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G = C2×F16  order 480 = 25·3·5

Direct product of C2 and F16

direct product, metabelian, soluble, monomial, A-group

Aliases: C2×F16, C25⋊C15, C24⋊C30, C22⋊A4⋊C10, C24⋊C5⋊2C6, (C2×C24⋊C5)⋊C3, (C2×C22⋊A4)⋊C5, SmallGroup(480,1190)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C2×F16
Chief series
C1 — C24 — C24⋊C5 — F16 — C2×F16
Lower central
C24 — C2×F16

Subgroups: 516 in 44 conjugacy classes, 10 normal (all characteristic)
C1, C2, C2 [×2], C3, C22 [×11], C5, C6, C23 [×11], C10, A4, C15, C24, C24 [×2], C2×A4, C30, C25, C22⋊A4, C24⋊C5, C2×C22⋊A4, C2×C24⋊C5, F16, C2×F16

Quotients:
C1, C2, C3, C5, C6, C10, C15, C30, F16, C2×F16

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f15=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=cde, fcf-1=cd=dc, fef-1=ce=ec, de=ed, fdf-1=b >

Permutation representations
►On 30 points - transitive group 30T108
Generators in S30
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 16)(15 17)

(1 18)(3 20)(5 22)(6 23)(7 24)(8 25)(12 29)(15 17)

(2 19)(5 22)(6 23)(8 25)(10 27)(11 28)(12 29)(13 30)

(1 18)(2 19)(4 21)(6 23)(7 24)(8 25)(9 26)(13 30)

(1 18)(2 19)(6 23)(9 26)(10 27)(12 29)(14 16)(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)
 
G:=sub<Sym(30)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,16)(15,17), (1,18)(3,20)(5,22)(6,23)(7,24)(8,25)(12,29)(15,17), (2,19)(5,22)(6,23)(8,25)(10,27)(11,28)(12,29)(13,30), (1,18)(2,19)(4,21)(6,23)(7,24)(8,25)(9,26)(13,30), (1,18)(2,19)(6,23)(9,26)(10,27)(12,29)(14,16)(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)>;
 
G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,16)(15,17), (1,18)(3,20)(5,22)(6,23)(7,24)(8,25)(12,29)(15,17), (2,19)(5,22)(6,23)(8,25)(10,27)(11,28)(12,29)(13,30), (1,18)(2,19)(4,21)(6,23)(7,24)(8,25)(9,26)(13,30), (1,18)(2,19)(6,23)(9,26)(10,27)(12,29)(14,16)(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) );
 
G=PermutationGroup([(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,16),(15,17)], [(1,18),(3,20),(5,22),(6,23),(7,24),(8,25),(12,29),(15,17)], [(2,19),(5,22),(6,23),(8,25),(10,27),(11,28),(12,29),(13,30)], [(1,18),(2,19),(4,21),(6,23),(7,24),(8,25),(9,26),(13,30)], [(1,18),(2,19),(6,23),(9,26),(10,27),(12,29),(14,16),(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)])
 
G:=TransitiveGroup(30,108);
 

Matrix representation ►G ⊆ GL15(ℤ)

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

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],[-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],[0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0] >;
 

32 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D6A6B10A10B10C10D15A···15H30A···30H
order1222335555661010101015···1530···30
size11151516161616161616161616161616···1616···16

32 irreducible representations

dim111111111515
type++++
imageC1C2C3C5C6C10C15C30F16C2×F16
kernelC2×F16F16C2×C24⋊C5C2×C22⋊A4C24⋊C5C22⋊A4C25C24C2C1
# reps1124248811

In GAP, Magma, Sage, TeX 

C_2\times F_{16}
 
% in TeX
 
G:=Group("C2xF16");
 
// GroupNames label
 
G:=SmallGroup(480,1190);
 
// by ID
 
G=gap.SmallGroup(480,1190);
 
# by ID
 
G:=PCGroup([7,-2,-3,-5,-2,2,2,2,4630,2117,7886,1243,4422,2959,10303,1245]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^15=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=c*d*e,f*c*f^-1=c*d=d*c,f*e*f^-1=c*e=e*c,d*e=e*d,f*d*f^-1=b>;
 
// generators/relations
 

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