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G = C22.F9order 288 = 25·32

The non-split extension by C22 of F9 acting via F9/C32⋊C4=C2

metabelian, soluble, monomial

Aliases: C22.F9, C62.2C8, C322M5(2), C2.F92C2, C2.6(C2×F9), C3⋊Dic3.2C8, C322C8.6C4, C322C8.6C22, (C3×C6).6(C2×C8), (C2×C3⋊Dic3).5C4, C3⋊Dic3.4(C2×C4), (C2×C322C8).10C2, SmallGroup(288,866)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C22.F9
C1C32C3×C6C3⋊Dic3C322C8C2.F9 — C22.F9
C32C3×C6 — C22.F9
C1C2C22

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

2C2
4C3
9C4
9C4
4C6
4C6
4C6
9C2×C4
9C8
9C8
4C2×C6
12Dic3
12Dic3
2C3×C6
9C16
9C16
9C2×C8
12C2×Dic3
9M5(2)

Character table of C22.F9

 class 12A2B34A4B4C6A6B6C8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 11289918888999918181818181818181818
ρ1111111111111111111111111    trivial
ρ211-1111-1-11-11111-1-11-1-1-1111-1    linear of order 2
ρ311-1111-1-11-11111-1-1-1111-1-1-11    linear of order 2
ρ41111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1i-iii-i-ii-i    linear of order 4
ρ611-1111-1-11-1-1-1-1-111ii-i-i-i-iii    linear of order 4
ρ711-1111-1-11-1-1-1-1-111-i-iiiii-i-i    linear of order 4
ρ81111111111-1-1-1-1-1-1-ii-i-iii-ii    linear of order 4
ρ911-11-1-11-11-1-i-iiii-iζ85ζ83ζ85ζ8ζ83ζ87ζ8ζ87    linear of order 8
ρ101111-1-1-1111-i-iii-iiζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83    linear of order 8
ρ111111-1-1-1111ii-i-ii-iζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85    linear of order 8
ρ121111-1-1-1111-i-iii-iiζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87    linear of order 8
ρ1311-11-1-11-11-1ii-i-i-iiζ83ζ85ζ83ζ87ζ85ζ8ζ87ζ8    linear of order 8
ρ1411-11-1-11-11-1-i-iiii-iζ8ζ87ζ8ζ85ζ87ζ83ζ85ζ83    linear of order 8
ρ1511-11-1-11-11-1ii-i-i-iiζ87ζ8ζ87ζ83ζ8ζ85ζ83ζ85    linear of order 8
ρ161111-1-1-1111ii-i-ii-iζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8    linear of order 8
ρ172-2022i-2i00-2088583870000000000    complex lifted from M5(2)
ρ182-2022i-2i00-2085887830000000000    complex lifted from M5(2)
ρ192-202-2i2i00-2083878850000000000    complex lifted from M5(2)
ρ202-202-2i2i00-2087838580000000000    complex lifted from M5(2)
ρ21888-1000-1-1-100000000000000    orthogonal lifted from F9
ρ2288-8-10001-1100000000000000    orthogonal lifted from C2×F9
ρ238-80-100031-300000000000000    symplectic faithful, Schur index 2
ρ248-80-1000-31300000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C22.F9
On 48 points
Generators in S48
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(2 33 32)(3 34 17)(4 18 35)(6 20 37)(7 21 38)(8 39 22)(10 41 24)(11 42 25)(12 26 43)(14 28 45)(15 29 46)(16 47 30)
(1 48 31)(3 34 17)(4 35 18)(5 19 36)(7 21 38)(8 22 39)(9 40 23)(11 42 25)(12 43 26)(13 27 44)(15 29 46)(16 30 47)
(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 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)

G:=sub<Sym(48)| (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,33,32)(3,34,17)(4,18,35)(6,20,37)(7,21,38)(8,39,22)(10,41,24)(11,42,25)(12,26,43)(14,28,45)(15,29,46)(16,47,30), (1,48,31)(3,34,17)(4,35,18)(5,19,36)(7,21,38)(8,22,39)(9,40,23)(11,42,25)(12,43,26)(13,27,44)(15,29,46)(16,30,47), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)>;

G:=Group( (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,33,32)(3,34,17)(4,18,35)(6,20,37)(7,21,38)(8,39,22)(10,41,24)(11,42,25)(12,26,43)(14,28,45)(15,29,46)(16,47,30), (1,48,31)(3,34,17)(4,35,18)(5,19,36)(7,21,38)(8,22,39)(9,40,23)(11,42,25)(12,43,26)(13,27,44)(15,29,46)(16,30,47), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48) );

G=PermutationGroup([(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(2,33,32),(3,34,17),(4,18,35),(6,20,37),(7,21,38),(8,39,22),(10,41,24),(11,42,25),(12,26,43),(14,28,45),(15,29,46),(16,47,30)], [(1,48,31),(3,34,17),(4,35,18),(5,19,36),(7,21,38),(8,22,39),(9,40,23),(11,42,25),(12,43,26),(13,27,44),(15,29,46),(16,30,47)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)])

Matrix representation of C22.F9 in GL10(𝔽97)

1000000000
09600000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
96000000000
09600000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000010000
00505096960000
00000009600
00000019600
00000004701
0000005009696
,
1000000000
0100000000
00096000000
00196000000
00047010000
0050096960000
00000096100
00000096000
00000047010
00000047001
,
0100000000
50000000000
0000001000
0000000100
0000000010
0000000001
00009610000
00505095960000
004084700000
008274700000

G:=sub<GL(10,GF(97))| [1,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,50,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,96,96,47,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,96,96,47,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,96,47,47,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,50,40,8,0,0,0,0,0,0,0,50,8,27,0,0,0,0,0,0,96,95,47,47,0,0,0,0,0,0,1,96,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C22.F9 in GAP, Magma, Sage, TeX

C_2^2.F_9
% in TeX

G:=Group("C2^2.F9");
// GroupNames label

G:=SmallGroup(288,866);
// by ID

G=gap.SmallGroup(288,866);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,58,80,4037,2371,362,10982,3156,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=1,e^8=b,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

Subgroup lattice of C22.F9 in TeX
Character table of C22.F9 in TeX

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