direct product, metabelian, supersoluble, monomial
Aliases: C3×C3⋊Q16, C32⋊6Q16, C12.36D6, Dic6.2C6, C3⋊C8.C6, C4.4(S3×C6), C3⋊2(C3×Q16), C12.4(C2×C6), C6.10(C3×D4), (C3×C6).31D4, (C3×Q8).5C6, Q8.3(C3×S3), (C3×Q8).10S3, C6.32(C3⋊D4), (C3×Dic6).3C2, (Q8×C32).1C2, (C3×C12).11C22, (C3×C3⋊C8).2C2, C2.7(C3×C3⋊D4), SmallGroup(144,83)
Series: Derived ►Chief ►Lower central ►Upper central
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
(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 He3⋊6Q16 Dic18.C6 He3⋊7Q16 C32.CSU2(𝔽3) C32⋊CSU2(𝔽3) C32⋊2CSU2(𝔽3)
C3×C3⋊Q16 is a maximal quotient of
He3⋊6Q16 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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