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G = F16⋊C2  order 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

Derived series
C1 — C24 — C24⋊C5 — F16⋊C2
Chief series
C1 — C24 — C24⋊C5 — F16 — F16⋊C2

Subgroups: 472 in 31 conjugacy classes, 7 normal (all characteristic)
C1, C2 [×2], C3, C4, C22 [×4], C5, C6, C2×C4, D4 [×2], C23 [×2], D5, A4, C15, C22⋊C4, C2×D4, C24, C2×A4, C3×D5, C22≀C2, C22⋊A4, C24⋊C5, C24⋊C6, C24⋊D5, F16, F16⋊C2

Quotients:
C1, C2, C3, C6, D5, C3×D5, F16⋊C2

Generators and relations
 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 >

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

(1 3)(2 14)(4 15)(5 12)(6 7)(8 13)(9 11)(10 16)

(1 14)(2 3)(4 9)(5 7)(6 12)(8 16)(10 13)(11 15)

(1 7)(2 12)(3 6)(4 8)(5 14)(9 16)(10 11)(13 15)

(2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(3 6)(4 10)(5 14)(8 11)(9 15)(13 16)
 
G:=sub<Sym(16)| (1,13)(2,16)(3,8)(4,6)(5,11)(7,15)(9,12)(10,14), (1,3)(2,14)(4,15)(5,12)(6,7)(8,13)(9,11)(10,16), (1,14)(2,3)(4,9)(5,7)(6,12)(8,16)(10,13)(11,15), (1,7)(2,12)(3,6)(4,8)(5,14)(9,16)(10,11)(13,15), (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (3,6)(4,10)(5,14)(8,11)(9,15)(13,16)>;
 
G:=Group( (1,13)(2,16)(3,8)(4,6)(5,11)(7,15)(9,12)(10,14), (1,3)(2,14)(4,15)(5,12)(6,7)(8,13)(9,11)(10,16), (1,14)(2,3)(4,9)(5,7)(6,12)(8,16)(10,13)(11,15), (1,7)(2,12)(3,6)(4,8)(5,14)(9,16)(10,11)(13,15), (2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (3,6)(4,10)(5,14)(8,11)(9,15)(13,16) );
 
G=PermutationGroup([(1,13),(2,16),(3,8),(4,6),(5,11),(7,15),(9,12),(10,14)], [(1,3),(2,14),(4,15),(5,12),(6,7),(8,13),(9,11),(10,16)], [(1,14),(2,3),(4,9),(5,7),(6,12),(8,16),(10,13),(11,15)], [(1,7),(2,12),(3,6),(4,8),(5,14),(9,16),(10,11),(13,15)], [(2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(3,6),(4,10),(5,14),(8,11),(9,15),(13,16)])
 
G:=TransitiveGroup(16,777);
 

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

(1 19)(2 20)(3 11)(4 7)(6 16)(9 14)(10 15)(12 17)

(2 15)(3 16)(4 7)(5 18)(6 11)(8 13)(10 20)(12 17)

(1 9)(2 15)(3 11)(5 8)(6 16)(10 20)(13 18)(14 19)

(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)

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

(1 18)(5 22)(8 25)(9 26)(11 28)(13 30)(14 16)(15 17)

(1 18)(4 21)(5 22)(7 24)(9 26)(10 27)(11 28)(12 29)

(2 19)(3 20)(4 21)(5 22)(9 26)(12 29)(13 30)(15 17)

(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)(16 25)(17 29)(19 22)(20 26)(21 30)(24 27)
 
G:=sub<Sym(30)| (3,20)(4,21)(6,23)(8,25)(9,26)(10,27)(11,28)(15,17), (1,18)(5,22)(8,25)(9,26)(11,28)(13,30)(14,16)(15,17), (1,18)(4,21)(5,22)(7,24)(9,26)(10,27)(11,28)(12,29), (2,19)(3,20)(4,21)(5,22)(9,26)(12,29)(13,30)(15,17), (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)(16,25)(17,29)(19,22)(20,26)(21,30)(24,27)>;
 
G:=Group( (3,20)(4,21)(6,23)(8,25)(9,26)(10,27)(11,28)(15,17), (1,18)(5,22)(8,25)(9,26)(11,28)(13,30)(14,16)(15,17), (1,18)(4,21)(5,22)(7,24)(9,26)(10,27)(11,28)(12,29), (2,19)(3,20)(4,21)(5,22)(9,26)(12,29)(13,30)(15,17), (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)(16,25)(17,29)(19,22)(20,26)(21,30)(24,27) );
 
G=PermutationGroup([(3,20),(4,21),(6,23),(8,25),(9,26),(10,27),(11,28),(15,17)], [(1,18),(5,22),(8,25),(9,26),(11,28),(13,30),(14,16),(15,17)], [(1,18),(4,21),(5,22),(7,24),(9,26),(10,27),(11,28),(12,29)], [(2,19),(3,20),(4,21),(5,22),(9,26),(12,29),(13,30),(15,17)], [(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),(16,25),(17,29),(19,22),(20,26),(21,30),(24,27)])
 
G:=TransitiveGroup(30,112);
 

►On 30 points - transitive group 30T116
Generators in S30
(3 16)(4 17)(6 19)(8 21)(9 22)(10 23)(11 24)(15 28)

(1 29)(5 18)(8 21)(9 22)(11 24)(13 26)(14 27)(15 28)

(1 29)(4 17)(5 18)(7 20)(9 22)(10 23)(11 24)(12 25)

(2 30)(3 16)(4 17)(5 18)(9 22)(12 25)(13 26)(15 28)

(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 29)(2 18)(3 22)(4 26)(5 30)(6 19)(7 23)(8 27)(9 16)(10 20)(11 24)(12 28)(13 17)(14 21)(15 25)
 
G:=sub<Sym(30)| (3,16)(4,17)(6,19)(8,21)(9,22)(10,23)(11,24)(15,28), (1,29)(5,18)(8,21)(9,22)(11,24)(13,26)(14,27)(15,28), (1,29)(4,17)(5,18)(7,20)(9,22)(10,23)(11,24)(12,25), (2,30)(3,16)(4,17)(5,18)(9,22)(12,25)(13,26)(15,28), (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), (1,29)(2,18)(3,22)(4,26)(5,30)(6,19)(7,23)(8,27)(9,16)(10,20)(11,24)(12,28)(13,17)(14,21)(15,25)>;
 
G:=Group( (3,16)(4,17)(6,19)(8,21)(9,22)(10,23)(11,24)(15,28), (1,29)(5,18)(8,21)(9,22)(11,24)(13,26)(14,27)(15,28), (1,29)(4,17)(5,18)(7,20)(9,22)(10,23)(11,24)(12,25), (2,30)(3,16)(4,17)(5,18)(9,22)(12,25)(13,26)(15,28), (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), (1,29)(2,18)(3,22)(4,26)(5,30)(6,19)(7,23)(8,27)(9,16)(10,20)(11,24)(12,28)(13,17)(14,21)(15,25) );
 
G=PermutationGroup([(3,16),(4,17),(6,19),(8,21),(9,22),(10,23),(11,24),(15,28)], [(1,29),(5,18),(8,21),(9,22),(11,24),(13,26),(14,27),(15,28)], [(1,29),(4,17),(5,18),(7,20),(9,22),(10,23),(11,24),(12,25)], [(2,30),(3,16),(4,17),(5,18),(9,22),(12,25),(13,26),(15,28)], [(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)], [(1,29),(2,18),(3,22),(4,26),(5,30),(6,19),(7,23),(8,27),(9,16),(10,20),(11,24),(12,28),(13,17),(14,21),(15,25)])
 
G:=TransitiveGroup(30,116);
 

Matrix representation ►G ⊆ 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] >;
 

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+√5/200ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5    complex lifted from C3×D5
ρ10220-1+√-3-1-√-30-1+√5/2-1-√5/200ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52    complex lifted from C3×D5
ρ11220-1-√-3-1+√-30-1+√5/2-1-√5/200ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52    complex lifted from C3×D5
ρ12220-1+√-3-1-√-30-1-√5/2-1+√5/200ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5    complex lifted from C3×D5
ρ1315-1-300100000000    orthogonal faithful
ρ1415-1300-100000000    orthogonal faithful

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
 

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