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## G = C3×C3⋊Q16order 144 = 24·32

### Direct product of C3 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C3⋊Q16
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×Dic6 — C3×C3⋊Q16
 Lower central C3 — C6 — C12 — C3×C3⋊Q16
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for C3×C3⋊Q16
G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Smallest permutation representation of C3×C3⋊Q16
On 48 points
Generators in S48
(1 11 46)(2 12 47)(3 13 48)(4 14 41)(5 15 42)(6 16 43)(7 9 44)(8 10 45)(17 34 28)(18 35 29)(19 36 30)(20 37 31)(21 38 32)(22 39 25)(23 40 26)(24 33 27)
(1 11 46)(2 47 12)(3 13 48)(4 41 14)(5 15 42)(6 43 16)(7 9 44)(8 45 10)(17 34 28)(18 29 35)(19 36 30)(20 31 37)(21 38 32)(22 25 39)(23 40 26)(24 27 33)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 34 13 38)(10 33 14 37)(11 40 15 36)(12 39 16 35)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)

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

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

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

C3×C3⋊Q16 is a maximal subgroup of
D12.11D6  Dic6.9D6  Dic6.10D6  Dic6.22D6  D12.13D6  D12.14D6  D12.15D6  C3×S3×Q16  He36Q16  Dic18.C6  He37Q16  C32.CSU2(𝔽3)  C32⋊CSU2(𝔽3)  C322CSU2(𝔽3)
C3×C3⋊Q16 is a maximal quotient of
He36Q16  Dic18.C6

36 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C ··· 12M 12N 12O 24A 24B 24C 24D order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 8 8 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 1 1 2 2 2 2 4 12 1 1 2 2 2 6 6 2 2 4 ··· 4 12 12 6 6 6 6

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - - image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 Q16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×Q16 C3×C3⋊D4 C3⋊Q16 C3×C3⋊Q16 kernel C3×C3⋊Q16 C3×C3⋊C8 C3×Dic6 Q8×C32 C3⋊Q16 C3⋊C8 Dic6 C3×Q8 C3×Q8 C3×C6 C12 C32 Q8 C6 C6 C4 C3 C2 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 2 4 4 1 2

Matrix representation of C3×C3⋊Q16 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 4 4 6 1 5 6 3 5 5 6 1 3 4 5 4
,
 5 0 0 5 5 4 1 0 1 2 0 0 0 1 0 5
,
 2 1 2 5 2 0 2 2 6 4 1 5 6 5 4 4
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[5,5,1,0,0,4,2,1,0,1,0,0,5,0,0,5],[2,2,6,6,1,0,4,5,2,2,1,4,5,2,5,4] >;

C3×C3⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC3:Q16");
// GroupNames label

G:=SmallGroup(144,83);
// by ID

G=gap.SmallGroup(144,83);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,151,867,441,69,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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