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## G = C3×C3⋊D16order 288 = 25·32

### Direct product of C3 and C3⋊D16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C3⋊D16
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×D24 — C3×C3⋊D16
 Lower central C3 — C6 — C12 — C24 — C3×C3⋊D16
 Upper central C1 — C6 — C12 — C24 — C3×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 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F ··· 6M 6N 6O 8A 8B 12A 12B 12C 12D 12E 16A 16B 16C 16D 24A 24B 24C 24D 24E ··· 24J 48A ··· 48H order 1 2 2 2 3 3 3 3 3 4 6 6 6 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 16 16 16 16 24 24 24 24 24 ··· 24 48 ··· 48 size 1 1 8 24 1 1 2 2 2 2 1 1 2 2 2 8 ··· 8 24 24 2 2 2 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 ··· 6

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D8 C3×S3 C3⋊D4 C3×D4 D16 S3×C6 C3×D8 C3×C3⋊D4 C3×D16 D4⋊S3 C3⋊D16 C3×D4⋊S3 C3×C3⋊D16 kernel C3×C3⋊D16 C3×C3⋊C16 C3×D24 C32×D8 C3⋊D16 C3⋊C16 D24 C3×D8 C3×D8 C3×C12 C24 C3×C6 D8 C12 C12 C32 C8 C6 C4 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 4 2 4 4 8 1 2 2 4

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 6 2 6 3 6 3 3 2 0 0 2 0 5 5 1 1
,
 0 1 6 2 4 0 0 3 4 4 5 2 5 4 3 2
,
 1 0 6 3 0 2 3 2 0 3 2 2 0 1 1 2
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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