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G = C24⋊C9  order 144 = 24·32

2nd semidirect product of C24 and C9 acting via C9/C3=C3

metabelian, soluble, monomial, A-group

Aliases: C24⋊2C9, (C2×C6).3A4, C22⋊(C3.A4), C3.(C22⋊A4), (C23×C6).2C3, SmallGroup(144,111)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C24⋊C9
 G = < a,b,c,d,e | a2=b2=c2=d2=e9=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a, ece-1=cd=dc, ede-1=c >

Subgroups: 171 in 61 conjugacy classes, 15 normal (5 characteristic)
C1, C2, C3, C22, C22, C6, C23, C9, C2×C6, C2×C6, C24, C22×C6, C3.A4, C23×C6, C24⋊C9
Quotients: C1, C3, C9, A4, C3.A4, C22⋊A4, C24⋊C9

Character table of C24⋊C9

 class 12A2B2C2D2E3A3B6A6B6C6D6E6F6G6H6I6J9A9B9C9D9E9F
 size 133333113333333333161616161616
ρ1111111111111111111111111    trivial
ρ2111111111111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ3111111111111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ4111111ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ92ζ98ζ95ζ94ζ9ζ97    linear of order 9
ρ5111111ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ94ζ97ζ9ζ98ζ92ζ95    linear of order 9
ρ6111111ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ98ζ95ζ92ζ97ζ94ζ9    linear of order 9
ρ7111111ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ95ζ92ζ98ζ9ζ97ζ94    linear of order 9
ρ8111111ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ97ζ9ζ94ζ95ζ98ζ92    linear of order 9
ρ9111111ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ9ζ94ζ97ζ92ζ95ζ98    linear of order 9
ρ103-1-1-13-133-1-13-1-1-1-1-13-1000000    orthogonal lifted from A4
ρ113-13-1-1-1333-1-1-1-1-13-1-1-1000000    orthogonal lifted from A4
ρ1233-1-1-1-133-1-1-13-1-1-1-1-13000000    orthogonal lifted from A4
ρ133-1-13-1-133-13-1-1-1-1-13-1-1000000    orthogonal lifted from A4
ρ143-1-1-1-1333-1-1-1-133-1-1-1-1000000    orthogonal lifted from A4
ρ1533-1-1-1-1-3+3√-3/2-3-3√-3/2ζ6ζ6ζ6-3-3√-3/2ζ6ζ65ζ65ζ65ζ65-3+3√-3/2000000    complex lifted from C3.A4
ρ163-1-13-1-1-3+3√-3/2-3-3√-3/2ζ6-3-3√-3/2ζ6ζ6ζ6ζ65ζ65-3+3√-3/2ζ65ζ65000000    complex lifted from C3.A4
ρ173-13-1-1-1-3+3√-3/2-3-3√-3/2-3-3√-3/2ζ6ζ6ζ6ζ6ζ65-3+3√-3/2ζ65ζ65ζ65000000    complex lifted from C3.A4
ρ183-1-1-13-1-3-3√-3/2-3+3√-3/2ζ65ζ65-3+3√-3/2ζ65ζ65ζ6ζ6ζ6-3-3√-3/2ζ6000000    complex lifted from C3.A4
ρ193-1-1-1-13-3-3√-3/2-3+3√-3/2ζ65ζ65ζ65ζ65-3+3√-3/2-3-3√-3/2ζ6ζ6ζ6ζ6000000    complex lifted from C3.A4
ρ203-1-13-1-1-3-3√-3/2-3+3√-3/2ζ65-3+3√-3/2ζ65ζ65ζ65ζ6ζ6-3-3√-3/2ζ6ζ6000000    complex lifted from C3.A4
ρ213-1-1-13-1-3+3√-3/2-3-3√-3/2ζ6ζ6-3-3√-3/2ζ6ζ6ζ65ζ65ζ65-3+3√-3/2ζ65000000    complex lifted from C3.A4
ρ2233-1-1-1-1-3-3√-3/2-3+3√-3/2ζ65ζ65ζ65-3+3√-3/2ζ65ζ6ζ6ζ6ζ6-3-3√-3/2000000    complex lifted from C3.A4
ρ233-13-1-1-1-3-3√-3/2-3+3√-3/2-3+3√-3/2ζ65ζ65ζ65ζ65ζ6-3-3√-3/2ζ6ζ6ζ6000000    complex lifted from C3.A4
ρ243-1-1-1-13-3+3√-3/2-3-3√-3/2ζ6ζ6ζ6ζ6-3-3√-3/2-3+3√-3/2ζ65ζ65ζ65ζ65000000    complex lifted from C3.A4

Smallest permutation representation of C24⋊C9
►On 36 points
Generators in S36
(2 24)(3 25)(5 27)(6 19)(8 21)(9 22)(10 30)(11 31)(13 33)(14 34)(16 36)(17 28)

(1 23)(3 25)(4 26)(6 19)(7 20)(9 22)(11 31)(12 32)(14 34)(15 35)(17 28)(18 29)

(1 12)(3 14)(4 15)(6 17)(7 18)(9 11)(19 28)(20 29)(22 31)(23 32)(25 34)(26 35)

(1 12)(2 13)(4 15)(5 16)(7 18)(8 10)(20 29)(21 30)(23 32)(24 33)(26 35)(27 36)

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

C24⋊C9 is a maximal subgroup of
 C24⋊C18  C24⋊D9  A4×C3.A4  C3.A42  C24⋊3- 1+2  C24⋊23- 1+2  C9×C22⋊A4  C24⋊43- 1+2  C62.A4
C24⋊C9 is a maximal quotient of
 C22⋊(Q8⋊C9)  2+ 1+4⋊2C9  C24⋊C27

Matrix representation of C24⋊C9 ►in GL6(𝔽19)

100000
0180000
14018000
000100
0000180
0008018
,
1800000
0180000
591000
0001800
0000180
00011181
,
1800000
0180000
591000
000100
000010
000001
,
1800000
010000
01018000
000100
000010
000001
,
010000
141017000
009000
000010
00041018
0001099

G:=sub<GL(6,GF(19))| [1,0,14,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,8,0,0,0,0,18,0,0,0,0,0,0,18],[18,0,5,0,0,0,0,18,9,0,0,0,0,0,1,0,0,0,0,0,0,18,0,11,0,0,0,0,18,18,0,0,0,0,0,1],[18,0,5,0,0,0,0,18,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,0,0,0,0,0,0,1,10,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,14,0,0,0,0,1,10,0,0,0,0,0,17,9,0,0,0,0,0,0,0,4,10,0,0,0,1,10,9,0,0,0,0,18,9] >;
 

C24⋊C9 in GAP, Magma, Sage, TeX 

C_2^4\rtimes C_9
 
% in TeX
 
G:=Group("C2^4:C9");
 
// GroupNames label
 
G:=SmallGroup(144,111);
 
// by ID
 
G=gap.SmallGroup(144,111);
 
# by ID
 
G:=PCGroup([6,-3,-3,-2,2,-2,2,18,326,651,2164,3893]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^9=1,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

Export

Character table of C24⋊C9 in TeX

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