C5xD4xA4 - GroupNames

G = C₅×D₄×A₄ order 480 = 2⁵·3·5

Direct product of C₅, D₄ and A₄

direct product, metabelian, soluble, monomial

Aliases: C₅×D₄×A₄, C₂⁴⋊₃C₃₀, C₄⋊(C₁₀×A₄), C₂₀⋊₃(C₂×A₄), (C₂²×C₄)⋊C₃₀, (C₂²×D₄)⋊C₁₅, C₂²⋊(D₄×C₁₅), (A₄×C₂₀)⋊₇C₂, (C₄×A₄)⋊₃C₁₀, (C₂²×C₂₀)⋊₃C₆, (C₂³×C₁₀)⋊₂C₆, C₂²⋊₂(C₁₀×A₄), (C₂²×A₄)⋊₁C₁₀, C₂³.₇(C₂×C₃₀), C₁₀.₁₃(C₂²×A₄), (C₁₀×A₄).₂₄C₂², (D₄×C₂×C₁₀)⋊C₃, (A₄×C₂×C₁₀)⋊₁C₂, C₂.₂(A₄×C₂×C₁₀), (C₂×C₁₀)⋊₃(C₂×A₄), (C₂×C₁₀)⋊₆(C₃×D₄), (C₂×A₄).₇(C₂×C₁₀), (C₂²×C₁₀).₃₀(C₂×C₆), SmallGroup(480,1127)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₅×D₄×A₄

Chief series

C₁ — C₂² — C₂³ — C₂²×C₁₀ — C₁₀×A₄ — A₄×C₂×C₁₀ — C₅×D₄×A₄

Lower central

C₂² — C₂³ — C₅×D₄×A₄

Upper central

C₁ — C₁₀ — C₅×D₄

Generators and relations for C₅×D₄×A₄
G = < a,b,c,d,e,f | a⁵=b⁴=c²=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 416 in 132 conjugacy classes, 36 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₅, C₆, C₂×C₄, D₄, D₄, C₂³, C₂³, C₁₀, C₁₀, C₁₂, A₄, C₂×C₆, C₁₅, C₂²×C₄, C₂×D₄, C₂⁴, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₃×D₄, C₂×A₄, C₂×A₄, C₃₀, C₂²×D₄, C₂×C₂₀, C₅×D₄, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₄×A₄, C₂²×A₄, C₆₀, C₅×A₄, C₂×C₃₀, C₂²×C₂₀, D₄×C₁₀, C₂³×C₁₀, D₄×A₄, D₄×C₁₅, C₁₀×A₄, C₁₀×A₄, D₄×C₂×C₁₀, A₄×C₂₀, A₄×C₂×C₁₀, C₅×D₄×A₄
Quotients: C₁, C₂, C₃, C₂², C₅, C₆, D₄, C₁₀, A₄, C₂×C₆, C₁₅, C₂×C₁₀, C₃×D₄, C₂×A₄, C₃₀, C₅×D₄, C₂²×A₄, C₅×A₄, C₂×C₃₀, D₄×A₄, D₄×C₁₅, C₁₀×A₄, A₄×C₂×C₁₀, C₅×D₄×A₄

Smallest permutation representation of C₅×D₄×A₄
►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 43 13 37)(2 44 14 38)(3 45 15 39)(4 41 11 40)(5 42 12 36)(6 31 60 30)(7 32 56 26)(8 33 57 27)(9 34 58 28)(10 35 59 29)(16 52 22 46)(17 53 23 47)(18 54 24 48)(19 55 25 49)(20 51 21 50)

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

(1 13)(2 14)(3 15)(4 11)(5 12)(6 60)(7 56)(8 57)(9 58)(10 59)(26 32)(27 33)(28 34)(29 35)(30 31)(36 42)(37 43)(38 44)(39 45)(40 41)

(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 26 16)(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)

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

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

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

100 conjugacy classes

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

100 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	3	3	3	3	3	3	6	6
type	+	+	+										+				+	+	+				+
image	C₁	C₂	C₂	C₃	C₅	C₆	C₆	C₁₀	C₁₀	C₁₅	C₃₀	C₃₀	D₄	C₃×D₄	C₅×D₄	D₄×C₁₅	A₄	C₂×A₄	C₂×A₄	C₅×A₄	C₁₀×A₄	C₁₀×A₄	D₄×A₄	C₅×D₄×A₄
kernel	C₅×D₄×A₄	A₄×C₂₀	A₄×C₂×C₁₀	D₄×C₂×C₁₀	D₄×A₄	C₂²×C₂₀	C₂³×C₁₀	C₄×A₄	C₂²×A₄	C₂²×D₄	C₂²×C₄	C₂⁴	C₅×A₄	C₂×C₁₀	A₄	C₂²	C₅×D₄	C₂₀	C₂×C₁₀	D₄	C₄	C₂²	C₅	C₁
# reps	1	1	2	2	4	2	4	4	8	8	8	16	1	2	4	8	1	1	2	4	4	8	1	4

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

1	0	0	0	0
0	1	0	0	0
0	0	34	0	0
0	0	0	34	0
0	0	0	0	34

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

0	60	0	0	0
60	0	0	0	0
0	0	60	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	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

47	0	0	0	0
0	47	0	0	0
0	0	0	0	47
0	0	47	0	0
0	0	0	47	0

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,34,0,0,0,0,0,34,0,0,0,0,0,34],[0,1,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[0,60,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[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],[47,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,47,0,0,47,0,0] >;

C₅×D₄×A₄ in GAP, Magma, Sage, TeX

C_5\times D_4\times A_4

% in TeX

G:=Group("C5xD4xA4");

// GroupNames label

G:=SmallGroup(480,1127);

// by ID

G=gap.SmallGroup(480,1127);

# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,2,869,2539,4430]);

// Polycyclic

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

// generators/relations

G = C5×D4×A4 order 480 = 25·3·5

Direct product of C5, D4 and A4

G = C₅×D₄×A₄ order 480 = 2⁵·3·5

Direct product of C₅, D₄ and A₄