C₃⋊₁(C₄×F₅), C₅⋊₂(C₄×Dic₃), (C₄×D₅).₆S₃, D₅.₂(C₄×S₃), C₆.₁₀(C₂×F₅), C₃₀.₁₀(C₂×C₄), (C₃×Dic₅)⋊₄C₄, (D₅×C₁₂).₉C₂, C₁₀.₃(C₂×Dic₃), (C₆×D₅).₁₉C₂², C₂.₂(C₂×C₃⋊F₅), (C₂×C₃⋊F₅).₄C₂, (C₃×D₅).₂(C₂×C₄), SmallGroup(240,120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₅ — C₄×C₃⋊F₅

Chief series

C₁ — C₅ — C₁₅ — C₃×D₅ — C₆×D₅ — C₂×C₃⋊F₅ — C₄×C₃⋊F₅

Lower central

C₁₅ — C₄×C₃⋊F₅

Upper central

C₁ — C₄

Generators and relations for C₄×C₃⋊F₅
G = < a,b,c,d | a⁴=b³=c⁵=d⁴=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1, dcd^-1=c³ >

Subgroups: 240 in 60 conjugacy classes, 29 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₅, C₆, C₆, C₂×C₄, D₅, C₁₀, Dic₃, C₁₂, C₁₂, C₂×C₆, C₁₅, C₄², Dic₅, C₂₀, F₅, D₁₀, C₂×Dic₃, C₂×C₁₂, C₃×D₅, C₃₀, C₄×D₅, C₂×F₅, C₄×Dic₃, C₃×Dic₅, C₆₀, C₃⋊F₅, C₆×D₅, C₄×F₅, D₅×C₁₂, C₂×C₃⋊F₅, C₄×C₃⋊F₅
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Dic₃, D₆, C₄², F₅, C₄×S₃, C₂×Dic₃, C₂×F₅, C₄×Dic₃, C₃⋊F₅, C₄×F₅, C₂×C₃⋊F₅, C₄×C₃⋊F₅

Smallest permutation representation of C₄×C₃⋊F₅
►On 60 points

Generators in S₆₀

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

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

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

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

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

C₄×C₃⋊F₅ is a maximal subgroup of
C₃₀.C₄² C₃₀.₄C₄² D₁₂⋊₄F₅ D₆₀⋊₂C₄ C₂₄⋊F₅ Dic₁₀⋊Dic₃ D₂₀⋊₂Dic₃ Dic₆⋊₅F₅ (C₄×S₃)⋊F₅ C₄×S₃×F₅ D₆₀⋊₃C₄ (C₂×C₁₂)⋊₆F₅
C₄×C₃⋊F₅ is a maximal quotient of
C₂₄⋊F₅ C₃₀.₁₁C₄² D₁₀.₁₀D₁₂

36 conjugacy classes

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	···	4L	5	6A	6B	6C	10	12A	12B	12C	12D	15A	15B	20A	20B	30A	30B	60A	60B	60C	60D
order	1	2	2	2	3	4	4	4	4	4	···	4	5	6	6	6	10	12	12	12	12	15	15	20	20	30	30	60	60	60	60
size	1	1	5	5	2	1	1	5	5	15	···	15	4	2	10	10	4	2	2	10	10	4	4	4	4	4	4	4	4	4	4

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+				+	-	-	+		+	+
image	C₁	C₂	C₂	C₄	C₄	C₄	S₃	Dic₃	Dic₃	D₆	C₄×S₃	F₅	C₂×F₅	C₃⋊F₅	C₄×F₅	C₂×C₃⋊F₅	C₄×C₃⋊F₅
kernel	C₄×C₃⋊F₅	D₅×C₁₂	C₂×C₃⋊F₅	C₃×Dic₅	C₆₀	C₃⋊F₅	C₄×D₅	Dic₅	C₂₀	D₁₀	D₅	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	1	2	2	2	8	1	1	1	1	4	1	1	2	2	2	4

Matrix representation of C₄×C₃⋊F₅ ►in GL₄(𝔽₆₁) generated by

50	0	0	0
0	50	0	0
0	0	50	0
0	0	0	50

27	0	55	55
6	33	6	0
0	6	33	6
55	55	0	27

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

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

G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,50,0,0,0,0,50],[27,6,0,55,0,33,6,55,55,6,33,0,55,0,6,27],[0,0,0,60,1,0,0,60,0,1,0,60,0,0,1,60],[1,0,0,60,0,0,1,60,0,0,0,60,0,1,0,60] >;

C₄×C₃⋊F₅ in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes F_5

% in TeX

G:=Group("C4xC3:F5");

// GroupNames label

G:=SmallGroup(240,120);

// by ID

G=gap.SmallGroup(240,120);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,55,964,5189,1745]);

// Polycyclic

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

// generators/relations

G = C4×C3⋊F5 order 240 = 24·3·5

Direct product of C4 and C3⋊F5

G = C₄×C₃⋊F₅ order 240 = 2⁴·3·5

Direct product of C₄ and C₃⋊F₅