Q16xF5 - GroupNames

G = Q₁₆×F₅ order 320 = 2⁶·5

Direct product of Q₁₆ and F₅

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

Aliases: Q₁₆×F₅, Dic₂₀⋊₆C₄, C₅⋊(C₄×Q₁₆), (C₅×Q₁₆)⋊₅C₄, C₅⋊Q₁₆⋊₃C₄, (C₈×F₅).₁C₂, C₂.₂₃(D₄×F₅), C₈.₁₅(C₂×F₅), Q₈.₃(C₂×F₅), (Q₈×F₅).₁C₂, C₄₀.₁₃(C₂×C₄), C₁₀.₂₂(C₄×D₄), (C₂×F₅).₁₂D₄, D₅.D₈.₂C₂, D₅.₂(C₂×Q₁₆), (D₅×Q₁₆).₅C₂, C₄⋊F₅.₅C₂², C₄.₉(C₂²×F₅), D₅.₄(C₄○D₈), D₁₀.₆₇(C₂×D₄), C₂₀.₉(C₂²×C₄), Q₈⋊F₅.₁C₂, (Q₈×D₅).₆C₂², D₅⋊C₈.₁₃C₂², Dic₁₀.₅(C₂×C₄), (C₈×D₅).₂₃C₂², (C₄×D₅).₃₁C₂³, (C₄×F₅).₁₃C₂², Dic₅.₅(C₄○D₄), (C₅×Q₈).₃(C₂×C₄), C₅⋊₂C₈.₁₁(C₂×C₄), SmallGroup(320,1076)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — Q₁₆×F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₄×F₅ — Q₈×F₅ — Q₁₆×F₅

Lower central

C₅ — C₁₀ — C₂₀ — Q₁₆×F₅

Upper central

C₁ — C₂ — C₄ — Q₁₆

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

Subgroups: 418 in 110 conjugacy classes, 44 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₈, C₂×C₄, Q₈, Q₈, D₅, C₁₀, C₄², C₄⋊C₄, C₂×C₈, Q₁₆, Q₁₆, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂₀, F₅, F₅, D₁₀, C₄×C₈, Q₈⋊C₄, C₂.D₈, C₄×Q₈, C₂×Q₁₆, C₅⋊₂C₈, C₄₀, C₅⋊C₈, Dic₁₀, Dic₁₀, C₄×D₅, C₄×D₅, C₅×Q₈, C₂×F₅, C₂×F₅, C₄×Q₁₆, C₈×D₅, Dic₂₀, C₅⋊Q₁₆, C₅×Q₁₆, D₅⋊C₈, C₄×F₅, C₄×F₅, C₄⋊F₅, C₄⋊F₅, Q₈×D₅, C₈×F₅, D₅.D₈, Q₈⋊F₅, D₅×Q₁₆, Q₈×F₅, Q₁₆×F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, Q₁₆, C₂²×C₄, C₂×D₄, C₄○D₄, F₅, C₄×D₄, C₂×Q₁₆, C₄○D₈, C₂×F₅, C₄×Q₁₆, C₂²×F₅, D₄×F₅, Q₁₆×F₅

Smallest permutation representation of Q₁₆×F₅
►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 38 21 34)(18 37 22 33)(19 36 23 40)(20 35 24 39)(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 34 77 66)(2 11 35 78 67)(3 12 36 79 68)(4 13 37 80 69)(5 14 38 73 70)(6 15 39 74 71)(7 16 40 75 72)(8 9 33 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)

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

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	8	8
type	+	+	+	+	+	+				+		-		+	+	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	C₄○D₄	Q₁₆	C₄○D₈	F₅	C₂×F₅	C₂×F₅	D₄×F₅	Q₁₆×F₅
kernel	Q₁₆×F₅	C₈×F₅	D₅.D₈	Q₈⋊F₅	D₅×Q₁₆	Q₈×F₅	Dic₂₀	C₅⋊Q₁₆	C₅×Q₁₆	C₂×F₅	Dic₅	F₅	D₅	Q₁₆	C₈	Q₈	C₂	C₁
# reps	1	1	1	2	1	2	2	4	2	2	2	4	4	1	1	2	1	2

Matrix representation of Q₁₆×F₅ ►in GL₆(𝔽₄₁)

0	17	0	0	0	0
12	17	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

9	0	0	0	0	0
9	32	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	0	0	0	40
0	0	1	0	0	40
0	0	0	1	0	40
0	0	0	0	1	40

40	0	0	0	0	0
0	40	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))| [0,12,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

Q₁₆×F₅ in GAP, Magma, Sage, TeX

Q_{16}\times F_5

% in TeX

G:=Group("Q16xF5");

// GroupNames label

G:=SmallGroup(320,1076);

// by ID

G=gap.SmallGroup(320,1076);

# by ID

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

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

// generators/relations

G = Q16×F5 order 320 = 26·5

Direct product of Q16 and F5

G = Q₁₆×F₅ order 320 = 2⁶·5

Direct product of Q₁₆ and F₅