C5xD4:4D4 - GroupNames

G = C₅×D₄⋊₄D₄ order 320 = 2⁶·5

Direct product of C₅ and D₄⋊₄D₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₅×D₄⋊₄D₄, 2₊¹⁺⁴⋊₁C₁₀, D₄⋊₄(C₅×D₄), C₄≀C₂⋊₁C₁₀, Q₈⋊₄(C₅×D₄), (C₅×D₄)⋊₂₂D₄, (C₅×Q₈)⋊₂₂D₄, C₄⋊₁D₄⋊₁C₁₀, C₈⋊C₂²⋊₁C₁₀, C₄²⋊₂(C₂×C₁₀), C₄.₂₇(D₄×C₁₀), C₂³.₅(C₅×D₄), (C₄×C₂₀)⋊₃₅C₂², C₄.D₄⋊₁C₁₀, C₂₀.₃₈₈(C₂×D₄), (C₂²×C₁₀).₅D₄, (D₄×C₁₀)⋊₂₈C₂², M₄(2)⋊₁(C₂×C₁₀), C₂².₁₄(D₄×C₁₀), C₁₀.₁₀₀C₂²≀C₂, (C₂×C₂₀).₆₀₉C₂³, (C₅×2₊¹⁺⁴)⋊₇C₂, (C₅×M₄(2))⋊₁₇C₂², (C₅×C₄≀C₂)⋊₉C₂, (C₂×D₄)⋊₂(C₂×C₁₀), (C₅×C₈⋊C₂²)⋊₈C₂, (C₅×C₄⋊₁D₄)⋊₁₁C₂, C₄○D₄.₁(C₂×C₁₀), (C₅×C₄.D₄)⋊₇C₂, C₂.₁₄(C₅×C₂²≀C₂), (C₂×C₁₀).₄₀₉(C₂×D₄), (C₂×C₄).₄(C₂²×C₁₀), (C₅×C₄○D₄).₃₁C₂², SmallGroup(320,954)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₅×D₄⋊₄D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂×C₂₀ — D₄×C₁₀ — C₅×C₈⋊C₂² — C₅×D₄⋊₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₅×D₄⋊₄D₄

Upper central

C₁ — C₁₀ — C₂×C₂₀ — C₅×D₄⋊₄D₄

Generators and relations for C₅×D₄⋊₄D₄
G = < a,b,c,d,e | a⁵=b⁴=c²=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=b^-1c, ece=bc, ede=d^-1 >

Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₁₀, C₁₀, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, C₂₀, C₂₀, C₂×C₁₀, C₂×C₁₀, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₄₀, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×D₄, C₅×Q₈, C₂²×C₁₀, C₂²×C₁₀, D₄⋊₄D₄, C₄×C₂₀, C₅×M₄(2), C₅×D₈, C₅×SD₁₆, D₄×C₁₀, D₄×C₁₀, C₅×C₄○D₄, C₅×C₄○D₄, C₅×C₄.D₄, C₅×C₄≀C₂, C₅×C₄⋊₁D₄, C₅×C₈⋊C₂², C₅×2₊¹⁺⁴, C₅×D₄⋊₄D₄
Quotients: C₁, C₂, C₂², C₅, D₄, C₂³, C₁₀, C₂×D₄, C₂×C₁₀, C₂²≀C₂, C₅×D₄, C₂²×C₁₀, D₄⋊₄D₄, D₄×C₁₀, C₅×C₂²≀C₂, C₅×D₄⋊₄D₄

Smallest permutation representation of C₅×D₄⋊₄D₄
►On 40 points

Generators in S₄₀

(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)

(1 31 21 26)(2 32 22 27)(3 33 23 28)(4 34 24 29)(5 35 25 30)(6 16 36 11)(7 17 37 12)(8 18 38 13)(9 19 39 14)(10 20 40 15)

(1 16)(2 17)(3 18)(4 19)(5 20)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(26 36)(27 37)(28 38)(29 39)(30 40)

(6 11 36 16)(7 12 37 17)(8 13 38 18)(9 14 39 19)(10 15 40 20)

(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)(26 31)(27 32)(28 33)(29 34)(30 35)

G:=sub<Sym(40)| (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), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15), (1,16)(2,17)(3,18)(4,19)(5,20)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(26,36)(27,37)(28,38)(29,39)(30,40), (6,11,36,16)(7,12,37,17)(8,13,38,18)(9,14,39,19)(10,15,40,20), (6,16)(7,17)(8,18)(9,19)(10,20)(11,36)(12,37)(13,38)(14,39)(15,40)(26,31)(27,32)(28,33)(29,34)(30,35)>;

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), (1,31,21,26)(2,32,22,27)(3,33,23,28)(4,34,24,29)(5,35,25,30)(6,16,36,11)(7,17,37,12)(8,18,38,13)(9,19,39,14)(10,20,40,15), (1,16)(2,17)(3,18)(4,19)(5,20)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(26,36)(27,37)(28,38)(29,39)(30,40), (6,11,36,16)(7,12,37,17)(8,13,38,18)(9,14,39,19)(10,15,40,20), (6,16)(7,17)(8,18)(9,19)(10,20)(11,36)(12,37)(13,38)(14,39)(15,40)(26,31)(27,32)(28,33)(29,34)(30,35) );

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)], [(1,31,21,26),(2,32,22,27),(3,33,23,28),(4,34,24,29),(5,35,25,30),(6,16,36,11),(7,17,37,12),(8,18,38,13),(9,19,39,14),(10,20,40,15)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(26,36),(27,37),(28,38),(29,39),(30,40)], [(6,11,36,16),(7,12,37,17),(8,13,38,18),(9,14,39,19),(10,15,40,20)], [(6,16),(7,17),(8,18),(9,19),(10,20),(11,36),(12,37),(13,38),(14,39),(15,40),(26,31),(27,32),(28,33),(29,34),(30,35)]])

80 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	5A	5B	5C	5D	8A	8B	10A	10B	10C	10D	10E	10F	10G	10H	10I	···	10X	10Y	10Z	10AA	10AB	20A	···	20H	20I	···	20X	40A	···	40H
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	5	5	5	5	8	8	10	10	10	10	10	10	10	10	10	···	10	10	10	10	10	20	···	20	20	···	20	40	···	40
size	1	1	2	4	4	4	4	8	2	2	4	4	4	4	1	1	1	1	8	8	1	1	1	1	2	2	2	2	4	···	4	8	8	8	8	2	···	2	4	···	4	8	···	8

80 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+				+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₅	C₁₀	C₁₀	C₁₀	C₁₀	C₁₀	D₄	D₄	D₄	C₅×D₄	C₅×D₄	C₅×D₄	D₄⋊₄D₄	C₅×D₄⋊₄D₄
kernel	C₅×D₄⋊₄D₄	C₅×C₄.D₄	C₅×C₄≀C₂	C₅×C₄⋊₁D₄	C₅×C₈⋊C₂²	C₅×2₊¹⁺⁴	D₄⋊₄D₄	C₄.D₄	C₄≀C₂	C₄⋊₁D₄	C₈⋊C₂²	2₊¹⁺⁴	C₅×D₄	C₅×Q₈	C₂²×C₁₀	D₄	Q₈	C₂³	C₅	C₁
# reps	1	1	2	1	2	1	4	4	8	4	8	4	2	2	2	8	8	8	2	8

Matrix representation of C₅×D₄⋊₄D₄ ►in GL₆(𝔽₄₁)

16	0	0	0	0	0
0	16	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

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	1	0	0
0	0	40	0	0	0
0	0	40	40	1	2
0	0	0	1	40	40

40	1	0	0	0	0
0	1	0	0	0	0
0	0	40	40	1	2
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	40	1	1

40	1	0	0	0	0
39	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	1	1	40	39
0	0	0	0	1	1

40	0	0	0	0	0
39	1	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	1	1	40	39
0	0	0	40	0	1

G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,40,0,0,0,1,0,40,1,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,1,1,0,0,0,0,0,0,40,0,0,0,0,0,40,0,1,40,0,0,1,1,0,1,0,0,2,0,0,1],[40,39,0,0,0,0,1,1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,39,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,40,1,40,0,0,0,0,40,0,0,0,0,0,39,1] >;

C₅×D₄⋊₄D₄ in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes_4D_4

% in TeX

G:=Group("C5xD4:4D4");

// GroupNames label

G:=SmallGroup(320,954);

// by ID

G=gap.SmallGroup(320,954);

# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,7004,3511,1768,172,5052]);

// Polycyclic

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

// generators/relations

G = C5×D4⋊4D4 order 320 = 26·5

Direct product of C5 and D4⋊4D4

G = C₅×D₄⋊₄D₄ order 320 = 2⁶·5

Direct product of C₅ and D₄⋊₄D₄