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G = C4.3F9order 288 = 25·32

2nd central extension by C4 of F9

metabelian, soluble, monomial, A-group

Aliases: C4.3F9, C3⋊S3⋊C16, C2.F93C2, C2.1(C2×F9), (C3×C12).2C8, C321(C2×C16), C322C8.1C4, C322C8.3C22, (C4×C3⋊S3).1C4, (C2×C3⋊S3).1C8, (C3×C6).1(C2×C8), C3⋊S33C8.8C2, C3⋊Dic3.1(C2×C4), SmallGroup(288,861)

Series: Derived Chief Lower central Upper central

C1C32 — C4.3F9
C1C32C3×C6C3⋊Dic3C322C8C2.F9 — C4.3F9
C32 — C4.3F9
C1C4

Generators and relations for C4.3F9
 G = < a,b,c,d | a4=b3=c3=1, d8=a2, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

9C2
9C2
4C3
9C4
9C22
4C6
12S3
12S3
9C8
9C2×C4
9C8
4C12
12Dic3
12D6
9C16
9C16
9C2×C8
12C4×S3
9C2×C16

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

36 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 8A···8H12A12B16A···16P
order12223444468···8121216···16
size11998119989···9889···9

36 irreducible representations

dim11111111888
type+++++
imageC1C2C2C4C4C8C8C16F9C2×F9C4.3F9
kernelC4.3F9C2.F9C3⋊S33C8C322C8C4×C3⋊S3C3×C12C2×C3⋊S3C3⋊S3C4C2C1
# reps121224416112

Matrix representation of C4.3F9 in GL9(𝔽97)

7500000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000010000
01196960000
01100969600
000001000
000000001
01100009696
,
100000000
0096000000
0196000000
0096010000
01096960000
0096000100
01000969600
0096000010
0096000001
,
7000000000
0000096100
01100959600
0000096010
0000096001
0001096000
0000196000
0000096000
0010096000

G:=sub<GL(9,GF(97))| [75,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,96,96,96,0,96,0,96,96,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[70,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,96,95,96,96,96,96,96,96,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C4.3F9 in GAP, Magma, Sage, TeX

C_4._3F_9
% in TeX

G:=Group("C4.3F9");
// GroupNames label

G:=SmallGroup(288,861);
// by ID

G=gap.SmallGroup(288,861);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,120,58,80,4037,2371,362,10982,3156,1203]);
// 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,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C4.3F9 in TeX

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