SD16xF5

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

Aliases: SD₁₆×F₅, C₅⋊(C₄×SD₁₆), C₈⋊₅(C₂×F₅), C₄₀⋊₄(C₂×C₄), Q₈⋊D₅⋊₁C₄, (C₈×F₅)⋊₄C₂, (Q₈×F₅)⋊₁C₂, Q₈⋊₁(C₂×F₅), C₄₀⋊C₂⋊₃C₄, D₄.D₅⋊₃C₄, C₄₀⋊C₄⋊₄C₂, D₄.₃(C₂×F₅), (D₄×F₅).₁C₂, C₂.₁₉(D₄×F₅), Q₈⋊F₅⋊₁C₂, (C₅×SD₁₆)⋊₃C₄, D₂₀.₁(C₂×C₄), C₁₀.₁₈(C₄×D₄), (C₂×F₅).₁₁D₄, C₄⋊F₅.₃C₂², C₄.₅(C₂²×F₅), Dic₁₀⋊₁(C₂×C₄), D₅.₃(C₄○D₈), D₁₀.₆₅(C₂×D₄), D₂₀⋊C₄.₁C₂, C₂₀.₅(C₂²×C₄), D₅.₂(C₂×SD₁₆), (D₅×SD₁₆).₂C₂, (D₄×D₅).₇C₂², (Q₈×D₅).₄C₂², D₅⋊C₈.₁₁C₂², (C₄×D₅).₂₇C₂³, (C₈×D₅).₂₇C₂², (C₄×F₅).₁₁C₂², Dic₅.₃(C₄○D₄), (C₅×Q₈)⋊₁(C₂×C₄), C₅⋊₂C₈⋊₁₃(C₂×C₄), (C₅×D₄).₃(C₂×C₄), SmallGroup(320,1072)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — SD₁₆×F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₄×F₅ — D₄×F₅ — SD₁₆×F₅

Lower central

C₅ — C₁₀ — C₂₀ — SD₁₆×F₅

Upper central

C₁ — C₂ — C₄ — SD₁₆

Subgroups: 546 in 122 conjugacy classes, 44 normal (42 characteristic)
C₁, C₂, C₂^[×4], C₄, C₄^[×8], C₂²^[×5], C₅, C₈, C₈^[×2], C₂×C₄^[×8], D₄, D₄^[×2], Q₈, Q₈^[×2], C₂³, D₅^[×2], D₅, C₁₀, C₁₀, C₄²^[×2], C₂²⋊C₄, C₄⋊C₄^[×3], C₂×C₈^[×2], SD₁₆, SD₁₆^[×3], C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂₀, F₅^[×2], F₅^[×3], D₁₀, D₁₀^[×3], C₂×C₁₀, C₄×C₈, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₄×D₄, C₄×Q₈, C₂×SD₁₆, C₅⋊₂C₈, C₄₀, C₅⋊C₈, Dic₁₀, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, C₅⋊D₄, C₅×D₄, C₅×Q₈, C₂×F₅^[×2], C₂×F₅^[×4], C₂²×D₅, C₄×SD₁₆, C₈×D₅, C₄₀⋊C₂, D₄.D₅, Q₈⋊D₅, C₅×SD₁₆, D₅⋊C₈, C₄×F₅, C₄×F₅, C₄⋊F₅^[×2], C₄⋊F₅, C₂²⋊F₅, D₄×D₅, Q₈×D₅, C₂²×F₅, C₈×F₅, C₄₀⋊C₄, D₂₀⋊C₄, Q₈⋊F₅, D₅×SD₁₆, D₄×F₅, Q₈×F₅, SD₁₆×F₅

Quotients:
C₁, C₂^[×7], C₄^[×4], C₂²^[×7], C₂×C₄^[×6], D₄^[×2], C₂³, SD₁₆^[×2], C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×SD₁₆, C₄○D₈, C₂×F₅^[×3], C₄×SD₁₆, C₂²×F₅, D₄×F₅, SD₁₆×F₅

Generators and relations
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=a³, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c³ >

Smallest permutation representation
►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)

(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₄₁)

15	15	0	0	0	0
26	15	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

40	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	40
0	0	1	0	0	40
0	0	0	1	0	40
0	0	0	0	1	40

32	0	0	0	0	0
0	32	0	0	0	0
0	0	0	0	1	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(41))| [15,26,0,0,0,0,15,15,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

35 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	···	4N	5	8A	8B	8C	···	8H	10A	10B	20A	20B	40A	40B
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	···	4	5	8	8	8	···	8	10	10	20	20	40	40
size	1	1	4	5	5	20	2	4	5	5	5	5	10	10	10	20	···	20	4	2	2	10	···	10	4	16	8	16	8	8

35 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4	8	8
type	+	+	+	+	+	+	+	+					+				+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	C₄○D₄	SD₁₆	C₄○D₈	F₅	C₂×F₅	C₂×F₅	C₂×F₅	D₄×F₅	SD₁₆×F₅
kernel	SD₁₆×F₅	C₈×F₅	C₄₀⋊C₄	D₂₀⋊C₄	Q₈⋊F₅	D₅×SD₁₆	D₄×F₅	Q₈×F₅	C₄₀⋊C₂	D₄.D₅	Q₈⋊D₅	C₅×SD₁₆	C₂×F₅	Dic₅	F₅	D₅	SD₁₆	C₈	D₄	Q₈	C₂	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	1	1	1	1	1	2

In GAP, Magma, Sage, TeX

SD_{16}\times F_5

% in TeX

G:=Group("SD16xF5");

// GroupNames label

G:=SmallGroup(320,1072);

// by ID

G=gap.SmallGroup(320,1072);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,851,438,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;

// generators/relations

G = SD₁₆×F₅ order 320 = 2⁶·5

Direct product of SD₁₆ and F₅

G = SD16×F5 order 320 = 26·5

Direct product of SD16 and F5

G = SD₁₆×F₅ order 320 = 2⁶·5

Direct product of SD₁₆ and F₅