direct product, metabelian, supersoluble, monomial
Aliases: C2×C9⋊D12, D6⋊6D18, C18⋊2D12, Dic9⋊4D6, C62.69D6, C9⋊3(C2×D12), (C3×C18)⋊3D4, C6⋊1(C9⋊D4), (C6×Dic9)⋊8C2, (C2×Dic9)⋊4S3, (S3×C6).33D6, (C2×C6).22D18, (C2×C18).22D6, (C22×S3)⋊3D9, (S3×C18)⋊9C22, C22.17(S3×D9), C6.24(C22×D9), (C3×C18).24C23, (C6×C18).18C22, C18.24(C22×S3), (C3×Dic9)⋊7C22, C6.21(C3⋊D12), (C3×C9)⋊6(C2×D4), (S3×C2×C18)⋊5C2, C6.43(C2×S32), (C2×C6).28S32, C3⋊1(C2×C9⋊D4), (S3×C2×C6).5S3, C2.24(C2×S3×D9), (C22×C9⋊S3)⋊2C2, (C2×C9⋊S3)⋊6C22, C3.2(C2×C3⋊D12), C32.3(C2×C3⋊D4), (C3×C6).55(C3⋊D4), (C3×C6).92(C22×S3), SmallGroup(432,312)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C9⋊D12
G = < a,b,c,d | a2=b9=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1356 in 194 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C3×C9, Dic9, D18, C2×C18, C2×C18, C3×Dic3, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×D12, C2×C3⋊D4, S3×C9, C9⋊S3, C3×C18, C3×C18, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C3⋊D12, C6×Dic3, S3×C2×C6, C22×C3⋊S3, C3×Dic9, S3×C18, S3×C18, C2×C9⋊S3, C2×C9⋊S3, C6×C18, C2×C9⋊D4, C2×C3⋊D12, C9⋊D12, C6×Dic9, S3×C2×C18, C22×C9⋊S3, C2×C9⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C3⋊D4, C22×S3, D18, S32, C2×D12, C2×C3⋊D4, C9⋊D4, C22×D9, C3⋊D12, C2×S32, S3×D9, C2×C9⋊D4, C2×C3⋊D12, C9⋊D12, C2×S3×D9, C2×C9⋊D12
►On 72 points
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 49)(11 50)(12 51)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 25)(21 26)(22 27)(23 28)(24 29)(37 72)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)
(1 62 13 9 70 21 5 66 17)(2 18 67 6 22 71 10 14 63)(3 64 15 11 72 23 7 68 19)(4 20 69 8 24 61 12 16 65)(25 46 59 29 38 51 33 42 55)(26 56 43 34 52 39 30 60 47)(27 48 49 31 40 53 35 44 57)(28 58 45 36 54 41 32 50 37)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 72)(22 71)(23 70)(24 69)(25 38)(26 37)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(49 57)(50 56)(51 55)(52 54)(58 60)
G:=sub<Sym(72)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71), (1,62,13,9,70,21,5,66,17)(2,18,67,6,22,71,10,14,63)(3,64,15,11,72,23,7,68,19)(4,20,69,8,24,61,12,16,65)(25,46,59,29,38,51,33,42,55)(26,56,43,34,52,39,30,60,47)(27,48,49,31,40,53,35,44,57)(28,58,45,36,54,41,32,50,37), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(25,38)(26,37)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(49,57)(50,56)(51,55)(52,54)(58,60)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,49)(11,50)(12,51)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,25)(21,26)(22,27)(23,28)(24,29)(37,72)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71), (1,62,13,9,70,21,5,66,17)(2,18,67,6,22,71,10,14,63)(3,64,15,11,72,23,7,68,19)(4,20,69,8,24,61,12,16,65)(25,46,59,29,38,51,33,42,55)(26,56,43,34,52,39,30,60,47)(27,48,49,31,40,53,35,44,57)(28,58,45,36,54,41,32,50,37), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,72)(22,71)(23,70)(24,69)(25,38)(26,37)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(49,57)(50,56)(51,55)(52,54)(58,60) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,49),(11,50),(12,51),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,25),(21,26),(22,27),(23,28),(24,29),(37,72),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71)], [(1,62,13,9,70,21,5,66,17),(2,18,67,6,22,71,10,14,63),(3,64,15,11,72,23,7,68,19),(4,20,69,8,24,61,12,16,65),(25,46,59,29,38,51,33,42,55),(26,56,43,34,52,39,30,60,47),(27,48,49,31,40,53,35,44,57),(28,58,45,36,54,41,32,50,37)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,72),(22,71),(23,70),(24,69),(25,38),(26,37),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,39),(49,57),(50,56),(51,55),(52,54),(58,60)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18R | 18S | ··· | 18AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 54 | 54 | 2 | 2 | 4 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D6 | D9 | D12 | C3⋊D4 | D18 | D18 | C9⋊D4 | S32 | C3⋊D12 | C2×S32 | S3×D9 | C9⋊D12 | C2×S3×D9 |
| kernel | C2×C9⋊D12 | C9⋊D12 | C6×Dic9 | S3×C2×C18 | C22×C9⋊S3 | C2×Dic9 | S3×C2×C6 | C3×C18 | Dic9 | C2×C18 | S3×C6 | C62 | C22×S3 | C18 | C3×C6 | D6 | C2×C6 | C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 1 | 2 | 1 | 3 | 6 | 3 |
Matrix representation of C2×C9⋊D12 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 26 |
| 0 | 0 | 0 | 0 | 11 | 17 |
| 27 | 32 | 0 | 0 | 0 | 0 |
| 5 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 5 | 0 | 0 |
| 0 | 0 | 32 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 36 | 36 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,11,0,0,0,0,26,17],[27,5,0,0,0,0,32,32,0,0,0,0,0,0,10,32,0,0,0,0,5,5,0,0,0,0,0,0,1,36,0,0,0,0,0,36],[0,36,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,1] >;
C2×C9⋊D12 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes D_{12} % in TeX
G:=Group("C2xC9:D12"); // GroupNames label
G:=SmallGroup(432,312);
// by ID
G=gap.SmallGroup(432,312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,64,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^12=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