Q16:5F5 - GroupNames

G = Q₁₆⋊₅F₅ order 320 = 2⁶·5

The semidirect product of Q₁₆ and F₅ acting through Inn(Q₁₆)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄₀⋊₆C₄, Q₁₆⋊₅F₅, C₅⋊C₈.₆D₄, C₅⋊₃(C₈○D₈), Q₈⋊D₅⋊₃C₄, (C₈×F₅)⋊₃C₂, (C₅×Q₁₆)⋊₆C₄, Q₈.F₅⋊₃C₂, C₈.₁₆(C₂×F₅), C₂.₂₅(D₄×F₅), Q₈.₅(C₂×F₅), C₄₀.₁₄(C₂×C₄), Q₈⋊₂F₅⋊₃C₂, D₂₀.₅(C₂×C₄), C₁₀.₂₄(C₄×D₄), D₁₀.Q₈⋊₄C₂, C₄.₁₁(C₂²×F₅), D₁₀.₅(C₄○D₄), Q₈.D₁₀.₅C₂, C₄.F₅.₅C₂², C₂₀.₁₁(C₂²×C₄), D₅⋊C₈.₁₄C₂², Dic₅.₇₆(C₂×D₄), (C₄×D₅).₃₃C₂³, (C₈×D₅).₂₄C₂², (C₄×F₅).₁₄C₂², Q₈⋊₂D₅.₆C₂², (C₅×Q₈).₅(C₂×C₄), C₅⋊₂C₈.₁₃(C₂×C₄), SmallGroup(320,1078)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — Q₁₆⋊₅F₅

Chief series

C₁ — C₅ — C₁₀ — Dic₅ — C₄×D₅ — D₅⋊C₈ — Q₈.F₅ — Q₁₆⋊₅F₅

Lower central

C₅ — C₁₀ — C₂₀ — Q₁₆⋊₅F₅

Upper central

C₁ — C₂ — C₄ — Q₁₆

Generators and relations for Q₁₆⋊₅F₅
G = < a,b,c,d | a⁸=c⁵=d⁴=1, b²=a⁴, bab^-1=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a²b, dcd^-1=c³ >

Subgroups: 418 in 106 conjugacy classes, 40 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₈, C₂×C₄, D₄, Q₈, D₅, C₁₀, C₄², C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₄×C₈, C₄≀C₂, C₈.C₄, C₈○D₄, C₄○D₈, C₅⋊₂C₈, C₄₀, C₅⋊C₈, C₅⋊C₈, C₄×D₅, C₄×D₅, D₂₀, D₂₀, C₅×Q₈, C₂×F₅, C₈○D₈, C₈×D₅, D₄₀, Q₈⋊D₅, C₅×Q₁₆, D₅⋊C₈, D₅⋊C₈, C₄.F₅, C₄.F₅, C₄×F₅, Q₈⋊₂D₅, C₈×F₅, D₁₀.Q₈, Q₈⋊₂F₅, Q₈.D₁₀, Q₈.F₅, Q₁₆⋊₅F₅
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, Q₁₆⋊₅F₅

Smallest permutation representation of Q₁₆⋊₅F₅
►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 60 5 64)(2 59 6 63)(3 58 7 62)(4 57 8 61)(9 30 13 26)(10 29 14 25)(11 28 15 32)(12 27 16 31)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(41 74 45 78)(42 73 46 77)(43 80 47 76)(44 79 48 75)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)

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

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

35 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	4	4	4	8	8
type	+	+	+	+	+	+				+			+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₄○D₄	C₈○D₈	F₅	C₂×F₅	C₂×F₅	D₄×F₅	Q₁₆⋊₅F₅
kernel	Q₁₆⋊₅F₅	C₈×F₅	D₁₀.Q₈	Q₈⋊₂F₅	Q₈.D₁₀	Q₈.F₅	D₄₀	Q₈⋊D₅	C₅×Q₁₆	C₅⋊C₈	D₁₀	C₅	Q₁₆	C₈	Q₈	C₂	C₁
# reps	1	1	1	2	1	2	2	4	2	2	2	8	1	1	2	1	2

Matrix representation of Q₁₆⋊₅F₅ ►in GL₆(𝔽₄₁)

27	0	0	0	0	0
26	38	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	39	0	0	0	0
1	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	40	40	40	40
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0

1	0	0	0	0	0
4	9	0	0	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	40	40	40	40

G:=sub<GL(6,GF(41))| [27,26,0,0,0,0,0,38,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,4,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

Q₁₆⋊₅F₅ in GAP, Magma, Sage, TeX

Q_{16}\rtimes_5F_5

% in TeX

G:=Group("Q16:5F5");

// GroupNames label

G:=SmallGroup(320,1078);

// by ID

G=gap.SmallGroup(320,1078);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,136,851,438,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^8=c^5=d^4=1,b^2=a^4,b*a*b^-1=a^-1,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 = Q16⋊5F5 order 320 = 26·5

The semidirect product of Q16 and F5 acting through Inn(Q16)

G = Q₁₆⋊₅F₅ order 320 = 2⁶·5

The semidirect product of Q₁₆ and F₅ acting through Inn(Q₁₆)