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G = C2.(C42⋊C9)  order 288 = 25·32

The central stem extension by C2 of C42⋊C9

non-abelian, soluble

Aliases: C2.(C42⋊C9), C2.C42⋊C9, C22.(Q8⋊C9), C6.1(C42⋊C3), (C22×C6).7A4, C3.(C23.3A4), C23.3(C3.A4), (C2×C6).1SL2(𝔽3), (C3×C2.C42).C3, SmallGroup(288,3)

Series: Derived Chief Lower central Upper central

C1C2C2.C42 — C2.(C42⋊C9)
C1C2C23C2.C42C3×C2.C42 — C2.(C42⋊C9)
C2.C42 — C2.(C42⋊C9)
C1C6

Generators and relations for C2.(C42⋊C9)
 G = < a,b,c,d | a2=b4=c4=d9=1, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=bc-1, dcd-1=b-1c2 >

3C2
3C2
3C22
3C22
6C4
6C4
3C6
3C6
16C9
3C2×C4
3C2×C4
6C2×C4
6C2×C4
3C2×C6
3C2×C6
6C12
6C12
16C18
3C22×C4
3C2×C12
3C2×C12
6C2×C12
6C2×C12
4C3.A4
3C22×C12
4C2×C3.A4

Smallest permutation representation of C2.(C42⋊C9)
On 36 points
Generators in S36
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(19 33)(20 34)(21 35)(22 36)(23 28)(24 29)(25 30)(26 31)(27 32)
(2 33 18 19)(3 20)(5 36 12 22)(6 23)(8 30 15 25)(9 26)(10 34)(13 28)(16 31)(21 35)(24 29)(27 32)
(1 32)(3 20 10 34)(4 35)(6 23 13 28)(7 29)(9 26 16 31)(11 21)(14 24)(17 27)(19 33)(22 36)(25 30)
(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)

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

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

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

36 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F9A···9F12A···12H18A···18F
order12223344446666669···912···1218···18
size113311666611333316···166···616···16

36 irreducible representations

dim111222333366
type+-++
imageC1C3C9SL2(𝔽3)SL2(𝔽3)Q8⋊C9A4C3.A4C42⋊C3C42⋊C9C23.3A4C2.(C42⋊C9)
kernelC2.(C42⋊C9)C3×C2.C42C2.C42C2×C6C2×C6C22C22×C6C23C6C2C3C1
# reps126126124812

Matrix representation of C2.(C42⋊C9) in GL5(𝔽37)

360000
036000
00100
00010
00001
,
011000
100000
00600
00060
000036
,
2710000
110000
003100
00010
00006
,
2821000
120000
00080
000015
00700

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,10,0,0,0,11,0,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,36],[27,1,0,0,0,10,10,0,0,0,0,0,31,0,0,0,0,0,1,0,0,0,0,0,6],[28,12,0,0,0,21,0,0,0,0,0,0,0,0,7,0,0,8,0,0,0,0,0,15,0] >;

C2.(C42⋊C9) in GAP, Magma, Sage, TeX

C_2.(C_4^2\rtimes C_9)
% in TeX

G:=Group("C2.(C4^2:C9)");
// GroupNames label

G:=SmallGroup(288,3);
// by ID

G=gap.SmallGroup(288,3);
# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,21,380,268,2775,521,80,7564,10589]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^9=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations

Export

Subgroup lattice of C2.(C42⋊C9) in TeX

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