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G = C22⋊F9  order 288 = 25·32

The semidirect product of C22 and F9 acting via F9/C32⋊C4=C2

metabelian, soluble, monomial

Aliases: C22⋊F9, C62⋊1C8, (C2×F9)⋊2C2, C2.7(C2×F9), C32⋊C4.6D4, C32⋊2(C22⋊C8), C3⋊S3.5M4(2), (C2×C3⋊S3)⋊1C8, (C3×C6).7(C2×C8), (C2×C32⋊C4).6C4, (C22×C3⋊S3).4C4, C3⋊S3.3(C22⋊C4), (C22×C32⋊C4).4C2, (C2×C32⋊C4).11C22, (C2×C3⋊S3).3(C2×C4), SmallGroup(288,867)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C22⋊F9
Chief series
C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C2×F9 — C22⋊F9
Lower central
C32 — C3×C6 — C22⋊F9
Upper central
C1 — C2 — C22

Generators and relations for C22⋊F9
 G = < a,b,c,d,e | a2=b2=c3=d3=e8=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 452 in 66 conjugacy classes, 19 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C23, C32, D6, C2×C6, C2×C8, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22⋊C8, C32⋊C4, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, F9, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C2×F9, C22×C32⋊C4, C22⋊F9
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, F9, C2×F9, C22⋊F9

Character table of C22⋊F9

 class 12A2B2C2D2E34A4B4C4D4E4F6A6B6C8A8B8C8D8E8F8G8H
 size 11299188999918188881818181818181818
ρ1111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311-111-111111-1-1-11-1-1-1-1-11111    linear of order 2
ρ411-111-111111-1-1-11-11111-1-1-1-1    linear of order 2
ρ511-111-11-1-1-1-111-11-1ii-i-i-i-iii    linear of order 4
ρ611-111-11-1-1-1-111-11-1-i-iiiii-i-i    linear of order 4
ρ71111111-1-1-1-1-1-1111-i-iii-i-iii    linear of order 4
ρ81111111-1-1-1-1-1-1111ii-i-iii-i-i    linear of order 4
ρ911-1-1-111-iii-i-ii-11-1ζ87ζ83ζ85ζ8ζ83ζ87ζ8ζ85    linear of order 8
ρ1011-1-1-111i-i-iii-i-11-1ζ85ζ8ζ87ζ83ζ8ζ85ζ83ζ87    linear of order 8
ρ1111-1-1-111-iii-i-ii-11-1ζ83ζ87ζ8ζ85ζ87ζ83ζ85ζ8    linear of order 8
ρ1211-1-1-111i-i-iii-i-11-1ζ8ζ85ζ83ζ87ζ85ζ8ζ87ζ83    linear of order 8
ρ13111-1-1-11i-i-ii-ii111ζ8ζ85ζ83ζ87ζ8ζ85ζ83ζ87    linear of order 8
ρ14111-1-1-11-iii-ii-i111ζ83ζ87ζ8ζ85ζ83ζ87ζ8ζ85    linear of order 8
ρ15111-1-1-11-iii-ii-i111ζ87ζ83ζ85ζ8ζ87ζ83ζ85ζ8    linear of order 8
ρ16111-1-1-11i-i-ii-ii111ζ85ζ8ζ87ζ83ζ85ζ8ζ87ζ83    linear of order 8
ρ172-20-2202-2-222000-2000000000    orthogonal lifted from D4
ρ182-20-220222-2-2000-2000000000    orthogonal lifted from D4
ρ192-202-202-2i2i-2i2i000-2000000000    complex lifted from M4(2)
ρ202-202-2022i-2i2i-2i000-2000000000    complex lifted from M4(2)
ρ21888000-1000000-1-1-100000000    orthogonal lifted from F9
ρ228-80000-100000031-300000000    orthogonal faithful
ρ2388-8000-10000001-1100000000    orthogonal lifted from C2×F9
ρ248-80000-1000000-31300000000    orthogonal faithful

Permutation representations of C22⋊F9
►On 24 points - transitive group 24T630
Generators in S24
(2 5)(4 7)(10 19)(12 21)(14 23)(16 17)

(1 8)(2 5)(3 6)(4 7)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)

(1 15 11)(3 9 13)(4 10 14)(6 18 22)(7 19 23)(8 24 20)

(1 11 15)(2 16 12)(4 10 14)(5 17 21)(7 19 23)(8 20 24)

(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
 
G:=sub<Sym(24)| (2,5)(4,7)(10,19)(12,21)(14,23)(16,17), (1,8)(2,5)(3,6)(4,7)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,15,11)(3,9,13)(4,10,14)(6,18,22)(7,19,23)(8,24,20), (1,11,15)(2,16,12)(4,10,14)(5,17,21)(7,19,23)(8,20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
 
G:=Group( (2,5)(4,7)(10,19)(12,21)(14,23)(16,17), (1,8)(2,5)(3,6)(4,7)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,15,11)(3,9,13)(4,10,14)(6,18,22)(7,19,23)(8,24,20), (1,11,15)(2,16,12)(4,10,14)(5,17,21)(7,19,23)(8,20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
 
G=PermutationGroup([[(2,5),(4,7),(10,19),(12,21),(14,23),(16,17)], [(1,8),(2,5),(3,6),(4,7),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(1,15,11),(3,9,13),(4,10,14),(6,18,22),(7,19,23),(8,24,20)], [(1,11,15),(2,16,12),(4,10,14),(5,17,21),(7,19,23),(8,20,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,630);
 

►On 24 points - transitive group 24T631
Generators in S24
(2 6)(4 8)(9 22)(11 24)(13 18)(15 20)

(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)

(2 13 22)(3 14 23)(4 24 15)(6 18 9)(7 19 10)(8 11 20)

(1 12 21)(3 14 23)(4 15 24)(5 17 16)(7 19 10)(8 20 11)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
 
G:=sub<Sym(24)| (2,6)(4,8)(9,22)(11,24)(13,18)(15,20), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (1,12,21)(3,14,23)(4,15,24)(5,17,16)(7,19,10)(8,20,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
 
G:=Group( (2,6)(4,8)(9,22)(11,24)(13,18)(15,20), (1,5)(2,6)(3,7)(4,8)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (1,12,21)(3,14,23)(4,15,24)(5,17,16)(7,19,10)(8,20,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
 
G=PermutationGroup([[(2,6),(4,8),(9,22),(11,24),(13,18),(15,20)], [(1,5),(2,6),(3,7),(4,8),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(2,13,22),(3,14,23),(4,24,15),(6,18,9),(7,19,10),(8,11,20)], [(1,12,21),(3,14,23),(4,15,24),(5,17,16),(7,19,10),(8,20,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,631);
 

Matrix representation of C22⋊F9 ►in GL8(ℤ)

10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
01000000
-1-1000000
00010000
00-1-10000
00001000
00000100
000000-1-1
00000010
,
10000000
01000000
00-1-10000
00100000
00000100
0000-1-100
000000-1-1
00000010
,
00001000
00000100
00000010
00000001
00100000
00010000
-10000000
11000000

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C22⋊F9 in GAP, Magma, Sage, TeX 

C_2^2\rtimes F_9
 
% in TeX
 
G:=Group("C2^2:F9");
 
// GroupNames label
 
G:=SmallGroup(288,867);
 
// by ID
 
G=gap.SmallGroup(288,867);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,100,4037,2371,201,10982,3156,622]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=e^8=1,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

Export

Character table of C22⋊F9 in TeX

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