C4xC5:S4 - GroupNames

G = C₄×C₅⋊S₄ order 480 = 2⁵·3·5

Direct product of C₄ and C₅⋊S₄

direct product, non-abelian, soluble, monomial

Aliases: C₄×C₅⋊S₄, C₂₀⋊₂S₄, C₂³.₂D₃₀, C₅⋊₃(C₄×S₄), A₄⋊₂(C₄×D₅), (C₄×A₄)⋊₂D₅, C₂²⋊(C₄×D₁₅), (A₄×C₂₀)⋊₂C₂, C₁₀.₁₇(C₂×S₄), A₄⋊Dic₅⋊₅C₂, (C₂²×C₂₀)⋊₃S₃, (C₂×A₄).₉D₁₀, (C₂²×C₄)⋊₁D₁₅, (C₁₀×A₄).₉C₂², (C₂²×C₁₀).₁₄D₆, C₂.₁(C₂×C₅⋊S₄), (C₂×C₅⋊S₄).₂C₂, (C₅×A₄)⋊₇(C₂×C₄), (C₂×C₁₀)⋊₇(C₄×S₃), SmallGroup(480,1025)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×A₄ — C₄×C₅⋊S₄

Chief series

C₁ — C₂² — C₂×C₁₀ — C₅×A₄ — C₁₀×A₄ — C₂×C₅⋊S₄ — C₄×C₅⋊S₄

Lower central

C₅×A₄ — C₄×C₅⋊S₄

Upper central

C₁ — C₄

Generators and relations for C₄×C₅⋊S₄
G = < a,b,c,d,e,f | a⁴=b⁵=c²=d²=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b^-1, ece^-1=fcf=cd=dc, ede^-1=c, df=fd, fef=e^-1 >

Subgroups: 864 in 112 conjugacy classes, 23 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, Dic₃, C₁₂, A₄, D₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, Dic₅, C₂₀, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₄×S₃, S₄, C₂×A₄, D₁₅, C₃₀, C₄×D₄, C₄×D₅, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂²×D₅, C₂²×C₁₀, A₄⋊C₄, C₄×A₄, C₂×S₄, Dic₁₅, C₆₀, C₅×A₄, D₃₀, C₄×Dic₅, C₁₀.D₄, D₁₀⋊C₄, C₂³.D₅, C₂×C₄×D₅, C₂×C₅⋊D₄, C₂²×C₂₀, C₄×S₄, C₄×D₁₅, C₅⋊S₄, C₁₀×A₄, C₄×C₅⋊D₄, A₄⋊Dic₅, A₄×C₂₀, C₂×C₅⋊S₄, C₄×C₅⋊S₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₅, D₆, D₁₀, C₄×S₃, S₄, D₁₅, C₄×D₅, C₂×S₄, D₃₀, C₄×S₄, C₄×D₁₅, C₅⋊S₄, C₂×C₅⋊S₄, C₄×C₅⋊S₄

Smallest permutation representation of C₄×C₅⋊S₄
►On 60 points

Generators in S₆₀

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

(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 13)(2 14)(3 15)(4 11)(5 12)(6 56)(7 57)(8 58)(9 59)(10 60)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)

(6 56)(7 57)(8 58)(9 59)(10 60)(16 21)(17 22)(18 23)(19 24)(20 25)(26 31)(27 32)(28 33)(29 34)(30 35)(46 51)(47 52)(48 53)(49 54)(50 55)

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

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

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

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

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

52 conjugacy classes

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

52 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	3	3	3	6	6	6
type	+	+	+	+		+	+	+	+		+		+		+	+		+	+
image	C₁	C₂	C₂	C₂	C₄	S₃	D₅	D₆	D₁₀	C₄×S₃	D₁₅	C₄×D₅	D₃₀	C₄×D₁₅	S₄	C₂×S₄	C₄×S₄	C₅⋊S₄	C₂×C₅⋊S₄	C₄×C₅⋊S₄
kernel	C₄×C₅⋊S₄	A₄⋊Dic₅	A₄×C₂₀	C₂×C₅⋊S₄	C₅⋊S₄	C₂²×C₂₀	C₄×A₄	C₂²×C₁₀	C₂×A₄	C₂×C₁₀	C₂²×C₄	A₄	C₂³	C₂²	C₂₀	C₁₀	C₅	C₄	C₂	C₁
# reps	1	1	1	1	4	1	2	1	2	2	4	4	4	8	2	2	4	2	2	4

Matrix representation of C₄×C₅⋊S₄ ►in GL₅(𝔽₆₁)

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

17	60	0	0	0
1	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

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

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

1	0	0	0	0
17	60	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,50],[17,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,17,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₄×C₅⋊S₄ in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes S_4

% in TeX

G:=Group("C4xC5:S4");

// GroupNames label

G:=SmallGroup(480,1025);

// by ID

G=gap.SmallGroup(480,1025);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,36,451,3364,10085,1286,5886,2232]);

// Polycyclic

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

// generators/relations

G = C4×C5⋊S4 order 480 = 25·3·5

Direct product of C4 and C5⋊S4

G = C₄×C₅⋊S₄ order 480 = 2⁵·3·5

Direct product of C₄ and C₅⋊S₄