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G = C4.F9order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: C4.F9, C321M5(2), C2.F91C2, C2.4(C2×F9), (C3×C12).3C8, C322C8.5C4, C322C8.4C22, (C4×C3⋊S3).2C4, (C2×C3⋊S3).2C8, (C3×C6).2(C2×C8), C3⋊S33C8.9C2, C3⋊Dic3.2(C2×C4), SmallGroup(288,862)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C4.F9
C1C32C3×C6C3⋊Dic3C322C8C2.F9 — C4.F9
C32C3×C6 — C4.F9
C1C2C4

Generators and relations for C4.F9
 G = < a,b,c,d | a4=b3=c3=1, d8=a2, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >

18C2
4C3
9C22
9C4
4C6
12S3
12S3
9C2×C4
9C8
9C8
4C12
12Dic3
12D6
2C3⋊S3
9C16
9C16
9C2×C8
12C4×S3
9M5(2)

Character table of C4.F9

 class 12A2B34A4B4C68A8B8C8D8E8F12A12B16A16B16C16D16E16F16G16H
 size 11188299899991818881818181818181818
ρ1111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311-11-11111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ411-11-11111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ511-11-1111-1-1-1-111-1-1-i-iiiii-i-i    linear of order 4
ρ611-11-1111-1-1-1-111-1-1ii-i-i-i-iii    linear of order 4
ρ711111111-1-1-1-1-1-111-i-iii-i-iii    linear of order 4
ρ811111111-1-1-1-1-1-111ii-i-iii-i-i    linear of order 4
ρ91111-1-1-11-i-iiii-i-1-1ζ8ζ85ζ83ζ87ζ85ζ8ζ87ζ83    linear of order 8
ρ101111-1-1-11ii-i-i-ii-1-1ζ87ζ83ζ85ζ8ζ83ζ87ζ8ζ85    linear of order 8
ρ111111-1-1-11-i-iiii-i-1-1ζ85ζ8ζ87ζ83ζ8ζ85ζ83ζ87    linear of order 8
ρ121111-1-1-11ii-i-i-ii-1-1ζ83ζ87ζ8ζ85ζ87ζ83ζ85ζ8    linear of order 8
ρ1311-111-1-11-i-iii-ii11ζ8ζ85ζ83ζ87ζ8ζ85ζ83ζ87    linear of order 8
ρ1411-111-1-11ii-i-ii-i11ζ83ζ87ζ8ζ85ζ83ζ87ζ8ζ85    linear of order 8
ρ1511-111-1-11ii-i-ii-i11ζ87ζ83ζ85ζ8ζ87ζ83ζ85ζ8    linear of order 8
ρ1611-111-1-11-i-iii-ii11ζ85ζ8ζ87ζ83ζ85ζ8ζ87ζ83    linear of order 8
ρ172-20202i-2i-28858387000000000000    complex lifted from M5(2)
ρ182-2020-2i2i-28783858000000000000    complex lifted from M5(2)
ρ192-20202i-2i-28588783000000000000    complex lifted from M5(2)
ρ202-2020-2i2i-28387885000000000000    complex lifted from M5(2)
ρ21880-1800-1000000-1-100000000    orthogonal lifted from F9
ρ22880-1-800-10000001100000000    orthogonal lifted from C2×F9
ρ238-80-10001000000-3i3i00000000    complex faithful
ρ248-80-100010000003i-3i00000000    complex faithful

Smallest permutation representation of C4.F9
On 48 points
Generators in S48
(1 13 9 5)(2 6 10 14)(3 15 11 7)(4 8 12 16)(17 46 25 38)(18 39 26 47)(19 48 27 40)(20 41 28 33)(21 34 29 42)(22 43 30 35)(23 36 31 44)(24 45 32 37)
(2 26 43)(3 27 44)(4 45 28)(6 47 30)(7 48 31)(8 32 33)(10 18 35)(11 19 36)(12 37 20)(14 39 22)(15 40 23)(16 24 41)
(1 25 42)(3 27 44)(4 28 45)(5 46 29)(7 48 31)(8 33 32)(9 17 34)(11 19 36)(12 20 37)(13 38 21)(15 40 23)(16 41 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 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)| (1,13,9,5)(2,6,10,14)(3,15,11,7)(4,8,12,16)(17,46,25,38)(18,39,26,47)(19,48,27,40)(20,41,28,33)(21,34,29,42)(22,43,30,35)(23,36,31,44)(24,45,32,37), (2,26,43)(3,27,44)(4,45,28)(6,47,30)(7,48,31)(8,32,33)(10,18,35)(11,19,36)(12,37,20)(14,39,22)(15,40,23)(16,24,41), (1,25,42)(3,27,44)(4,28,45)(5,46,29)(7,48,31)(8,33,32)(9,17,34)(11,19,36)(12,20,37)(13,38,21)(15,40,23)(16,41,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)>;

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

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

Matrix representation of C4.F9 in GL10(𝔽97)

22000000000
757500000000
00960000000
00096000000
00009600000
00000960000
00000096000
00000009600
00000000960
00000000096
,
1000000000
0100000000
0010000000
0001000000
0000010000
001196960000
00000009600
00000019600
00000009601
000000109696
,
1000000000
0100000000
00096000000
00196000000
00096010000
001096960000
00000096100
00000096000
00000096010
00000096001
,
969500000000
74100000000
0000001000
0000000100
0000000010
0000000001
00009610000
001195960000
0033339600000
0033329600000

G:=sub<GL(10,GF(97))| [22,75,0,0,0,0,0,0,0,0,0,75,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,96,96,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,96,96,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,96,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,74,0,0,0,0,0,0,0,0,95,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,33,33,0,0,0,0,0,0,0,1,33,32,0,0,0,0,0,0,96,95,96,96,0,0,0,0,0,0,1,96,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C4.F9 in GAP, Magma, Sage, TeX

C_4.F_9
% in TeX

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

G:=SmallGroup(288,862);
// by ID

G=gap.SmallGroup(288,862);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,120,58,80,4037,2371,362,10982,3156,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^3=1,d^8=a^2,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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