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G = C3×C3⋊D16order 288 = 25·32

Direct product of C3 and C3⋊D16

direct product, metabelian, supersoluble, monomial

Aliases: C3×C3⋊D16, D243C6, C326D16, C24.52D6, C3⋊C161C6, (C3×D8)⋊1C6, (C3×D8)⋊5S3, D81(C3×S3), C32(C3×D16), C8.4(S3×C6), C6.8(C3×D8), (C3×D24)⋊7C2, C24.2(C2×C6), (C3×C6).30D8, C12.3(C3×D4), (C3×C12).41D4, (C32×D8)⋊1C2, C6.30(D4⋊S3), C12.83(C3⋊D4), (C3×C24).13C22, (C3×C3⋊C16)⋊4C2, C2.4(C3×D4⋊S3), C4.1(C3×C3⋊D4), SmallGroup(288,260)

Series: Derived Chief Lower central Upper central

C1C24 — C3×C3⋊D16
C1C3C6C12C24C3×C24C3×D24 — C3×C3⋊D16
C3C6C12C24 — C3×C3⋊D16
C1C6C12C24C3×D8

Generators and relations for C3×C3⋊D16
 G = < a,b,c,d | a3=b3=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 250 in 67 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], S3, C6 [×2], C6 [×6], C8, D4 [×2], C32, C12 [×2], C12, D6, C2×C6 [×5], C16, D8, D8, C3×S3, C3×C6, C3×C6, C24 [×2], C24, D12, C3×D4 [×5], D16, C3×C12, S3×C6, C62, C3⋊C16, C48, D24, C3×D8 [×2], C3×D8 [×2], C3×C24, C3×D12, D4×C32, C3⋊D16, C3×D16, C3×C3⋊C16, C3×D24, C32×D8, C3×C3⋊D16
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, D16, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, C3⋊D16, C3×D16, C3×D4⋊S3, C3×C3⋊D16

Smallest permutation representation of C3×C3⋊D16
On 48 points
Generators in S48
(1 44 28)(2 45 29)(3 46 30)(4 47 31)(5 48 32)(6 33 17)(7 34 18)(8 35 19)(9 36 20)(10 37 21)(11 38 22)(12 39 23)(13 40 24)(14 41 25)(15 42 26)(16 43 27)
(1 44 28)(2 29 45)(3 46 30)(4 31 47)(5 48 32)(6 17 33)(7 34 18)(8 19 35)(9 36 20)(10 21 37)(11 38 22)(12 23 39)(13 40 24)(14 25 41)(15 42 26)(16 27 43)
(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 22)(18 21)(19 20)(23 32)(24 31)(25 30)(26 29)(27 28)(33 38)(34 37)(35 36)(39 48)(40 47)(41 46)(42 45)(43 44)

G:=sub<Sym(48)| (1,44,28)(2,45,29)(3,46,30)(4,47,31)(5,48,32)(6,33,17)(7,34,18)(8,35,19)(9,36,20)(10,37,21)(11,38,22)(12,39,23)(13,40,24)(14,41,25)(15,42,26)(16,43,27), (1,44,28)(2,29,45)(3,46,30)(4,31,47)(5,48,32)(6,17,33)(7,34,18)(8,19,35)(9,36,20)(10,21,37)(11,38,22)(12,23,39)(13,40,24)(14,25,41)(15,42,26)(16,27,43), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,38)(34,37)(35,36)(39,48)(40,47)(41,46)(42,45)(43,44)>;

G:=Group( (1,44,28)(2,45,29)(3,46,30)(4,47,31)(5,48,32)(6,33,17)(7,34,18)(8,35,19)(9,36,20)(10,37,21)(11,38,22)(12,39,23)(13,40,24)(14,41,25)(15,42,26)(16,43,27), (1,44,28)(2,29,45)(3,46,30)(4,31,47)(5,48,32)(6,17,33)(7,34,18)(8,19,35)(9,36,20)(10,21,37)(11,38,22)(12,23,39)(13,40,24)(14,25,41)(15,42,26)(16,27,43), (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), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,38)(34,37)(35,36)(39,48)(40,47)(41,46)(42,45)(43,44) );

G=PermutationGroup([(1,44,28),(2,45,29),(3,46,30),(4,47,31),(5,48,32),(6,33,17),(7,34,18),(8,35,19),(9,36,20),(10,37,21),(11,38,22),(12,39,23),(13,40,24),(14,41,25),(15,42,26),(16,43,27)], [(1,44,28),(2,29,45),(3,46,30),(4,31,47),(5,48,32),(6,17,33),(7,34,18),(8,19,35),(9,36,20),(10,21,37),(11,38,22),(12,23,39),(13,40,24),(14,25,41),(15,42,26),(16,27,43)], [(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)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,22),(18,21),(19,20),(23,32),(24,31),(25,30),(26,29),(27,28),(33,38),(34,37),(35,36),(39,48),(40,47),(41,46),(42,45),(43,44)])

54 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F···6M6N6O8A8B12A12B12C12D12E16A16B16C16D24A24B24C24D24E···24J48A···48H
order1222333334666666···666881212121212161616162424242424···2448···48
size11824112222112228···824242222444666622224···46···6

54 irreducible representations

dim111111112222222222224444
type+++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D8C3×S3C3⋊D4C3×D4D16S3×C6C3×D8C3×C3⋊D4C3×D16D4⋊S3C3⋊D16C3×D4⋊S3C3×C3⋊D16
kernelC3×C3⋊D16C3×C3⋊C16C3×D24C32×D8C3⋊D16C3⋊C16D24C3×D8C3×D8C3×C12C24C3×C6D8C12C12C32C8C6C4C3C6C3C2C1
# reps111122221112222424481224

Matrix representation of C3×C3⋊D16 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
6263
6332
0020
5511
,
0162
4003
4452
5432
,
1063
0232
0322
0112
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,6,0,5,2,3,0,5,6,3,2,1,3,2,0,1],[0,4,4,5,1,0,4,4,6,0,5,3,2,3,2,2],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2] >;

C3×C3⋊D16 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes D_{16}
% in TeX

G:=Group("C3xC3:D16");
// GroupNames label

G:=SmallGroup(288,260);
// by ID

G=gap.SmallGroup(288,260);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,1011,514,192,2524,1271,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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