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G = C24⋊C18  order 288 = 25·32

1st semidirect product of C24 and C18 acting via C18/C3=C6

metabelian, soluble, monomial

Aliases: C24⋊1C18, C22≀C2⋊C9, C24⋊C9⋊1C2, (C22×C6).1A4, (C23×C6).1C6, C3.(C24⋊C6), C23⋊1(C3.A4), (C2×C6).6(C2×A4), (C3×C22≀C2).C3, C22.2(C2×C3.A4), SmallGroup(288,73)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C24⋊C18
Chief series
C1 — C22 — C24 — C23×C6 — C24⋊C9 — C24⋊C18
Lower central
C24 — C24⋊C18
Upper central
C1 — C3

Generators and relations for C24⋊C18
 G = < a,b,c,d,e | a2=b2=c2=d2=e18=1, ab=ba, ac=ca, ad=da, eae-1=db=bd, bc=cb, ebe-1=abcd, ede-1=cd=dc, ece-1=d >

Subgroups: 270 in 62 conjugacy classes, 12 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, D4, C23, C23, C9, C12, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C18, C2×C12, C3×D4, C22×C6, C22×C6, C22≀C2, C3.A4, C3×C22⋊C4, C6×D4, C23×C6, C2×C3.A4, C3×C22≀C2, C24⋊C9, C24⋊C18
Quotients: C1, C2, C3, C6, C9, A4, C18, C2×A4, C3.A4, C2×C3.A4, C24⋊C6, C24⋊C18

Character table of C24⋊C18

 class 12A2B2C2D3A3B46A6B6C6D6E6F6G6H9A9B9C9D9E9F12A12B18A18B18C18D18E18F
 size 134661112334466661616161616161212161616161616
ρ1111111111111111111111111111111    trivial
ρ211-11111-111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111111111111111ζ32ζ32ζ3ζ3ζ3ζ3211ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ411-11111-111-1-11111ζ3ζ3ζ32ζ32ζ32ζ3-1-1ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ51111111111111111ζ3ζ3ζ32ζ32ζ32ζ311ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ611-11111-111-1-11111ζ32ζ32ζ3ζ3ζ3ζ32-1-1ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ711-111ζ3ζ32-1ζ32ζ3ζ65ζ6ζ32ζ32ζ3ζ3ζ94ζ97ζ98ζ95ζ92ζ9ζ6ζ65-ζ97-ζ98-ζ95-ζ94-ζ9-ζ92    linear of order 18
ρ811111ζ32ζ31ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ98ζ95ζ97ζ9ζ94ζ92ζ3ζ32ζ95ζ97ζ9ζ98ζ92ζ94    linear of order 9
ρ911-111ζ3ζ32-1ζ32ζ3ζ65ζ6ζ32ζ32ζ3ζ3ζ97ζ9ζ95ζ92ζ98ζ94ζ6ζ65-ζ9-ζ95-ζ92-ζ97-ζ94-ζ98    linear of order 18
ρ1011111ζ32ζ31ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ95ζ92ζ9ζ94ζ97ζ98ζ3ζ32ζ92ζ9ζ94ζ95ζ98ζ97    linear of order 9
ρ1111-111ζ32ζ3-1ζ3ζ32ζ6ζ65ζ3ζ3ζ32ζ32ζ98ζ95ζ97ζ9ζ94ζ92ζ65ζ6-ζ95-ζ97-ζ9-ζ98-ζ92-ζ94    linear of order 18
ρ1211-111ζ3ζ32-1ζ32ζ3ζ65ζ6ζ32ζ32ζ3ζ3ζ9ζ94ζ92ζ98ζ95ζ97ζ6ζ65-ζ94-ζ92-ζ98-ζ9-ζ97-ζ95    linear of order 18
ρ1311-111ζ32ζ3-1ζ3ζ32ζ6ζ65ζ3ζ3ζ32ζ32ζ95ζ92ζ9ζ94ζ97ζ98ζ65ζ6-ζ92-ζ9-ζ94-ζ95-ζ98-ζ97    linear of order 18
ρ1411-111ζ32ζ3-1ζ3ζ32ζ6ζ65ζ3ζ3ζ32ζ32ζ92ζ98ζ94ζ97ζ9ζ95ζ65ζ6-ζ98-ζ94-ζ97-ζ92-ζ95-ζ9    linear of order 18
ρ1511111ζ3ζ321ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ9ζ94ζ92ζ98ζ95ζ97ζ32ζ3ζ94ζ92ζ98ζ9ζ97ζ95    linear of order 9
ρ1611111ζ3ζ321ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ97ζ9ζ95ζ92ζ98ζ94ζ32ζ3ζ9ζ95ζ92ζ97ζ94ζ98    linear of order 9
ρ1711111ζ32ζ31ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ92ζ98ζ94ζ97ζ9ζ95ζ3ζ32ζ98ζ94ζ97ζ92ζ95ζ9    linear of order 9
ρ1811111ζ3ζ321ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ94ζ97ζ98ζ95ζ92ζ9ζ32ζ3ζ97ζ98ζ95ζ94ζ9ζ92    linear of order 9
ρ1933-3-1-133133-3-3-1-1-1-100000011000000    orthogonal lifted from C2×A4
ρ20333-1-133-13333-1-1-1-1000000-1-1000000    orthogonal lifted from A4
ρ2133-3-1-1-3+3√-3/2-3-3√-3/21-3-3√-3/2-3+3√-3/23-3√-3/23+3√-3/2ζ6ζ6ζ65ζ65000000ζ32ζ3000000    complex lifted from C2×C3.A4
ρ22333-1-1-3-3√-3/2-3+3√-3/2-1-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/2ζ65ζ65ζ6ζ6000000ζ65ζ6000000    complex lifted from C3.A4
ρ23333-1-1-3+3√-3/2-3-3√-3/2-1-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/2ζ6ζ6ζ65ζ65000000ζ6ζ65000000    complex lifted from C3.A4
ρ2433-3-1-1-3-3√-3/2-3+3√-3/21-3+3√-3/2-3-3√-3/23+3√-3/23-3√-3/2ζ65ζ65ζ6ζ6000000ζ3ζ32000000    complex lifted from C2×C3.A4
ρ256-20-22660-2-2002-22-200000000000000    orthogonal lifted from C24⋊C6
ρ266-202-2660-2-200-22-2200000000000000    orthogonal lifted from C24⋊C6
ρ276-20-22-3-3√-3-3+3√-301-√-31+√-300-1+√-31-√-3-1-√-31+√-300000000000000    complex faithful
ρ286-202-2-3-3√-3-3+3√-301-√-31+√-3001-√-3-1+√-31+√-3-1-√-300000000000000    complex faithful
ρ296-202-2-3+3√-3-3-3√-301+√-31-√-3001+√-3-1-√-31-√-3-1+√-300000000000000    complex faithful
ρ306-20-22-3+3√-3-3-3√-301+√-31-√-300-1-√-31+√-3-1+√-31-√-300000000000000    complex faithful

Smallest permutation representation of C24⋊C18
►On 36 points
Generators in S36
(2 35)(3 27)(5 29)(6 21)(8 23)(9 33)(10 30)(12 32)(13 24)(15 26)(16 36)(18 20)

(1 25)(2 35)(4 19)(5 29)(7 31)(8 23)(11 22)(12 32)(14 34)(15 26)(17 28)(18 20)

(2 15)(3 16)(5 18)(6 10)(8 12)(9 13)(20 29)(21 30)(23 32)(24 33)(26 35)(27 36)

(1 14)(2 15)(4 17)(5 18)(7 11)(8 12)(19 28)(20 29)(22 31)(23 32)(25 34)(26 35)

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

Matrix representation of C24⋊C18 ►in GL6(𝔽37)

100000
010000
00363500
000100
00003635
000001
,
36350000
010000
00363500
000100
000010
000001
,
100000
010000
0036000
0003600
0000360
0000036
,
3600000
0360000
0036000
0003600
000010
000001
,
0000260
00001111
1000000
27270000
0010000
00272700

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,35,1,0,0,0,0,0,0,36,0,0,0,0,0,35,1],[36,0,0,0,0,0,35,1,0,0,0,0,0,0,36,0,0,0,0,0,35,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,10,27,0,0,0,0,0,27,0,0,0,0,0,0,10,27,0,0,0,0,0,27,26,11,0,0,0,0,0,11,0,0,0,0] >;
 

C24⋊C18 in GAP, Magma, Sage, TeX 

C_2^4\rtimes C_{18}
 
% in TeX
 
G:=Group("C2^4:C18");
 
// GroupNames label
 
G:=SmallGroup(288,73);
 
// by ID
 
G=gap.SmallGroup(288,73);
 
# by ID
 
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,2523,514,6304,956,3036,5305]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^18=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=d*b=b*d,b*c=c*b,e*b*e^-1=a*b*c*d,e*d*e^-1=c*d=d*c,e*c*e^-1=d>;
 
// generators/relations
 

Export

Character table of C24⋊C18 in TeX

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