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G = F16⋊C2order 480 = 25·3·5

The semidirect product of F16 and C2 acting faithfully

non-abelian, soluble, monomial

Aliases: F16⋊C2, C24⋊C5⋊C6, C24⋊D5⋊C3, C22⋊A4⋊D5, C24⋊(C3×D5), SmallGroup(480,1188)

Series: Derived Chief Lower central Upper central

C1C24C24⋊C5 — F16⋊C2
C1C24C24⋊C5F16 — F16⋊C2
C24⋊C5 — F16⋊C2
C1

Generators and relations for F16⋊C2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, ab=ba, ac=ca, ad=da, eae-1=bcd, faf=abc, ebe-1=bc=cb, fbf=ede-1=bd=db, fcf=cd=dc, ece-1=a, df=fd, fef=e4 >

15C2
20C2
16C3
16C5
5C22
15C22
15C22
30C22
30C4
80C6
16D5
16C15
5C23
15C23
15C2×C4
30D4
30D4
20A4
16C3×D5
15C2×D4
15C22⋊C4
20C2×A4
5C22≀C2
5C24⋊C6

Character table of F16⋊C2

 class 12A2B3A3B45A5B6A6B15A15B15C15D
 size 115201616603232808032323232
ρ111111111111111    trivial
ρ211-111-111-1-11111    linear of order 2
ρ3111ζ3ζ32111ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ411-1ζ32ζ3-111ζ6ζ65ζ32ζ32ζ3ζ3    linear of order 6
ρ511-1ζ3ζ32-111ζ65ζ6ζ3ζ3ζ32ζ32    linear of order 6
ρ6111ζ32ζ3111ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ7220220-1-5/2-1+5/200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ8220220-1+5/2-1-5/200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ9220-1--3-1+-30-1-5/2-1+5/200ζ32ζ5432ζ5ζ32ζ5332ζ52ζ3ζ533ζ52ζ3ζ543ζ5    complex lifted from C3×D5
ρ10220-1+-3-1--30-1+5/2-1-5/200ζ3ζ533ζ52ζ3ζ543ζ5ζ32ζ5432ζ5ζ32ζ5332ζ52    complex lifted from C3×D5
ρ11220-1--3-1+-30-1+5/2-1-5/200ζ32ζ5332ζ52ζ32ζ5432ζ5ζ3ζ543ζ5ζ3ζ533ζ52    complex lifted from C3×D5
ρ12220-1+-3-1--30-1-5/2-1+5/200ζ3ζ543ζ5ζ3ζ533ζ52ζ32ζ5332ζ52ζ32ζ5432ζ5    complex lifted from C3×D5
ρ1315-1-300100000000    orthogonal faithful
ρ1415-1300-100000000    orthogonal faithful

Permutation representations of F16⋊C2
On 16 points: primitive, doubly transitive - transitive group 16T777
Generators in S16
(1 2)(3 14)(4 11)(5 6)(7 12)(8 10)(9 15)(13 16)
(1 7)(2 12)(3 6)(4 8)(5 14)(9 16)(10 11)(13 15)
(1 3)(2 14)(4 15)(5 12)(6 7)(8 13)(9 11)(10 16)
(1 11)(2 4)(3 9)(5 13)(6 16)(7 10)(8 12)(14 15)
(2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)

G:=sub<Sym(16)| (1,2)(3,14)(4,11)(5,6)(7,12)(8,10)(9,15)(13,16), (1,7)(2,12)(3,6)(4,8)(5,14)(9,16)(10,11)(13,15), (1,3)(2,14)(4,15)(5,12)(6,7)(8,13)(9,11)(10,16), (1,11)(2,4)(3,9)(5,13)(6,16)(7,10)(8,12)(14,15), (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)>;

G:=Group( (1,2)(3,14)(4,11)(5,6)(7,12)(8,10)(9,15)(13,16), (1,7)(2,12)(3,6)(4,8)(5,14)(9,16)(10,11)(13,15), (1,3)(2,14)(4,15)(5,12)(6,7)(8,13)(9,11)(10,16), (1,11)(2,4)(3,9)(5,13)(6,16)(7,10)(8,12)(14,15), (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15) );

G=PermutationGroup([(1,2),(3,14),(4,11),(5,6),(7,12),(8,10),(9,15),(13,16)], [(1,7),(2,12),(3,6),(4,8),(5,14),(9,16),(10,11),(13,15)], [(1,3),(2,14),(4,15),(5,12),(6,7),(8,13),(9,11),(10,16)], [(1,11),(2,4),(3,9),(5,13),(6,16),(7,10),(8,12),(14,15)], [(2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,5),(3,9),(4,13),(7,10),(8,14),(12,15)])

G:=TransitiveGroup(16,777);

On 20 points - transitive group 20T122
Generators in S20
(1 12)(3 9)(4 10)(5 16)(6 11)(7 17)(14 19)(15 20)
(1 17)(3 14)(4 15)(5 6)(7 12)(9 19)(10 20)(11 16)
(1 17)(2 13)(4 10)(5 11)(6 16)(7 12)(8 18)(15 20)
(2 18)(3 19)(4 10)(5 6)(8 13)(9 14)(11 16)(15 20)
(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)
(2 5)(3 4)(6 18)(8 11)(9 15)(10 19)(13 16)(14 20)

G:=sub<Sym(20)| (1,12)(3,9)(4,10)(5,16)(6,11)(7,17)(14,19)(15,20), (1,17)(3,14)(4,15)(5,6)(7,12)(9,19)(10,20)(11,16), (1,17)(2,13)(4,10)(5,11)(6,16)(7,12)(8,18)(15,20), (2,18)(3,19)(4,10)(5,6)(8,13)(9,14)(11,16)(15,20), (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (2,5)(3,4)(6,18)(8,11)(9,15)(10,19)(13,16)(14,20)>;

G:=Group( (1,12)(3,9)(4,10)(5,16)(6,11)(7,17)(14,19)(15,20), (1,17)(3,14)(4,15)(5,6)(7,12)(9,19)(10,20)(11,16), (1,17)(2,13)(4,10)(5,11)(6,16)(7,12)(8,18)(15,20), (2,18)(3,19)(4,10)(5,6)(8,13)(9,14)(11,16)(15,20), (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20), (2,5)(3,4)(6,18)(8,11)(9,15)(10,19)(13,16)(14,20) );

G=PermutationGroup([(1,12),(3,9),(4,10),(5,16),(6,11),(7,17),(14,19),(15,20)], [(1,17),(3,14),(4,15),(5,6),(7,12),(9,19),(10,20),(11,16)], [(1,17),(2,13),(4,10),(5,11),(6,16),(7,12),(8,18),(15,20)], [(2,18),(3,19),(4,10),(5,6),(8,13),(9,14),(11,16),(15,20)], [(1,2,3,4,5),(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)], [(2,5),(3,4),(6,18),(8,11),(9,15),(10,19),(13,16),(14,20)])

G:=TransitiveGroup(20,122);

On 30 points - transitive group 30T112
Generators in S30
(3 23)(4 24)(6 26)(8 28)(9 29)(10 30)(11 16)(15 20)
(1 21)(5 25)(8 28)(9 29)(11 16)(13 18)(14 19)(15 20)
(1 21)(4 24)(5 25)(7 27)(9 29)(10 30)(11 16)(12 17)
(2 22)(3 23)(4 24)(5 25)(9 29)(12 17)(13 18)(15 20)
(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)
(2 5)(3 9)(4 13)(7 10)(8 14)(12 15)(17 20)(18 24)(19 28)(22 25)(23 29)(27 30)

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

G:=Group( (3,23)(4,24)(6,26)(8,28)(9,29)(10,30)(11,16)(15,20), (1,21)(5,25)(8,28)(9,29)(11,16)(13,18)(14,19)(15,20), (1,21)(4,24)(5,25)(7,27)(9,29)(10,30)(11,16)(12,17), (2,22)(3,23)(4,24)(5,25)(9,29)(12,17)(13,18)(15,20), (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), (2,5)(3,9)(4,13)(7,10)(8,14)(12,15)(17,20)(18,24)(19,28)(22,25)(23,29)(27,30) );

G=PermutationGroup([(3,23),(4,24),(6,26),(8,28),(9,29),(10,30),(11,16),(15,20)], [(1,21),(5,25),(8,28),(9,29),(11,16),(13,18),(14,19),(15,20)], [(1,21),(4,24),(5,25),(7,27),(9,29),(10,30),(11,16),(12,17)], [(2,22),(3,23),(4,24),(5,25),(9,29),(12,17),(13,18),(15,20)], [(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)], [(2,5),(3,9),(4,13),(7,10),(8,14),(12,15),(17,20),(18,24),(19,28),(22,25),(23,29),(27,30)])

G:=TransitiveGroup(30,112);

On 30 points - transitive group 30T116
Generators in S30
(3 21)(4 22)(6 24)(8 26)(9 27)(10 28)(11 29)(15 18)
(1 19)(5 23)(8 26)(9 27)(11 29)(13 16)(14 17)(15 18)
(1 19)(4 22)(5 23)(7 25)(9 27)(10 28)(11 29)(12 30)
(2 20)(3 21)(4 22)(5 23)(9 27)(12 30)(13 16)(15 18)
(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)
(1 19)(2 23)(3 27)(4 16)(5 20)(6 24)(7 28)(8 17)(9 21)(10 25)(11 29)(12 18)(13 22)(14 26)(15 30)

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

G:=Group( (3,21)(4,22)(6,24)(8,26)(9,27)(10,28)(11,29)(15,18), (1,19)(5,23)(8,26)(9,27)(11,29)(13,16)(14,17)(15,18), (1,19)(4,22)(5,23)(7,25)(9,27)(10,28)(11,29)(12,30), (2,20)(3,21)(4,22)(5,23)(9,27)(12,30)(13,16)(15,18), (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), (1,19)(2,23)(3,27)(4,16)(5,20)(6,24)(7,28)(8,17)(9,21)(10,25)(11,29)(12,18)(13,22)(14,26)(15,30) );

G=PermutationGroup([(3,21),(4,22),(6,24),(8,26),(9,27),(10,28),(11,29),(15,18)], [(1,19),(5,23),(8,26),(9,27),(11,29),(13,16),(14,17),(15,18)], [(1,19),(4,22),(5,23),(7,25),(9,27),(10,28),(11,29),(12,30)], [(2,20),(3,21),(4,22),(5,23),(9,27),(12,30),(13,16),(15,18)], [(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)], [(1,19),(2,23),(3,27),(4,16),(5,20),(6,24),(7,28),(8,17),(9,21),(10,25),(11,29),(12,18),(13,22),(14,26),(15,30)])

G:=TransitiveGroup(30,116);

Matrix representation of F16⋊C2 in GL15(ℤ)

100000000000000
0-10000000000000
00-1000000000000
000100000000000
000010000000000
00000-1000000000
000000-100000000
0000000-10000000
00000000-1000000
000000000100000
0000000000-10000
000000000001000
000000000000100
0000000000000-10
000000000000001
,
-100000000000000
010000000000000
001000000000000
000-100000000000
000010000000000
000001000000000
000000-100000000
0000000-10000000
000000001000000
000000000100000
0000000000-10000
00000000000-1000
000000000000-100
0000000000000-10
000000000000001
,
-100000000000000
0-10000000000000
00-1000000000000
000100000000000
0000-10000000000
000001000000000
000000100000000
0000000-10000000
000000001000000
000000000-100000
0000000000-10000
00000000000-1000
000000000000100
000000000000010
000000000000001
,
100000000000000
010000000000000
00-1000000000000
000-100000000000
000010000000000
000001000000000
000000-100000000
0000000-10000000
00000000-1000000
000000000-100000
000000000010000
00000000000-1000
000000000000100
000000000000010
00000000000000-1
,
000000100000000
000000010000000
000000001000000
000000000100000
000001000000000
000000000001000
000000000000100
000000000000010
000000000000001
000000000010000
010000000000000
001000000000000
000100000000000
000010000000000
100000000000000
,
-100000000000000
0000-10000000000
000-100000000000
00-1000000000000
0-10000000000000
00000-1000000000
000000000-100000
00000000-1000000
0000000-10000000
000000-100000000
0000000000-10000
00000000000000-1
0000000000000-10
000000000000-100
00000000000-1000

G:=sub<GL(15,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0] >;

F16⋊C2 in GAP, Magma, Sage, TeX

F_{16}\rtimes C_2
% in TeX

G:=Group("F16:C2");
// GroupNames label

G:=SmallGroup(480,1188);
// by ID

G=gap.SmallGroup(480,1188);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,2,2,506,2523,4210,717,6836,1768,13865,1902,2749,7356,4423,755]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=b*c*d,f*a*f=a*b*c,e*b*e^-1=b*c=c*b,f*b*f=e*d*e^-1=b*d=d*b,f*c*f=c*d=d*c,e*c*e^-1=a,d*f=f*d,f*e*f=e^4>;
// generators/relations

Export

Subgroup lattice of F16⋊C2 in TeX
Character table of F16⋊C2 in TeX

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