SD16:3F5 - GroupNames

G = SD₁₆⋊₃F₅ order 320 = 2⁶·5

The semidirect product of SD₁₆ and F₅ acting through Inn(SD₁₆)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD₁₆⋊₃F₅, C₅⋊C₈.₅D₄, D₄⋊D₅⋊₃C₄, C₅⋊₂(C₈○D₈), (C₈×F₅)⋊₅C₂, C₄₀⋊C₂⋊₄C₄, C₅⋊Q₁₆⋊₁C₄, D₄.F₅⋊₃C₂, Q₈.F₅⋊₁C₂, D₄.₅(C₂×F₅), C₈.₁₇(C₂×F₅), C₂.₂₁(D₄×F₅), Q₈.₁(C₂×F₅), C₄₀.₁₆(C₂×C₄), D₄⋊F₅⋊₃C₂, Q₈⋊₂F₅⋊₁C₂, (C₅×SD₁₆)⋊₄C₄, D₂₀.₃(C₂×C₄), C₁₀.₂₀(C₄×D₄), C₄₀.C₄⋊₄C₂, C₄.₇(C₂²×F₅), C₂₀.₇(C₂²×C₄), D₁₀.₃(C₄○D₄), C₄.F₅.₃C₂², D₅⋊C₈.₁₂C₂², Dic₁₀.₃(C₂×C₄), Dic₅.₇₄(C₂×D₄), (C₈×D₅).₂₈C₂², (C₄×D₅).₂₉C₂³, (C₄×F₅).₁₂C₂², SD₁₆⋊₃D₅.₂C₂, D₄⋊₂D₅.₆C₂², Q₈⋊₂D₅.₄C₂², C₅⋊₂C₈.₉(C₂×C₄), (C₅×D₄).₅(C₂×C₄), (C₅×Q₈).₁(C₂×C₄), SmallGroup(320,1074)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — SD₁₆⋊₃F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₄×D₅ — D₅⋊C₈ — D₄.F₅ — SD₁₆⋊₃F₅

Lower central

C₅ — C₁₀ — C₂₀ — SD₁₆⋊₃F₅

Upper central

C₁ — C₂ — C₄ — SD₁₆

Generators and relations for SD₁₆⋊₃F₅
G = < a,b,c,d | a⁸=b²=c⁵=d⁴=1, bab=a³, ac=ca, ad=da, bc=cb, dbd^-1=a²b, dcd^-1=c³ >

Subgroups: 402 in 106 conjugacy classes, 40 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₈, C₂×C₄, D₄, D₄, Q₈, Q₈, D₅, C₁₀, C₁₀, C₄², C₂×C₈, M₄(2), D₈, SD₁₆, SD₁₆, Q₁₆, C₄○D₄, Dic₅, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₄×C₈, C₄≀C₂, C₈.C₄, C₈○D₄, C₄○D₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₅⋊C₈, Dic₁₀, C₄×D₅, C₄×D₅, D₂₀, D₂₀, C₂×Dic₅, C₅⋊D₄, C₅×D₄, C₅×Q₈, C₂×F₅, C₈○D₈, C₈×D₅, C₄₀⋊C₂, D₄⋊D₅, C₅⋊Q₁₆, C₅×SD₁₆, D₅⋊C₈, D₅⋊C₈, C₄.F₅, C₄.F₅, C₄×F₅, C₂×C₅⋊C₈, C₂².F₅, D₄⋊₂D₅, Q₈⋊₂D₅, C₈×F₅, C₄₀.C₄, D₄⋊F₅, Q₈⋊₂F₅, SD₁₆⋊₃D₅, D₄.F₅, Q₈.F₅, SD₁₆⋊₃F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×F₅, C₈○D₈, C₂²×F₅, D₄×F₅, SD₁₆⋊₃F₅

Smallest permutation representation of SD₁₆⋊₃F₅
►On 80 points

Generators in S₈₀

(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 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)

(1 59)(2 62)(3 57)(4 60)(5 63)(6 58)(7 61)(8 64)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 36)(18 39)(19 34)(20 37)(21 40)(22 35)(23 38)(24 33)(41 77)(42 80)(43 75)(44 78)(45 73)(46 76)(47 79)(48 74)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)

(1 12 33 79 71)(2 13 34 80 72)(3 14 35 73 65)(4 15 36 74 66)(5 16 37 75 67)(6 9 38 76 68)(7 10 39 77 69)(8 11 40 78 70)(17 48 56 60 30)(18 41 49 61 31)(19 42 50 62 32)(20 43 51 63 25)(21 44 52 64 26)(22 45 53 57 27)(23 46 54 58 28)(24 47 55 59 29)

(9 38 68 76)(10 39 69 77)(11 40 70 78)(12 33 71 79)(13 34 72 80)(14 35 65 73)(15 36 66 74)(16 37 67 75)(17 50 44 28)(18 51 45 29)(19 52 46 30)(20 53 47 31)(21 54 48 32)(22 55 41 25)(23 56 42 26)(24 49 43 27)(57 59 61 63)(58 60 62 64)

G:=sub<Sym(80)| (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)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,59)(2,62)(3,57)(4,60)(5,63)(6,58)(7,61)(8,64)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,36)(18,39)(19,34)(20,37)(21,40)(22,35)(23,38)(24,33)(41,77)(42,80)(43,75)(44,78)(45,73)(46,76)(47,79)(48,74)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,12,33,79,71)(2,13,34,80,72)(3,14,35,73,65)(4,15,36,74,66)(5,16,37,75,67)(6,9,38,76,68)(7,10,39,77,69)(8,11,40,78,70)(17,48,56,60,30)(18,41,49,61,31)(19,42,50,62,32)(20,43,51,63,25)(21,44,52,64,26)(22,45,53,57,27)(23,46,54,58,28)(24,47,55,59,29), (9,38,68,76)(10,39,69,77)(11,40,70,78)(12,33,71,79)(13,34,72,80)(14,35,65,73)(15,36,66,74)(16,37,67,75)(17,50,44,28)(18,51,45,29)(19,52,46,30)(20,53,47,31)(21,54,48,32)(22,55,41,25)(23,56,42,26)(24,49,43,27)(57,59,61,63)(58,60,62,64)>;

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)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,59)(2,62)(3,57)(4,60)(5,63)(6,58)(7,61)(8,64)(9,28)(10,31)(11,26)(12,29)(13,32)(14,27)(15,30)(16,25)(17,36)(18,39)(19,34)(20,37)(21,40)(22,35)(23,38)(24,33)(41,77)(42,80)(43,75)(44,78)(45,73)(46,76)(47,79)(48,74)(49,69)(50,72)(51,67)(52,70)(53,65)(54,68)(55,71)(56,66), (1,12,33,79,71)(2,13,34,80,72)(3,14,35,73,65)(4,15,36,74,66)(5,16,37,75,67)(6,9,38,76,68)(7,10,39,77,69)(8,11,40,78,70)(17,48,56,60,30)(18,41,49,61,31)(19,42,50,62,32)(20,43,51,63,25)(21,44,52,64,26)(22,45,53,57,27)(23,46,54,58,28)(24,47,55,59,29), (9,38,68,76)(10,39,69,77)(11,40,70,78)(12,33,71,79)(13,34,72,80)(14,35,65,73)(15,36,66,74)(16,37,67,75)(17,50,44,28)(18,51,45,29)(19,52,46,30)(20,53,47,31)(21,54,48,32)(22,55,41,25)(23,56,42,26)(24,49,43,27)(57,59,61,63)(58,60,62,64) );

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),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,59),(2,62),(3,57),(4,60),(5,63),(6,58),(7,61),(8,64),(9,28),(10,31),(11,26),(12,29),(13,32),(14,27),(15,30),(16,25),(17,36),(18,39),(19,34),(20,37),(21,40),(22,35),(23,38),(24,33),(41,77),(42,80),(43,75),(44,78),(45,73),(46,76),(47,79),(48,74),(49,69),(50,72),(51,67),(52,70),(53,65),(54,68),(55,71),(56,66)], [(1,12,33,79,71),(2,13,34,80,72),(3,14,35,73,65),(4,15,36,74,66),(5,16,37,75,67),(6,9,38,76,68),(7,10,39,77,69),(8,11,40,78,70),(17,48,56,60,30),(18,41,49,61,31),(19,42,50,62,32),(20,43,51,63,25),(21,44,52,64,26),(22,45,53,57,27),(23,46,54,58,28),(24,47,55,59,29)], [(9,38,68,76),(10,39,69,77),(11,40,70,78),(12,33,71,79),(13,34,72,80),(14,35,65,73),(15,36,66,74),(16,37,67,75),(17,50,44,28),(18,51,45,29),(19,52,46,30),(20,53,47,31),(21,54,48,32),(22,55,41,25),(23,56,42,26),(24,49,43,27),(57,59,61,63),(58,60,62,64)]])

35 conjugacy classes

class	1	2A	2B	2C	2D	4A	4B	4C	4D	4E	4F	4G	4H	4I	5	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	8K	8L	8M	8N	10A	10B	20A	20B	40A	40B
order	1	2	2	2	2	4	4	4	4	4	4	4	4	4	5	8	8	8	8	8	8	8	8	8	8	8	8	8	8	10	10	20	20	40	40
size	1	1	4	10	20	2	4	5	5	10	10	10	10	20	4	2	2	5	5	5	5	10	10	10	10	20	20	20	20	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	4	4	4	4	8	8
type	+	+	+	+	+	+	+	+					+			+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	C₄○D₄	C₈○D₈	F₅	C₂×F₅	C₂×F₅	C₂×F₅	D₄×F₅	SD₁₆⋊₃F₅
kernel	SD₁₆⋊₃F₅	C₈×F₅	C₄₀.C₄	D₄⋊F₅	Q₈⋊₂F₅	SD₁₆⋊₃D₅	D₄.F₅	Q₈.F₅	C₄₀⋊C₂	D₄⋊D₅	C₅⋊Q₁₆	C₅×SD₁₆	C₅⋊C₈	D₁₀	C₅	SD₁₆	C₈	D₄	Q₈	C₂	C₁
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	2	8	1	1	1	1	1	2

Matrix representation of SD₁₆⋊₃F₅ ►in GL₆(𝔽₄₁)

3	0	0	0	0	0
0	27	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

0	38	0	0	0	0
27	0	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

1	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))| [3,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,27,0,0,0,0,38,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

SD₁₆⋊₃F₅ in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_3F_5

% in TeX

G:=Group("SD16:3F5");

// GroupNames label

G:=SmallGroup(320,1074);

// by ID

G=gap.SmallGroup(320,1074);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,136,851,438,102,6278,1595]);

// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;

// generators/relations

G = SD16⋊3F5 order 320 = 26·5

The semidirect product of SD16 and F5 acting through Inn(SD16)

G = SD₁₆⋊₃F₅ order 320 = 2⁶·5

The semidirect product of SD₁₆ and F₅ acting through Inn(SD₁₆)