D4xC3:F5 - GroupNames

G = D₄×C₃⋊F₅ order 480 = 2⁵·3·5

Direct product of D₄ and C₃⋊F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄×C₃⋊F₅, D₂₀⋊₃Dic₃, C₅⋊(D₄×Dic₃), C₃⋊₄(D₄×F₅), C₆₀⋊₅(C₂×C₄), C₁₅⋊₂₃(C₄×D₄), (C₃×D₄)⋊₃F₅, C₂₀⋊(C₂×Dic₃), C₁₂⋊₃(C₂×F₅), C₅⋊D₄⋊Dic₃, (D₄×C₁₅)⋊₃C₄, (C₃×D₂₀)⋊₃C₄, D₁₀⋊(C₂×Dic₃), C₆₀⋊C₄⋊₆C₂, D₅.₄(S₃×D₄), (D₄×D₅).₃S₃, Dic₅⋊(C₂×Dic₃), (C₅×D₄)⋊₃Dic₃, (C₄×D₅).₄₀D₆, C₆.₃₉(C₂²×F₅), C₃₀.₇₇(C₂²×C₄), (C₂²×D₅).₄₀D₆, (C₆×D₅).₆₃C₂³, D₁₀.D₆⋊₇C₂, D₅.₅(D₄⋊₂S₃), (D₅×C₁₂).₇₃C₂², D₁₀.₄₈(C₂²×S₃), C₁₀.₈(C₂²×Dic₃), C₄⋊₁(C₂×C₃⋊F₅), (C₄×C₃⋊F₅)⋊₈C₂, (C₂×C₆)⋊₃(C₂×F₅), (C₃×D₄×D₅).₃C₂, (C₂×C₃₀)⋊₅(C₂×C₄), C₂²⋊₂(C₂×C₃⋊F₅), (C₃×C₅⋊D₄)⋊₁C₄, (C₂²×C₃⋊F₅)⋊₄C₂, (C₆×D₅)⋊₁₅(C₂×C₄), C₂.₉(C₂²×C₃⋊F₅), (C₃×D₅).₁₁(C₂×D₄), (C₂×C₁₀)⋊₃(C₂×Dic₃), (D₅×C₂×C₆).₈₈C₂², (C₂×C₃⋊F₅).₁₆C₂², (C₃×Dic₅)⋊₁₁(C₂×C₄), (C₃×D₅).₁₂(C₄○D₄), SmallGroup(480,1067)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃₀ — D₄×C₃⋊F₅

Chief series

C₁ — C₅ — C₁₅ — C₃×D₅ — C₆×D₅ — C₂×C₃⋊F₅ — C₂²×C₃⋊F₅ — D₄×C₃⋊F₅

Lower central

C₁₅ — C₃₀ — D₄×C₃⋊F₅

Upper central

C₁ — C₂ — D₄

Generators and relations for D₄×C₃⋊F₅
G = < a,b,c,d,e | a⁴=b²=c³=d⁵=e⁴=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d³ >

Subgroups: 988 in 188 conjugacy classes, 63 normal (35 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₅, C₆, C₆, C₂×C₄, D₄, D₄, C₂³, D₅, D₅, C₁₀, C₁₀, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, D₁₀, C₂×C₁₀, C₂×Dic₃, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂²×C₆, C₃×D₅, C₃×D₅, C₃₀, C₃₀, C₄×D₄, C₄×D₅, D₂₀, C₅⋊D₄, C₅×D₄, C₂×F₅, C₂²×D₅, C₄×Dic₃, C₄⋊Dic₃, C₆.D₄, C₂²×Dic₃, C₆×D₄, C₃×Dic₅, C₆₀, C₃⋊F₅, C₃⋊F₅, C₆×D₅, C₆×D₅, C₆×D₅, C₂×C₃₀, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, D₄×D₅, C₂²×F₅, D₄×Dic₃, D₅×C₁₂, C₃×D₂₀, C₃×C₅⋊D₄, D₄×C₁₅, C₂×C₃⋊F₅, C₂×C₃⋊F₅, C₂×C₃⋊F₅, D₅×C₂×C₆, D₄×F₅, C₄×C₃⋊F₅, C₆₀⋊C₄, D₁₀.D₆, C₃×D₄×D₅, C₂²×C₃⋊F₅, D₄×C₃⋊F₅
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, Dic₃, D₆, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₂×Dic₃, C₂²×S₃, C₄×D₄, C₂×F₅, S₃×D₄, D₄⋊₂S₃, C₂²×Dic₃, C₃⋊F₅, C₂²×F₅, D₄×Dic₃, C₂×C₃⋊F₅, D₄×F₅, C₂²×C₃⋊F₅, D₄×C₃⋊F₅

Smallest permutation representation of D₄×C₃⋊F₅
►On 60 points

Generators in S₆₀

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

(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	8	8
type	+	+	+	+	+	+				+	+	+	-	-	-	+		+	+	+	+	-				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₄	D₆	Dic₃	Dic₃	Dic₃	D₆	C₄○D₄	F₅	C₂×F₅	C₂×F₅	S₃×D₄	D₄⋊₂S₃	C₃⋊F₅	C₂×C₃⋊F₅	C₂×C₃⋊F₅	D₄×F₅	D₄×C₃⋊F₅
kernel	D₄×C₃⋊F₅	C₄×C₃⋊F₅	C₆₀⋊C₄	D₁₀.D₆	C₃×D₄×D₅	C₂²×C₃⋊F₅	C₃×D₂₀	C₃×C₅⋊D₄	D₄×C₁₅	D₄×D₅	C₃⋊F₅	C₄×D₅	D₂₀	C₅⋊D₄	C₅×D₄	C₂²×D₅	C₃×D₅	C₃×D₄	C₁₂	C₂×C₆	D₅	D₅	D₄	C₄	C₂²	C₃	C₁
# reps	1	1	1	2	1	2	2	4	2	1	2	1	1	2	1	2	2	1	1	2	1	1	2	2	4	1	2

Matrix representation of D₄×C₃⋊F₅ ►in GL₆(𝔽₆₁)

60	36	0	0	0	0
44	1	0	0	0	0
0	0	60	0	0	0
0	0	0	60	0	0
0	0	0	0	60	0
0	0	0	0	0	60

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	33	0	6	6
0	0	55	27	55	0
0	0	0	55	27	55
0	0	6	6	0	33

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

50	0	0	0	0	0
0	50	0	0	0	0
0	0	0	6	34	6
0	0	28	0	55	55
0	0	55	55	0	28
0	0	6	34	6	0

G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,36,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,55,0,6,0,0,0,27,55,6,0,0,6,55,27,0,0,0,6,0,55,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,1,0,0,60,0,0,0,1,0,60,0,0,0,0,1,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,28,55,6,0,0,6,0,55,34,0,0,34,55,0,6,0,0,6,55,28,0] >;

D₄×C₃⋊F₅ in GAP, Magma, Sage, TeX

D_4\times C_3\rtimes F_5

% in TeX

G:=Group("D4xC3:F5");

// GroupNames label

G:=SmallGroup(480,1067);

// by ID

G=gap.SmallGroup(480,1067);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,2693,14118,2379]);

// Polycyclic

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

// generators/relations

G = D4×C3⋊F5 order 480 = 25·3·5

Direct product of D4 and C3⋊F5

G = D₄×C₃⋊F₅ order 480 = 2⁵·3·5

Direct product of D₄ and C₃⋊F₅