D5xA4:C4 - GroupNames

G = D₅×A₄⋊C₄ order 480 = 2⁵·3·5

Direct product of D₅ and A₄⋊C₄

direct product, non-abelian, soluble, monomial

Aliases: D₅×A₄⋊C₄, D₁₀.₇S₄, A₄⋊₃(C₄×D₅), (D₅×A₄)⋊₁C₄, C₂.₃(D₅×S₄), C₂²⋊(D₅×Dic₃), C₁₀.₁₃(C₂×S₄), A₄⋊Dic₅⋊₃C₂, (C₂×A₄).₅D₁₀, C₂³.₅(S₃×D₅), (C₂³×D₅).₁S₃, (C₂²×C₁₀).₅D₆, (C₁₀×A₄).₅C₂², (C₂²×D₅)⋊₁Dic₃, C₅⋊₂(C₂×A₄⋊C₄), (C₅×A₄⋊C₄)⋊₂C₂, (C₂×D₅×A₄).₁C₂, (C₅×A₄)⋊₆(C₂×C₄), (C₂×C₁₀)⋊₂(C₂×Dic₃), SmallGroup(480,979)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×A₄ — D₅×A₄⋊C₄

Chief series

C₁ — C₂² — C₂×C₁₀ — C₅×A₄ — C₁₀×A₄ — C₂×D₅×A₄ — D₅×A₄⋊C₄

Lower central

C₅×A₄ — D₅×A₄⋊C₄

Upper central

C₁ — C₂

Generators and relations for D₅×A₄⋊C₄
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e³=f⁴=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf^-1=cd=dc, ede^-1=c, df=fd, fef^-1=e^-1 >

Subgroups: 920 in 126 conjugacy classes, 25 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₂×C₄, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, Dic₃, A₄, C₂×C₆, C₁₅, C₂²⋊C₄, C₂²×C₄, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×Dic₃, C₂×A₄, C₂×A₄, C₃×D₅, C₃₀, C₂×C₂²⋊C₄, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, A₄⋊C₄, A₄⋊C₄, C₂²×A₄, C₅×Dic₃, Dic₁₅, C₅×A₄, C₆×D₅, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₂×C₄×D₅, C₂³×D₅, C₂×A₄⋊C₄, D₅×Dic₃, D₅×A₄, C₁₀×A₄, D₅×C₂²⋊C₄, C₅×A₄⋊C₄, A₄⋊Dic₅, C₂×D₅×A₄, D₅×A₄⋊C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₅, Dic₃, D₆, D₁₀, C₂×Dic₃, S₄, C₄×D₅, A₄⋊C₄, C₂×S₄, S₃×D₅, C₂×A₄⋊C₄, D₅×Dic₃, D₅×S₄, D₅×A₄⋊C₄

Smallest permutation representation of D₅×A₄⋊C₄
►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 5)(2 4)(6 9)(7 8)(11 15)(12 14)(16 17)(18 20)(21 25)(22 24)(26 27)(28 30)(31 35)(32 34)(36 37)(38 40)(41 45)(42 44)(46 47)(48 50)(51 55)(52 54)(56 57)(58 60)

(16 25)(17 21)(18 22)(19 23)(20 24)(26 35)(27 31)(28 32)(29 33)(30 34)(36 45)(37 41)(38 42)(39 43)(40 44)(46 55)(47 51)(48 52)(49 53)(50 54)

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

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

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

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

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

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3	4A	4B	4C	4D	4E	4F	4G	4H	5A	5B	6A	6B	6C	10A	10B	10C	10D	10E	10F	15A	15B	20A	···	20H	30A	30B
order	1	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	5	5	6	6	6	10	10	10	10	10	10	15	15	20	···	20	30	30
size	1	1	3	3	5	5	15	15	8	6	6	6	6	30	30	30	30	2	2	8	40	40	2	2	6	6	6	6	16	16	12	···	12	16	16

40 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	3	3	3	4	4	6	6
type	+	+	+	+		+	+	-	+	+		+		+	+	-	+
image	C₁	C₂	C₂	C₂	C₄	S₃	D₅	Dic₃	D₆	D₁₀	C₄×D₅	S₄	A₄⋊C₄	C₂×S₄	S₃×D₅	D₅×Dic₃	D₅×S₄	D₅×A₄⋊C₄
kernel	D₅×A₄⋊C₄	C₅×A₄⋊C₄	A₄⋊Dic₅	C₂×D₅×A₄	D₅×A₄	C₂³×D₅	A₄⋊C₄	C₂²×D₅	C₂²×C₁₀	C₂×A₄	A₄	D₁₀	D₅	C₁₀	C₂³	C₂²	C₂	C₁
# reps	1	1	1	1	4	1	2	2	1	2	4	2	4	2	2	2	4	4

Matrix representation of D₅×A₄⋊C₄ ►in GL₅(𝔽₆₁)

43	1	0	0	0
60	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

60	18	0	0	0
0	1	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	60	0
0	0	1	60	1

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

1	0	0	0	0
0	1	0	0	0
0	0	0	60	0
0	0	1	60	2
0	0	0	0	1

11	0	0	0	0
0	11	0	0	0
0	0	60	1	59
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(61))| [43,60,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,18,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,1,0,0,0,60,60,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,1,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,60,60,0,0,0,0,2,1],[11,0,0,0,0,0,11,0,0,0,0,0,60,0,0,0,0,1,1,0,0,0,59,0,1] >;

D₅×A₄⋊C₄ in GAP, Magma, Sage, TeX

D_5\times A_4\rtimes C_4

% in TeX

G:=Group("D5xA4:C4");

// GroupNames label

G:=SmallGroup(480,979);

// by ID

G=gap.SmallGroup(480,979);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,36,234,3364,5052,1286,2953,2232]);

// Polycyclic

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

// generators/relations

G = D5×A4⋊C4 order 480 = 25·3·5

Direct product of D5 and A4⋊C4

G = D₅×A₄⋊C₄ order 480 = 2⁵·3·5

Direct product of D₅ and A₄⋊C₄