direct product, metabelian, supersoluble, monomial
Aliases: C9×C6.D4, C62.19C12, (C2×C6)⋊4C36, (C6×C18)⋊1C4, C6.9(C2×C36), C6.11(D4×C9), (C2×C18)⋊3Dic3, (C2×C18).50D6, (C3×C18).38D4, C23.3(S3×C9), (C2×Dic3)⋊2C18, (Dic3×C18)⋊4C2, (C22×C18).5S3, (C22×C6).8C18, C62.52(C2×C6), C22.7(S3×C18), (C2×C62).23C6, (C6×Dic3).4C6, C6.33(C6×Dic3), C2.5(Dic3×C18), C22⋊3(C9×Dic3), C18.35(C3⋊D4), (C6×C18).25C22, C18.21(C2×Dic3), (C2×C6×C18).1C2, C3⋊2(C9×C22⋊C4), C2.3(C9×C3⋊D4), (C3×C9)⋊7(C22⋊C4), (C2×C6).84(S3×C6), (C3×C6).59(C3×D4), C6.49(C3×C3⋊D4), (C3×C6).54(C2×C12), (C2×C6).10(C2×C18), (C3×C18).31(C2×C4), (C3×C6.D4).C3, (C22×C6).21(C3×S3), (C2×C6).10(C3×Dic3), C3.4(C3×C6.D4), C32.3(C3×C22⋊C4), SmallGroup(432,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C6.D4
G = < a,b,c,d | a9=b6=c4=1, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 244 in 134 conjugacy classes, 57 normal (27 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, C36, C2×C18, C2×C18, C2×C18, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C3×C18, C3×C18, C3×C18, C2×C36, C22×C18, C22×C18, C6×Dic3, C2×C62, C9×Dic3, C6×C18, C6×C18, C6×C18, C9×C22⋊C4, C3×C6.D4, Dic3×C18, C2×C6×C18, C9×C6.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C9, Dic3, C12, D6, C2×C6, C22⋊C4, C18, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C36, C2×C18, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, S3×C9, C2×C36, D4×C9, C6×Dic3, C3×C3⋊D4, C9×Dic3, S3×C18, C9×C22⋊C4, C3×C6.D4, Dic3×C18, C9×C3⋊D4, C9×C6.D4
►On 72 points
(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)
(1 50 7 47 4 53)(2 51 8 48 5 54)(3 52 9 49 6 46)(10 19 13 22 16 25)(11 20 14 23 17 26)(12 21 15 24 18 27)(28 44 34 41 31 38)(29 45 35 42 32 39)(30 37 36 43 33 40)(55 65 58 68 61 71)(56 66 59 69 62 72)(57 67 60 70 63 64)
(1 57 38 14)(2 58 39 15)(3 59 40 16)(4 60 41 17)(5 61 42 18)(6 62 43 10)(7 63 44 11)(8 55 45 12)(9 56 37 13)(19 49 72 36)(20 50 64 28)(21 51 65 29)(22 52 66 30)(23 53 67 31)(24 54 68 32)(25 46 69 33)(26 47 70 34)(27 48 71 35)
(1 26 47 14)(2 27 48 15)(3 19 49 16)(4 20 50 17)(5 21 51 18)(6 22 52 10)(7 23 53 11)(8 24 54 12)(9 25 46 13)(28 60 41 64)(29 61 42 65)(30 62 43 66)(31 63 44 67)(32 55 45 68)(33 56 37 69)(34 57 38 70)(35 58 39 71)(36 59 40 72)
G:=sub<Sym(72)| (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), (1,50,7,47,4,53)(2,51,8,48,5,54)(3,52,9,49,6,46)(10,19,13,22,16,25)(11,20,14,23,17,26)(12,21,15,24,18,27)(28,44,34,41,31,38)(29,45,35,42,32,39)(30,37,36,43,33,40)(55,65,58,68,61,71)(56,66,59,69,62,72)(57,67,60,70,63,64), (1,57,38,14)(2,58,39,15)(3,59,40,16)(4,60,41,17)(5,61,42,18)(6,62,43,10)(7,63,44,11)(8,55,45,12)(9,56,37,13)(19,49,72,36)(20,50,64,28)(21,51,65,29)(22,52,66,30)(23,53,67,31)(24,54,68,32)(25,46,69,33)(26,47,70,34)(27,48,71,35), (1,26,47,14)(2,27,48,15)(3,19,49,16)(4,20,50,17)(5,21,51,18)(6,22,52,10)(7,23,53,11)(8,24,54,12)(9,25,46,13)(28,60,41,64)(29,61,42,65)(30,62,43,66)(31,63,44,67)(32,55,45,68)(33,56,37,69)(34,57,38,70)(35,58,39,71)(36,59,40,72)>;
G:=Group( (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), (1,50,7,47,4,53)(2,51,8,48,5,54)(3,52,9,49,6,46)(10,19,13,22,16,25)(11,20,14,23,17,26)(12,21,15,24,18,27)(28,44,34,41,31,38)(29,45,35,42,32,39)(30,37,36,43,33,40)(55,65,58,68,61,71)(56,66,59,69,62,72)(57,67,60,70,63,64), (1,57,38,14)(2,58,39,15)(3,59,40,16)(4,60,41,17)(5,61,42,18)(6,62,43,10)(7,63,44,11)(8,55,45,12)(9,56,37,13)(19,49,72,36)(20,50,64,28)(21,51,65,29)(22,52,66,30)(23,53,67,31)(24,54,68,32)(25,46,69,33)(26,47,70,34)(27,48,71,35), (1,26,47,14)(2,27,48,15)(3,19,49,16)(4,20,50,17)(5,21,51,18)(6,22,52,10)(7,23,53,11)(8,24,54,12)(9,25,46,13)(28,60,41,64)(29,61,42,65)(30,62,43,66)(31,63,44,67)(32,55,45,68)(33,56,37,69)(34,57,38,70)(35,58,39,71)(36,59,40,72) );
G=PermutationGroup([[(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)], [(1,50,7,47,4,53),(2,51,8,48,5,54),(3,52,9,49,6,46),(10,19,13,22,16,25),(11,20,14,23,17,26),(12,21,15,24,18,27),(28,44,34,41,31,38),(29,45,35,42,32,39),(30,37,36,43,33,40),(55,65,58,68,61,71),(56,66,59,69,62,72),(57,67,60,70,63,64)], [(1,57,38,14),(2,58,39,15),(3,59,40,16),(4,60,41,17),(5,61,42,18),(6,62,43,10),(7,63,44,11),(8,55,45,12),(9,56,37,13),(19,49,72,36),(20,50,64,28),(21,51,65,29),(22,52,66,30),(23,53,67,31),(24,54,68,32),(25,46,69,33),(26,47,70,34),(27,48,71,35)], [(1,26,47,14),(2,27,48,15),(3,19,49,16),(4,20,50,17),(5,21,51,18),(6,22,52,10),(7,23,53,11),(8,24,54,12),(9,25,46,13),(28,60,41,64),(29,61,42,65),(30,62,43,66),(31,63,44,67),(32,55,45,68),(33,56,37,69),(34,57,38,70),(35,58,39,71),(36,59,40,72)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AE | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12H | 18A | ··· | 18R | 18S | ··· | 18BT | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | S3 | D4 | Dic3 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3×Dic3 | S3×C6 | S3×C9 | D4×C9 | C3×C3⋊D4 | C9×Dic3 | S3×C18 | C9×C3⋊D4 |
| kernel | C9×C6.D4 | Dic3×C18 | C2×C6×C18 | C3×C6.D4 | C6×C18 | C6×Dic3 | C2×C62 | C6.D4 | C62 | C2×Dic3 | C22×C6 | C2×C6 | C22×C18 | C3×C18 | C2×C18 | C2×C18 | C22×C6 | C18 | C3×C6 | C2×C6 | C2×C6 | C23 | C6 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C9×C6.D4 ►in GL4(𝔽37) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 27 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [34,0,0,0,0,34,0,0,0,0,9,0,0,0,0,9],[27,0,0,0,0,11,0,0,0,0,11,0,0,0,0,27],[0,36,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,36,0,0,1,0,0,0,0,0,0,36,0,0,1,0] >;
C9×C6.D4 in GAP, Magma, Sage, TeX
C_9\times C_6.D_4
% in TeX
G:=Group("C9xC6.D4"); // GroupNames label
G:=SmallGroup(432,165);
// by ID
G=gap.SmallGroup(432,165);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,268,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^6=c^4=1,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
// generators/relations