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G = C42⋊C18order 288 = 25·32

1st semidirect product of C42 and C18 acting via C18/C3=C6

metabelian, soluble, monomial

Aliases: C421C18, C42⋊C91C2, C422C2⋊C9, (C4×C12).1C6, (C22×C6).2A4, C3.(C42⋊C6), C23.1(C3.A4), (C2×C6).7(C2×A4), (C3×C422C2).C3, C22.3(C2×C3.A4), SmallGroup(288,74)

Series: Derived Chief Lower central Upper central

C1C42 — C42⋊C18
C1C22C42C4×C12C42⋊C9 — C42⋊C18
C42 — C42⋊C18
C1C3

Generators and relations for C42⋊C18
 G = < a,b,c | a4=b4=c18=1, ab=ba, cac-1=b-1, cbc-1=a-1b >

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

Character table of C42⋊C18

 class 12A2B3A3B4A4B4C6A6B6C6D9A9B9C9D9E9F12A12B12C12D12E12F18A18B18C18D18E18F
 size 134116612334416161616161666661212161616161616
ρ1111111111111111111111111111111    trivial
ρ211-11111-111-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111111ζ3ζ3ζ32ζ32ζ32ζ3111111ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ4111111111111ζ32ζ32ζ3ζ3ζ3ζ32111111ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ511-11111-111-1-1ζ3ζ3ζ32ζ32ζ32ζ31111-1-1ζ65ζ6ζ6ζ65ζ65ζ6    linear of order 6
ρ611-11111-111-1-1ζ32ζ32ζ3ζ3ζ3ζ321111-1-1ζ6ζ65ζ65ζ6ζ6ζ65    linear of order 6
ρ7111ζ3ζ32111ζ3ζ32ζ32ζ3ζ98ζ95ζ97ζ9ζ94ζ92ζ32ζ32ζ3ζ3ζ32ζ3ζ95ζ97ζ9ζ98ζ92ζ94    linear of order 9
ρ8111ζ3ζ32111ζ3ζ32ζ32ζ3ζ92ζ98ζ94ζ97ζ9ζ95ζ32ζ32ζ3ζ3ζ32ζ3ζ98ζ94ζ97ζ92ζ95ζ9    linear of order 9
ρ911-1ζ32ζ311-1ζ32ζ3ζ65ζ6ζ94ζ97ζ98ζ95ζ92ζ9ζ3ζ3ζ32ζ32ζ65ζ697989594992    linear of order 18
ρ10111ζ32ζ3111ζ32ζ3ζ3ζ32ζ94ζ97ζ98ζ95ζ92ζ9ζ3ζ3ζ32ζ32ζ3ζ32ζ97ζ98ζ95ζ94ζ9ζ92    linear of order 9
ρ1111-1ζ3ζ3211-1ζ3ζ32ζ6ζ65ζ95ζ92ζ9ζ94ζ97ζ98ζ32ζ32ζ3ζ3ζ6ζ6592994959897    linear of order 18
ρ1211-1ζ32ζ311-1ζ32ζ3ζ65ζ6ζ9ζ94ζ92ζ98ζ95ζ97ζ3ζ3ζ32ζ32ζ65ζ694929899795    linear of order 18
ρ13111ζ32ζ3111ζ32ζ3ζ3ζ32ζ9ζ94ζ92ζ98ζ95ζ97ζ3ζ3ζ32ζ32ζ3ζ32ζ94ζ92ζ98ζ9ζ97ζ95    linear of order 9
ρ1411-1ζ3ζ3211-1ζ3ζ32ζ6ζ65ζ92ζ98ζ94ζ97ζ9ζ95ζ32ζ32ζ3ζ3ζ6ζ6598949792959    linear of order 18
ρ1511-1ζ3ζ3211-1ζ3ζ32ζ6ζ65ζ98ζ95ζ97ζ9ζ94ζ92ζ32ζ32ζ3ζ3ζ6ζ6595979989294    linear of order 18
ρ1611-1ζ32ζ311-1ζ32ζ3ζ65ζ6ζ97ζ9ζ95ζ92ζ98ζ94ζ3ζ3ζ32ζ32ζ65ζ699592979498    linear of order 18
ρ17111ζ3ζ32111ζ3ζ32ζ32ζ3ζ95ζ92ζ9ζ94ζ97ζ98ζ32ζ32ζ3ζ3ζ32ζ3ζ92ζ9ζ94ζ95ζ98ζ97    linear of order 9
ρ18111ζ32ζ3111ζ32ζ3ζ3ζ32ζ97ζ9ζ95ζ92ζ98ζ94ζ3ζ3ζ32ζ32ζ3ζ32ζ9ζ95ζ92ζ97ζ94ζ98    linear of order 9
ρ1933-333-1-1133-3-3000000-1-1-1-111000000    orthogonal lifted from C2×A4
ρ2033333-1-1-13333000000-1-1-1-1-1-1000000    orthogonal lifted from A4
ρ21333-3+3-3/2-3-3-3/2-1-1-1-3+3-3/2-3-3-3/2-3-3-3/2-3+3-3/2000000ζ6ζ6ζ65ζ65ζ6ζ65000000    complex lifted from C3.A4
ρ2233-3-3-3-3/2-3+3-3/2-1-11-3-3-3/2-3+3-3/23-3-3/23+3-3/2000000ζ65ζ65ζ6ζ6ζ3ζ32000000    complex lifted from C2×C3.A4
ρ23333-3-3-3/2-3+3-3/2-1-1-1-3-3-3/2-3+3-3/2-3+3-3/2-3-3-3/2000000ζ65ζ65ζ6ζ6ζ65ζ6000000    complex lifted from C3.A4
ρ2433-3-3+3-3/2-3-3-3/2-1-11-3+3-3/2-3-3-3/23+3-3/23-3-3/2000000ζ6ζ6ζ65ζ65ζ32ζ3000000    complex lifted from C2×C3.A4
ρ256-20662i-2i0-2-200000000-2i2i-2i2i00000000    complex lifted from C42⋊C6
ρ266-2066-2i2i0-2-2000000002i-2i2i-2i00000000    complex lifted from C42⋊C6
ρ276-20-3+3-3-3-3-3-2i2i01--31+-3000000004ζ3243ζ324ζ343ζ300000000    complex faithful
ρ286-20-3-3-3-3+3-32i-2i01+-31--30000000043ζ34ζ343ζ324ζ3200000000    complex faithful
ρ296-20-3+3-3-3-3-32i-2i01--31+-30000000043ζ324ζ3243ζ34ζ300000000    complex faithful
ρ306-20-3-3-3-3+3-3-2i2i01+-31--3000000004ζ343ζ34ζ3243ζ3200000000    complex faithful

Smallest permutation representation of C42⋊C18
On 72 points
Generators in S72
(1 27 36 10)(2 52 19 66)(3 58)(4 13 21 30)(5 37 22 69)(6 47)(7 33 24 16)(8 40 25 72)(9 64)(11 43 28 57)(12 53)(14 46 31 60)(15 70)(17 49 34 63)(18 41)(20 44)(23 61)(26 50)(29 67)(32 38)(35 55)(39 62 71 48)(42 51 56 65)(45 68 59 54)
(1 65 36 51)(2 57)(3 29 20 12)(4 68 21 54)(5 46)(6 15 23 32)(7 71 24 39)(8 63)(9 35 26 18)(10 56 27 42)(11 52)(13 59 30 45)(14 69)(16 62 33 48)(17 40)(19 43)(22 60)(25 49)(28 66)(31 37)(34 72)(38 47 70 61)(41 64 55 50)(44 53 58 67)
(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)

G:=sub<Sym(72)| (1,27,36,10)(2,52,19,66)(3,58)(4,13,21,30)(5,37,22,69)(6,47)(7,33,24,16)(8,40,25,72)(9,64)(11,43,28,57)(12,53)(14,46,31,60)(15,70)(17,49,34,63)(18,41)(20,44)(23,61)(26,50)(29,67)(32,38)(35,55)(39,62,71,48)(42,51,56,65)(45,68,59,54), (1,65,36,51)(2,57)(3,29,20,12)(4,68,21,54)(5,46)(6,15,23,32)(7,71,24,39)(8,63)(9,35,26,18)(10,56,27,42)(11,52)(13,59,30,45)(14,69)(16,62,33,48)(17,40)(19,43)(22,60)(25,49)(28,66)(31,37)(34,72)(38,47,70,61)(41,64,55,50)(44,53,58,67), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;

G:=Group( (1,27,36,10)(2,52,19,66)(3,58)(4,13,21,30)(5,37,22,69)(6,47)(7,33,24,16)(8,40,25,72)(9,64)(11,43,28,57)(12,53)(14,46,31,60)(15,70)(17,49,34,63)(18,41)(20,44)(23,61)(26,50)(29,67)(32,38)(35,55)(39,62,71,48)(42,51,56,65)(45,68,59,54), (1,65,36,51)(2,57)(3,29,20,12)(4,68,21,54)(5,46)(6,15,23,32)(7,71,24,39)(8,63)(9,35,26,18)(10,56,27,42)(11,52)(13,59,30,45)(14,69)(16,62,33,48)(17,40)(19,43)(22,60)(25,49)(28,66)(31,37)(34,72)(38,47,70,61)(41,64,55,50)(44,53,58,67), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );

G=PermutationGroup([(1,27,36,10),(2,52,19,66),(3,58),(4,13,21,30),(5,37,22,69),(6,47),(7,33,24,16),(8,40,25,72),(9,64),(11,43,28,57),(12,53),(14,46,31,60),(15,70),(17,49,34,63),(18,41),(20,44),(23,61),(26,50),(29,67),(32,38),(35,55),(39,62,71,48),(42,51,56,65),(45,68,59,54)], [(1,65,36,51),(2,57),(3,29,20,12),(4,68,21,54),(5,46),(6,15,23,32),(7,71,24,39),(8,63),(9,35,26,18),(10,56,27,42),(11,52),(13,59,30,45),(14,69),(16,62,33,48),(17,40),(19,43),(22,60),(25,49),(28,66),(31,37),(34,72),(38,47,70,61),(41,64,55,50),(44,53,58,67)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)])

Matrix representation of C42⋊C18 in GL6(𝔽37)

1350000
1360000
9116000
9110600
12270006
352700310
,
3100000
0310000
0003100
2906000
200001
12000360
,
1000022
00002611
81000027
8000027
230010036
230100036

G:=sub<GL(6,GF(37))| [1,1,9,9,12,35,35,36,11,11,27,27,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,31,0,0,0,0,6,0],[31,0,0,29,2,12,0,31,0,0,0,0,0,0,0,6,0,0,0,0,31,0,0,0,0,0,0,0,0,36,0,0,0,0,1,0],[1,0,8,8,23,23,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0,0,26,0,0,0,0,22,11,27,27,36,36] >;

C42⋊C18 in GAP, Magma, Sage, TeX

C_4^2\rtimes C_{18}
% in TeX

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

G:=SmallGroup(288,74);
// by ID

G=gap.SmallGroup(288,74);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,2523,514,360,6304,3476,102,3036,5305]);
// Polycyclic

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

Export

Subgroup lattice of C42⋊C18 in TeX
Character table of C42⋊C18 in TeX

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