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G = C422C18order 288 = 25·32

2nd semidirect product of C42 and C18 acting via C18/C3=C6

metabelian, soluble, monomial

Aliases: C422C18, C41D4⋊C9, C42⋊C93C2, (C4×C12).2C6, (C22×C6).3A4, C3.(C23.A4), C23.2(C3.A4), (C2×C6).8(C2×A4), (C3×C41D4).C3, C22.4(C2×C3.A4), SmallGroup(288,75)

Series: Derived Chief Lower central Upper central

C1C42 — C422C18
C1C22C42C4×C12C42⋊C9 — C422C18
C42 — C422C18
C1C3

Generators and relations for C422C18
 G = < a,b,c | a4=b4=c18=1, ab=ba, cac-1=a2b-1, cbc-1=a-1b-1 >

3C2
4C2
12C2
3C4
3C4
6C22
6C22
6C22
6C22
3C6
4C6
12C6
16C9
3C23
3C2×C4
6D4
6D4
6D4
6D4
3C12
3C12
6C2×C6
6C2×C6
6C2×C6
6C2×C6
16C18
3C2×D4
3C2×D4
3C2×C12
3C22×C6
6C3×D4
6C3×D4
6C3×D4
6C3×D4
4C3.A4
3C6×D4
3C6×D4
4C2×C3.A4

Character table of C422C18

 class 12A2B2C3A3B4A4B6A6B6C6D6E6F9A9B9C9D9E9F12A12B12C12D18A18B18C18D18E18F
 size 134121166334412121616161616166666161616161616
ρ1111111111111111111111111111111    trivial
ρ211-1-1111111-1-1-1-11111111111-1-1-1-1-1-1    linear of order 2
ρ311-1-1111111-1-1-1-1ζ3ζ32ζ32ζ3ζ3ζ321111ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ411111111111111ζ32ζ3ζ3ζ32ζ32ζ31111ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ511111111111111ζ3ζ32ζ32ζ3ζ3ζ321111ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ611-1-1111111-1-1-1-1ζ32ζ3ζ3ζ32ζ32ζ31111ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ711-1-1ζ32ζ311ζ32ζ3ζ65ζ6ζ6ζ65ζ97ζ92ζ98ζ94ζ9ζ95ζ32ζ32ζ3ζ399592979498    linear of order 18
ρ811-1-1ζ32ζ311ζ32ζ3ζ65ζ6ζ6ζ65ζ94ζ95ζ92ζ9ζ97ζ98ζ32ζ32ζ3ζ397989594992    linear of order 18
ρ911-1-1ζ32ζ311ζ32ζ3ζ65ζ6ζ6ζ65ζ9ζ98ζ95ζ97ζ94ζ92ζ32ζ32ζ3ζ394929899795    linear of order 18
ρ101111ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32ζ92ζ97ζ9ζ95ζ98ζ94ζ3ζ3ζ32ζ32ζ98ζ94ζ97ζ92ζ95ζ9    linear of order 9
ρ1111-1-1ζ3ζ3211ζ3ζ32ζ6ζ65ζ65ζ6ζ95ζ94ζ97ζ98ζ92ζ9ζ3ζ3ζ32ζ3292994959897    linear of order 18
ρ121111ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3ζ94ζ95ζ92ζ9ζ97ζ98ζ32ζ32ζ3ζ3ζ97ζ98ζ95ζ94ζ9ζ92    linear of order 9
ρ131111ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3ζ9ζ98ζ95ζ97ζ94ζ92ζ32ζ32ζ3ζ3ζ94ζ92ζ98ζ9ζ97ζ95    linear of order 9
ρ1411-1-1ζ3ζ3211ζ3ζ32ζ6ζ65ζ65ζ6ζ92ζ97ζ9ζ95ζ98ζ94ζ3ζ3ζ32ζ3298949792959    linear of order 18
ρ1511-1-1ζ3ζ3211ζ3ζ32ζ6ζ65ζ65ζ6ζ98ζ9ζ94ζ92ζ95ζ97ζ3ζ3ζ32ζ3295979989294    linear of order 18
ρ161111ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32ζ95ζ94ζ97ζ98ζ92ζ9ζ3ζ3ζ32ζ32ζ92ζ9ζ94ζ95ζ98ζ97    linear of order 9
ρ171111ζ3ζ3211ζ3ζ32ζ32ζ3ζ3ζ32ζ98ζ9ζ94ζ92ζ95ζ97ζ3ζ3ζ32ζ32ζ95ζ97ζ9ζ98ζ92ζ94    linear of order 9
ρ181111ζ32ζ311ζ32ζ3ζ3ζ32ζ32ζ3ζ97ζ92ζ98ζ94ζ9ζ95ζ32ζ32ζ3ζ3ζ9ζ95ζ92ζ97ζ94ζ98    linear of order 9
ρ19333-133-1-13333-1-1000000-1-1-1-1000000    orthogonal lifted from A4
ρ2033-3133-1-133-3-311000000-1-1-1-1000000    orthogonal lifted from C2×A4
ρ2133-31-3-3-3/2-3+3-3/2-1-1-3-3-3/2-3+3-3/23-3-3/23+3-3/2ζ32ζ3000000ζ6ζ6ζ65ζ65000000    complex lifted from C2×C3.A4
ρ22333-1-3+3-3/2-3-3-3/2-1-1-3+3-3/2-3-3-3/2-3-3-3/2-3+3-3/2ζ65ζ6000000ζ65ζ65ζ6ζ6000000    complex lifted from C3.A4
ρ23333-1-3-3-3/2-3+3-3/2-1-1-3-3-3/2-3+3-3/2-3+3-3/2-3-3-3/2ζ6ζ65000000ζ6ζ6ζ65ζ65000000    complex lifted from C3.A4
ρ2433-31-3+3-3/2-3-3-3/2-1-1-3+3-3/2-3-3-3/23+3-3/23-3-3/2ζ3ζ32000000ζ65ζ65ζ6ζ6000000    complex lifted from C2×C3.A4
ρ256-200662-2-2-20000000000-22-22000000    orthogonal lifted from C23.A4
ρ266-20066-22-2-200000000002-22-2000000    orthogonal lifted from C23.A4
ρ276-200-3-3-3-3+3-3-221+-31--30000000000-1--31+-3-1+-31--3000000    complex faithful
ρ286-200-3-3-3-3+3-32-21+-31--300000000001+-3-1--31--3-1+-3000000    complex faithful
ρ296-200-3+3-3-3-3-32-21--31+-300000000001--3-1+-31+-3-1--3000000    complex faithful
ρ306-200-3+3-3-3-3-3-221--31+-30000000000-1+-31--3-1--31+-3000000    complex faithful

Smallest permutation representation of C422C18
On 36 points
Generators in S36
(2 11 28 19)(3 12 29 20)(5 22 31 14)(6 23 32 15)(8 17 34 25)(9 18 35 26)
(1 36 27 10)(2 11 28 19)(3 29)(4 13 30 21)(5 22 31 14)(6 32)(7 24 33 16)(8 17 34 25)(9 35)(12 20)(15 23)(18 26)
(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)| (2,11,28,19)(3,12,29,20)(5,22,31,14)(6,23,32,15)(8,17,34,25)(9,18,35,26), (1,36,27,10)(2,11,28,19)(3,29)(4,13,30,21)(5,22,31,14)(6,32)(7,24,33,16)(8,17,34,25)(9,35)(12,20)(15,23)(18,26), (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( (2,11,28,19)(3,12,29,20)(5,22,31,14)(6,23,32,15)(8,17,34,25)(9,18,35,26), (1,36,27,10)(2,11,28,19)(3,29)(4,13,30,21)(5,22,31,14)(6,32)(7,24,33,16)(8,17,34,25)(9,35)(12,20)(15,23)(18,26), (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([(2,11,28,19),(3,12,29,20),(5,22,31,14),(6,23,32,15),(8,17,34,25),(9,18,35,26)], [(1,36,27,10),(2,11,28,19),(3,29),(4,13,30,21),(5,22,31,14),(6,32),(7,24,33,16),(8,17,34,25),(9,35),(12,20),(15,23),(18,26)], [(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 C422C18 in GL6(𝔽37)

100000
010000
000100
9036000
7000036
000010
,
1350000
1360000
0280100
92836000
73000360
73000036
,
34000022
00002611
01000025
0000025
0001003
0010003

G:=sub<GL(6,GF(37))| [1,0,0,9,7,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0],[1,1,0,9,7,7,35,36,28,28,30,30,0,0,0,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[34,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,26,0,0,0,0,22,11,25,25,3,3] >;

C422C18 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2C_{18}
% in TeX

G:=Group("C4^2:2C18");
// GroupNames label

G:=SmallGroup(288,75);
// by ID

G=gap.SmallGroup(288,75);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,6555,514,360,3784,3476,102,3036,5305]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^18=1,a*b=b*a,c*a*c^-1=a^2*b^-1,c*b*c^-1=a^-1*b^-1>;
// generators/relations

Export

Subgroup lattice of C422C18 in TeX
Character table of C422C18 in TeX

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