direct product, metabelian, supersoluble, monomial
Aliases: C9×D4⋊2S3, C36.48D6, Dic6⋊3C18, D4⋊2(S3×C9), (D4×C9)⋊5S3, (C4×S3)⋊2C18, (S3×C36)⋊8C2, (C3×D4)⋊3C18, C3⋊D4⋊2C18, C4.5(S3×C18), (C2×C18).7D6, (S3×C12).3C6, C12.54(S3×C6), C12.5(C2×C18), (C9×Dic6)⋊9C2, D6.2(C2×C18), (Dic3×C18)⋊9C2, (C2×Dic3)⋊3C18, C62.16(C2×C6), C6.6(C22×C18), C22.1(S3×C18), (C6×C18).1C22, (C3×Dic6).3C6, (S3×C18).6C22, (C3×C36).47C22, (C3×C18).33C23, C18.54(C22×S3), Dic3.3(C2×C18), (C6×Dic3).11C6, (D4×C32).10C6, (C9×Dic3).16C22, (D4×C3×C9)⋊10C2, C3⋊2(C9×C4○D4), (C2×C6).(C2×C18), C6.67(S3×C2×C6), C2.7(S3×C2×C18), (C9×C3⋊D4)⋊6C2, (C3×C9)⋊14(C4○D4), (S3×C6).8(C2×C6), (C2×C6).19(S3×C6), (C3×D4⋊2S3).C3, (C3×C3⋊D4).2C6, (C3×C12).32(C2×C6), (C3×D4).18(C3×S3), C3.4(C3×D4⋊2S3), C32.3(C3×C4○D4), (C3×C6).43(C22×C6), (C3×Dic3).13(C2×C6), SmallGroup(432,359)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D4⋊2S3
G = < a,b,c,d,e | a9=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 256 in 136 conjugacy classes, 69 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3×C12, S3×C6, C62, D4⋊2S3, C3×C4○D4, S3×C9, C3×C18, C3×C18, C2×C36, D4×C9, D4×C9, Q8×C9, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, D4×C32, C9×Dic3, C9×Dic3, C3×C36, S3×C18, C6×C18, C9×C4○D4, C3×D4⋊2S3, C9×Dic6, S3×C36, Dic3×C18, C9×C3⋊D4, D4×C3×C9, C9×D4⋊2S3
Quotients: C1, C2, C3, C22, S3, C6, C23, C9, D6, C2×C6, C4○D4, C18, C3×S3, C22×S3, C22×C6, C2×C18, S3×C6, D4⋊2S3, C3×C4○D4, S3×C9, C22×C18, S3×C2×C6, S3×C18, C9×C4○D4, C3×D4⋊2S3, S3×C2×C18, C9×D4⋊2S3
►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 31 20 38)(2 32 21 39)(3 33 22 40)(4 34 23 41)(5 35 24 42)(6 36 25 43)(7 28 26 44)(8 29 27 45)(9 30 19 37)(10 49 67 56)(11 50 68 57)(12 51 69 58)(13 52 70 59)(14 53 71 60)(15 54 72 61)(16 46 64 62)(17 47 65 63)(18 48 66 55)
(28 44)(29 45)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)(46 62)(47 63)(48 55)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 46)(8 47)(9 48)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 37)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 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), (1,31,20,38)(2,32,21,39)(3,33,22,40)(4,34,23,41)(5,35,24,42)(6,36,25,43)(7,28,26,44)(8,29,27,45)(9,30,19,37)(10,49,67,56)(11,50,68,57)(12,51,69,58)(13,52,70,59)(14,53,71,60)(15,54,72,61)(16,46,64,62)(17,47,65,63)(18,48,66,55), (28,44)(29,45)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43)(46,62)(47,63)(48,55)(49,56)(50,57)(51,58)(52,59)(53,60)(54,61), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,37)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)>;
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), (1,31,20,38)(2,32,21,39)(3,33,22,40)(4,34,23,41)(5,35,24,42)(6,36,25,43)(7,28,26,44)(8,29,27,45)(9,30,19,37)(10,49,67,56)(11,50,68,57)(12,51,69,58)(13,52,70,59)(14,53,71,60)(15,54,72,61)(16,46,64,62)(17,47,65,63)(18,48,66,55), (28,44)(29,45)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(36,43)(46,62)(47,63)(48,55)(49,56)(50,57)(51,58)(52,59)(53,60)(54,61), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,46)(8,47)(9,48)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,37)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72) );
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)], [(1,31,20,38),(2,32,21,39),(3,33,22,40),(4,34,23,41),(5,35,24,42),(6,36,25,43),(7,28,26,44),(8,29,27,45),(9,30,19,37),(10,49,67,56),(11,50,68,57),(12,51,69,58),(13,52,70,59),(14,53,71,60),(15,54,72,61),(16,46,64,62),(17,47,65,63),(18,48,66,55)], [(28,44),(29,45),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(36,43),(46,62),(47,63),(48,55),(49,56),(50,57),(51,58),(52,59),(53,60),(54,61)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,46),(8,47),(9,48),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,37),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6I | 6J | ··· | 6O | 6P | 6Q | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M | 18A | ··· | 18F | 18G | ··· | 18X | 18Y | ··· | 18AJ | 18AK | ··· | 18AP | 36A | ··· | 36F | 36G | ··· | 36R | 36S | ··· | 36X | 36Y | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | C18 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C3×C4○D4 | S3×C9 | S3×C18 | S3×C18 | C9×C4○D4 | D4⋊2S3 | C3×D4⋊2S3 | C9×D4⋊2S3 |
| kernel | C9×D4⋊2S3 | C9×Dic6 | S3×C36 | Dic3×C18 | C9×C3⋊D4 | D4×C3×C9 | C3×D4⋊2S3 | C3×Dic6 | S3×C12 | C6×Dic3 | C3×C3⋊D4 | D4×C32 | D4⋊2S3 | Dic6 | C4×S3 | C2×Dic3 | C3⋊D4 | C3×D4 | D4×C9 | C36 | C2×C18 | C3×C9 | C3×D4 | C12 | C2×C6 | C32 | D4 | C4 | C22 | C3 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 6 | 6 | 6 | 12 | 12 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 1 | 2 | 6 |
Matrix representation of C9×D4⋊2S3 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 21 | 2 | 0 | 0 |
| 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 16 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 6 | 26 |
| 15 | 12 | 0 | 0 |
| 6 | 22 | 0 | 0 |
| 0 | 0 | 8 | 9 |
| 0 | 0 | 30 | 29 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[21,1,0,0,2,16,0,0,0,0,1,0,0,0,0,1],[1,16,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,10,6,0,0,0,26],[15,6,0,0,12,22,0,0,0,0,8,30,0,0,9,29] >;
C9×D4⋊2S3 in GAP, Magma, Sage, TeX
C_9\times D_4\rtimes_2S_3
% in TeX
G:=Group("C9xD4:2S3"); // GroupNames label
G:=SmallGroup(432,359);
// by ID
G=gap.SmallGroup(432,359);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,192,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^4=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations