C5xA4:D4 - GroupNames

G = C₅×A₄⋊D₄ order 480 = 2⁵·3·5

Direct product of C₅ and A₄⋊D₄

direct product, non-abelian, soluble, monomial

Aliases: C₅×A₄⋊D₄, A₄⋊C₄⋊C₁₀, (C₂×C₁₀)⋊₃S₄, A₄⋊₂(C₅×D₄), (C₅×A₄)⋊₉D₄, (C₁₀×S₄)⋊₅C₂, (C₂×S₄)⋊₂C₁₀, C₂⁴⋊₁(C₅×S₃), C₂²⋊₂(C₅×S₄), C₂.₁₁(C₁₀×S₄), C₁₀.₃₇(C₂×S₄), (C₂³×C₁₀)⋊₁S₃, (C₂²×A₄)⋊₂C₁₀, C₂³.₅(S₃×C₁₀), (C₂²×C₁₀).₁₂D₆, (C₁₀×A₄).₂₂C₂², (A₄×C₂×C₁₀)⋊₆C₂, (C₅×A₄⋊C₄)⋊₄C₂, C₂²⋊(C₅×C₃⋊D₄), (C₂×C₁₀)⋊₃(C₃⋊D₄), (C₂×A₄).₅(C₂×C₁₀), SmallGroup(480,1023)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₂×A₄ — C₅×A₄⋊D₄

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₁₀×A₄ — C₁₀×S₄ — C₅×A₄⋊D₄

Lower central

A₄ — C₂×A₄ — C₅×A₄⋊D₄

Upper central

C₁ — C₁₀ — C₂×C₁₀

Generators and relations for C₅×A₄⋊D₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d³=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd^-1=ebe^-1=fbf=bc=cb, dcd^-1=b, ce=ec, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 468 in 124 conjugacy classes, 24 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, C₁₀, C₁₀, Dic₃, A₄, D₆, C₂×C₆, C₁₅, C₂²⋊C₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₃⋊D₄, S₄, C₂×A₄, C₂×A₄, C₅×S₃, C₃₀, C₂²≀C₂, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, A₄⋊C₄, C₂×S₄, C₂²×A₄, C₅×Dic₃, C₅×A₄, S₃×C₁₀, C₂×C₃₀, C₅×C₂²⋊C₄, D₄×C₁₀, C₂³×C₁₀, A₄⋊D₄, C₅×C₃⋊D₄, C₅×S₄, C₁₀×A₄, C₁₀×A₄, C₅×C₂²≀C₂, C₅×A₄⋊C₄, C₁₀×S₄, A₄×C₂×C₁₀, C₅×A₄⋊D₄
Quotients: C₁, C₂, C₂², C₅, S₃, D₄, C₁₀, D₆, C₂×C₁₀, C₃⋊D₄, S₄, C₅×S₃, C₅×D₄, C₂×S₄, S₃×C₁₀, A₄⋊D₄, C₅×C₃⋊D₄, C₅×S₄, C₁₀×S₄, C₅×A₄⋊D₄

Smallest permutation representation of C₅×A₄⋊D₄
►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)

(6 58)(7 59)(8 60)(9 56)(10 57)(11 18)(12 19)(13 20)(14 16)(15 17)(26 33)(27 34)(28 35)(29 31)(30 32)(41 47)(42 48)(43 49)(44 50)(45 46)

(1 39)(2 40)(3 36)(4 37)(5 38)(11 18)(12 19)(13 20)(14 16)(15 17)(21 51)(22 52)(23 53)(24 54)(25 55)(41 47)(42 48)(43 49)(44 50)(45 46)

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

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

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	3	4A	4B	4C	5A	5B	5C	5D	6A	6B	6C	10A	10B	10C	10D	10E	10F	10G	10H	10I	···	10P	10Q	10R	10S	10T	10U	10V	10W	10X	15A	15B	15C	15D	20A	···	20L	30A	···	30L
order	1	2	2	2	2	2	2	3	4	4	4	5	5	5	5	6	6	6	10	10	10	10	10	10	10	10	10	···	10	10	10	10	10	10	10	10	10	15	15	15	15	20	···	20	30	···	30
size	1	1	2	3	3	6	12	8	12	12	12	1	1	1	1	8	8	8	1	1	1	1	2	2	2	2	3	···	3	6	6	6	6	12	12	12	12	8	8	8	8	12	···	12	8	···	8

70 irreducible representations

dim	1	1	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	C₂	C₅	C₁₀	C₁₀	C₁₀	S₃	D₄	D₆	C₃⋊D₄	C₅×S₃	C₅×D₄	S₃×C₁₀	C₅×C₃⋊D₄	S₄	C₂×S₄	C₅×S₄	C₁₀×S₄	A₄⋊D₄	C₅×A₄⋊D₄
kernel	C₅×A₄⋊D₄	C₅×A₄⋊C₄	C₁₀×S₄	A₄×C₂×C₁₀	A₄⋊D₄	A₄⋊C₄	C₂×S₄	C₂²×A₄	C₂³×C₁₀	C₅×A₄	C₂²×C₁₀	C₂×C₁₀	C₂⁴	A₄	C₂³	C₂²	C₂×C₁₀	C₁₀	C₂²	C₂	C₅	C₁
# reps	1	1	1	1	4	4	4	4	1	1	1	2	4	4	4	8	2	2	8	8	1	4

Matrix representation of C₅×A₄⋊D₄ ►in GL₅(𝔽₆₁)

20	0	0	0	0
0	20	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

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

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	1	0	0	0
59	60	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	0	0	1

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

G:=sub<GL(5,GF(61))| [20,0,0,0,0,0,20,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,60,0,1,0,0,60,1,0],[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],[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],[1,59,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C₅×A₄⋊D₄ in GAP, Magma, Sage, TeX

C_5\times A_4\rtimes D_4

% in TeX

G:=Group("C5xA4:D4");

// GroupNames label

G:=SmallGroup(480,1023);

// by ID

G=gap.SmallGroup(480,1023);

# by ID

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

// Polycyclic

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

// generators/relations

G = C5×A4⋊D4 order 480 = 25·3·5

Direct product of C5 and A4⋊D4

G = C₅×A₄⋊D₄ order 480 = 2⁵·3·5

Direct product of C₅ and A₄⋊D₄