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G = C4⋊F9order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: C4⋊F9, (C3×C12)⋊2C8, C2.5(C2×F9), C321(C4⋊C8), C3⋊Dic32C8, (C2×F9).2C2, C32⋊C4.4D4, C32⋊C4.2Q8, C3⋊S3.1M4(2), (C4×C3⋊S3).4C4, (C3×C6).4(C2×C8), C3⋊S3.2(C4⋊C4), (C4×C32⋊C4).7C2, (C2×C32⋊C4).5C4, (C2×C32⋊C4).10C22, (C2×C3⋊S3).2(C2×C4), SmallGroup(288,864)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C4⋊F9
C1C32C3⋊S3C32⋊C4C2×C32⋊C4C2×F9 — C4⋊F9
C32C3×C6 — C4⋊F9
C1C2C4

Generators and relations for C4⋊F9
 G = < a,b,c,d | a4=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >

9C2
9C2
4C3
9C4
9C4
9C22
9C4
18C4
4C6
12S3
12S3
9C2×C4
9C2×C4
9C2×C4
18C8
18C8
4C12
12D6
12Dic3
9C2×C8
9C2×C8
9C42
12C4×S3
2C32⋊C4
9C4⋊C8
2F9
2F9

Character table of C4⋊F9

 class 12A2B2C34A4B4C4D4E4F4G4H68A8B8C8D8E8F8G8H12A12B
 size 11998299991818188181818181818181888
ρ1111111111111111111111111    trivial
ρ211111-11111-1-1-11-1111-1-1-11-1-1    linear of order 2
ρ311111-11111-1-1-111-1-1-1111-1-1-1    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-111    linear of order 2
ρ5111111-1-1-1-11-1-11i-iii-i-ii-i11    linear of order 4
ρ611111-1-1-1-1-1-1111ii-i-i-i-iii-1-1    linear of order 4
ρ711111-1-1-1-1-1-1111-i-iiiii-i-i-1-1    linear of order 4
ρ8111111-1-1-1-11-1-11-ii-i-iii-ii11    linear of order 4
ρ911-1-111i-ii-i-1-ii1ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ8311    linear of order 8
ρ1011-1-11-1-ii-ii1-ii1ζ83ζ85ζ83ζ87ζ85ζ8ζ87ζ8-1-1    linear of order 8
ρ1111-1-11-1-ii-ii1-ii1ζ87ζ8ζ87ζ83ζ8ζ85ζ83ζ85-1-1    linear of order 8
ρ1211-1-11-1i-ii-i1i-i1ζ85ζ83ζ85ζ8ζ83ζ87ζ8ζ87-1-1    linear of order 8
ρ1311-1-11-1i-ii-i1i-i1ζ8ζ87ζ8ζ85ζ87ζ83ζ85ζ83-1-1    linear of order 8
ρ1411-1-111-ii-ii-1i-i1ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ8511    linear of order 8
ρ1511-1-111-ii-ii-1i-i1ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ811    linear of order 8
ρ1611-1-111i-ii-i-1-ii1ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ8711    linear of order 8
ρ172-22-2202-2-22000-20000000000    orthogonal lifted from D4
ρ182-22-220-222-2000-20000000000    symplectic lifted from Q8, Schur index 2
ρ192-2-2220-2i-2i2i2i000-20000000000    complex lifted from M4(2)
ρ202-2-22202i2i-2i-2i000-20000000000    complex lifted from M4(2)
ρ218800-180000000-100000000-1-1    orthogonal lifted from F9
ρ228800-1-80000000-10000000011    orthogonal lifted from C2×F9
ρ238-800-100000000100000000-3i3i    complex faithful
ρ248-800-1000000001000000003i-3i    complex faithful

Smallest permutation representation of C4⋊F9
On 36 points
Generators in S36
(1 4 2 3)(5 32 15 23)(6 24 16 33)(7 34 17 25)(8 26 18 35)(9 36 19 27)(10 28 20 29)(11 30 13 21)(12 22 14 31)
(1 10 6)(2 20 16)(3 29 33)(4 28 24)(5 7 12)(8 11 9)(13 19 18)(14 15 17)(21 27 35)(22 32 34)(23 25 31)(26 30 36)
(1 11 7)(2 13 17)(3 21 25)(4 30 34)(5 6 8)(9 12 10)(14 20 19)(15 16 18)(22 28 36)(23 33 35)(24 26 32)(27 31 29)
(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)

G:=sub<Sym(36)| (1,4,2,3)(5,32,15,23)(6,24,16,33)(7,34,17,25)(8,26,18,35)(9,36,19,27)(10,28,20,29)(11,30,13,21)(12,22,14,31), (1,10,6)(2,20,16)(3,29,33)(4,28,24)(5,7,12)(8,11,9)(13,19,18)(14,15,17)(21,27,35)(22,32,34)(23,25,31)(26,30,36), (1,11,7)(2,13,17)(3,21,25)(4,30,34)(5,6,8)(9,12,10)(14,20,19)(15,16,18)(22,28,36)(23,33,35)(24,26,32)(27,31,29), (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)>;

G:=Group( (1,4,2,3)(5,32,15,23)(6,24,16,33)(7,34,17,25)(8,26,18,35)(9,36,19,27)(10,28,20,29)(11,30,13,21)(12,22,14,31), (1,10,6)(2,20,16)(3,29,33)(4,28,24)(5,7,12)(8,11,9)(13,19,18)(14,15,17)(21,27,35)(22,32,34)(23,25,31)(26,30,36), (1,11,7)(2,13,17)(3,21,25)(4,30,34)(5,6,8)(9,12,10)(14,20,19)(15,16,18)(22,28,36)(23,33,35)(24,26,32)(27,31,29), (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) );

G=PermutationGroup([(1,4,2,3),(5,32,15,23),(6,24,16,33),(7,34,17,25),(8,26,18,35),(9,36,19,27),(10,28,20,29),(11,30,13,21),(12,22,14,31)], [(1,10,6),(2,20,16),(3,29,33),(4,28,24),(5,7,12),(8,11,9),(13,19,18),(14,15,17),(21,27,35),(22,32,34),(23,25,31),(26,30,36)], [(1,11,7),(2,13,17),(3,21,25),(4,30,34),(5,6,8),(9,12,10),(14,20,19),(15,16,18),(22,28,36),(23,33,35),(24,26,32),(27,31,29)], [(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)])

Matrix representation of C4⋊F9 in GL10(𝔽73)

0100000000
72000000000
00720000000
00072000000
00007200000
00000720000
00000072000
00000007200
00000000720
00000000072
,
1000000000
0100000000
00000007210
00000007201
00000007200
00100007200
00010007200
00001007200
00000107200
00000017200
,
1000000000
0100000000
00072000000
00172000000
00072001000
00072100000
00072010000
00072000001
00072000100
00072000010
,
22000000000
05100000000
0000100000
0000000100
0000010000
0000000010
0010000000
0000000001
0001000000
0000001000

G:=sub<GL(10,GF(73))| [0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,72,72,72,72,72,72,72,72,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[22,0,0,0,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C4⋊F9 in GAP, Magma, Sage, TeX

C_4\rtimes F_9
% in TeX

G:=Group("C4:F9");
// GroupNames label

G:=SmallGroup(288,864);
// by ID

G=gap.SmallGroup(288,864);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,100,4037,2371,201,10982,3156,622]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C4⋊F9 in TeX
Character table of C4⋊F9 in TeX

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