C5xC4:S4 - GroupNames

G = C₅×C₄⋊S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₄⋊S₄

direct product, non-abelian, soluble, monomial

Aliases: C₅×C₄⋊S₄, C₂₀⋊₃S₄, C₄⋊(C₅×S₄), A₄⋊₁(C₅×D₄), (C₅×A₄)⋊₆D₄, C₂²⋊(C₅×D₁₂), (C₂×S₄)⋊₁C₁₀, (C₁₀×S₄)⋊₄C₂, (A₄×C₂₀)⋊₄C₂, (C₄×A₄)⋊₁C₁₀, (C₂×C₁₀)⋊₂D₁₂, C₂.₄(C₁₀×S₄), C₁₀.₂₉(C₂×S₄), (C₂²×C₂₀)⋊₄S₃, C₂³.₃(S₃×C₁₀), (C₂²×C₁₀).₁₀D₆, (C₁₀×A₄).₂₀C₂², (C₂²×C₄)⋊₂(C₅×S₃), (C₂×A₄).₃(C₂×C₁₀), SmallGroup(480,1015)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₂×A₄ — C₅×C₄⋊S₄

Chief series

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

Lower central

A₄ — C₂×A₄ — C₅×C₄⋊S₄

Upper central

C₁ — C₁₀ — C₂₀

Generators and relations for C₅×C₄⋊S₄
G = < a,b,c,d,e,f | a⁵=b⁴=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: 436 in 112 conjugacy classes, 24 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₁₂, A₄, D₆, C₁₅, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, D₁₂, S₄, C₂×A₄, C₅×S₃, C₃₀, C₄⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₄×A₄, C₂×S₄, C₆₀, C₅×A₄, S₃×C₁₀, C₅×C₂²⋊C₄, C₅×C₄⋊C₄, C₂²×C₂₀, D₄×C₁₀, C₄⋊S₄, C₅×D₁₂, C₅×S₄, C₁₀×A₄, C₅×C₄⋊D₄, A₄×C₂₀, C₁₀×S₄, C₅×C₄⋊S₄
Quotients: C₁, C₂, C₂², C₅, S₃, D₄, C₁₀, D₆, C₂×C₁₀, D₁₂, S₄, C₅×S₃, C₅×D₄, C₂×S₄, S₃×C₁₀, C₄⋊S₄, C₅×D₁₂, C₅×S₄, C₁₀×S₄, C₅×C₄⋊S₄

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

Generators in S₆₀

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

(1 45)(2 41)(3 42)(4 43)(5 44)(6 17)(7 18)(8 19)(9 20)(10 16)(21 51)(22 52)(23 53)(24 54)(25 55)(36 48)(37 49)(38 50)(39 46)(40 47)

(11 57)(12 58)(13 59)(14 60)(15 56)(21 51)(22 52)(23 53)(24 54)(25 55)(26 33)(27 34)(28 35)(29 31)(30 32)(36 48)(37 49)(38 50)(39 46)(40 47)

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

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	6	6
type	+	+	+				+	+	+	+					+	+			+
image	C₁	C₂	C₂	C₅	C₁₀	C₁₀	S₃	D₄	D₆	D₁₂	C₅×S₃	C₅×D₄	S₃×C₁₀	C₅×D₁₂	S₄	C₂×S₄	C₅×S₄	C₁₀×S₄	C₄⋊S₄	C₅×C₄⋊S₄
kernel	C₅×C₄⋊S₄	A₄×C₂₀	C₁₀×S₄	C₄⋊S₄	C₄×A₄	C₂×S₄	C₂²×C₂₀	C₅×A₄	C₂²×C₁₀	C₂×C₁₀	C₂²×C₄	A₄	C₂³	C₂²	C₂₀	C₁₀	C₄	C₂	C₅	C₁
# reps	1	1	2	4	4	8	1	1	1	2	4	4	4	8	2	2	8	8	1	4

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

20	0	0	0	0
0	20	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

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

30	57	0	0	0
4	30	0	0	0
0	0	0	0	1
0	0	1	0	0
0	0	0	1	0

31	57	0	0	0
57	30	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	0	0	1

G:=sub<GL(5,GF(61))| [20,0,0,0,0,0,20,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,60,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[30,4,0,0,0,57,30,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[31,57,0,0,0,57,30,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C₅×C₄⋊S₄ in GAP, Magma, Sage, TeX

C_5\times C_4\rtimes S_4

% in TeX

G:=Group("C5xC4:S4");

// GroupNames label

G:=SmallGroup(480,1015);

// by ID

G=gap.SmallGroup(480,1015);

# by ID

G:=PCGroup([7,-2,-2,-5,-2,-3,-2,2,309,148,2804,10085,285,5886,475]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^4=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 = C5×C4⋊S4 order 480 = 25·3·5

Direct product of C5 and C4⋊S4

G = C₅×C₄⋊S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₄⋊S₄