direct product, metabelian, supersoluble, monomial
Aliases: C2×Dic9⋊C6, C62.43D6, C9⋊3(C6×D4), C18⋊2(C3×D4), C9⋊D4⋊4C6, D18⋊3(C2×C6), C9⋊C12⋊2C22, C23⋊3(C9⋊C6), (C22×C18)⋊2C6, Dic9⋊2(C2×C6), (C2×Dic9)⋊4C6, (C22×D9)⋊3C6, (C2×C62).12S3, C18.10(C22×C6), 3- 1+2⋊3(C2×D4), (C2×3- 1+2)⋊2D4, (C23×3- 1+2)⋊1C2, (C2×3- 1+2).10C23, (C22×3- 1+2)⋊2C22, (C2×C9⋊D4)⋊C3, C6.48(S3×C2×C6), (C2×C9⋊C12)⋊4C2, (C2×C18)⋊4(C2×C6), C22⋊4(C2×C9⋊C6), C3.3(C6×C3⋊D4), C32.(C2×C3⋊D4), (C2×C9⋊C6)⋊3C22, (C22×C9⋊C6)⋊3C2, (C2×C6).66(S3×C6), C6.35(C3×C3⋊D4), C2.10(C22×C9⋊C6), (C3×C6).40(C3⋊D4), (C3×C6).38(C22×S3), (C22×C6).32(C3×S3), SmallGroup(432,379)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic9⋊C6
G = < a,b,c,d | a2=b18=d6=1, c2=b9, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b7, dcd-1=b9c >
Subgroups: 654 in 178 conjugacy classes, 62 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, 3- 1+2, Dic9, D18, D18, C2×C18, C2×C18, C2×C18, C3×Dic3, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, C9⋊C6, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C22×C18, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C9⋊C12, C2×C9⋊C6, C2×C9⋊C6, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, C2×C9⋊D4, C6×C3⋊D4, C2×C9⋊C12, Dic9⋊C6, C22×C9⋊C6, C23×3- 1+2, C2×Dic9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, S3×C6, C2×C3⋊D4, C6×D4, C9⋊C6, C3×C3⋊D4, S3×C2×C6, C2×C9⋊C6, C6×C3⋊D4, Dic9⋊C6, C22×C9⋊C6, C2×Dic9⋊C6
►On 72 points
(1 34)(2 35)(3 36)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 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 46 10 37)(2 45 11 54)(3 44 12 53)(4 43 13 52)(5 42 14 51)(6 41 15 50)(7 40 16 49)(8 39 17 48)(9 38 18 47)(19 61 28 70)(20 60 29 69)(21 59 30 68)(22 58 31 67)(23 57 32 66)(24 56 33 65)(25 55 34 64)(26 72 35 63)(27 71 36 62)
(1 34)(2 29 8 35 14 23)(3 24 15 36 9 30)(4 19)(5 32 11 20 17 26)(6 27 18 21 12 33)(7 22)(10 25)(13 28)(16 31)(37 64)(38 59 44 65 50 71)(39 72 51 66 45 60)(40 67)(41 62 47 68 53 56)(42 57 54 69 48 63)(43 70)(46 55)(49 58)(52 61)
G:=sub<Sym(72)| (1,34)(2,35)(3,36)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,46,10,37)(2,45,11,54)(3,44,12,53)(4,43,13,52)(5,42,14,51)(6,41,15,50)(7,40,16,49)(8,39,17,48)(9,38,18,47)(19,61,28,70)(20,60,29,69)(21,59,30,68)(22,58,31,67)(23,57,32,66)(24,56,33,65)(25,55,34,64)(26,72,35,63)(27,71,36,62), (1,34)(2,29,8,35,14,23)(3,24,15,36,9,30)(4,19)(5,32,11,20,17,26)(6,27,18,21,12,33)(7,22)(10,25)(13,28)(16,31)(37,64)(38,59,44,65,50,71)(39,72,51,66,45,60)(40,67)(41,62,47,68,53,56)(42,57,54,69,48,63)(43,70)(46,55)(49,58)(52,61)>;
G:=Group( (1,34)(2,35)(3,36)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72), (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), (1,46,10,37)(2,45,11,54)(3,44,12,53)(4,43,13,52)(5,42,14,51)(6,41,15,50)(7,40,16,49)(8,39,17,48)(9,38,18,47)(19,61,28,70)(20,60,29,69)(21,59,30,68)(22,58,31,67)(23,57,32,66)(24,56,33,65)(25,55,34,64)(26,72,35,63)(27,71,36,62), (1,34)(2,29,8,35,14,23)(3,24,15,36,9,30)(4,19)(5,32,11,20,17,26)(6,27,18,21,12,33)(7,22)(10,25)(13,28)(16,31)(37,64)(38,59,44,65,50,71)(39,72,51,66,45,60)(40,67)(41,62,47,68,53,56)(42,57,54,69,48,63)(43,70)(46,55)(49,58)(52,61) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72)], [(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)], [(1,46,10,37),(2,45,11,54),(3,44,12,53),(4,43,13,52),(5,42,14,51),(6,41,15,50),(7,40,16,49),(8,39,17,48),(9,38,18,47),(19,61,28,70),(20,60,29,69),(21,59,30,68),(22,58,31,67),(23,57,32,66),(24,56,33,65),(25,55,34,64),(26,72,35,63),(27,71,36,62)], [(1,34),(2,29,8,35,14,23),(3,24,15,36,9,30),(4,19),(5,32,11,20,17,26),(6,27,18,21,12,33),(7,22),(10,25),(13,28),(16,31),(37,64),(38,59,44,65,50,71),(39,72,51,66,45,60),(40,67),(41,62,47,68,53,56),(42,57,54,69,48,63),(43,70),(46,55),(49,58),(52,61)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | ··· | 6G | 6H | ··· | 6M | 6N | 6O | 6P | 6Q | 6R | 6S | 6T | 6U | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 18 | 18 | 2 | 3 | 3 | 18 | 18 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 6 | ··· | 6 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | C3×S3 | C3×D4 | C3⋊D4 | S3×C6 | C3×C3⋊D4 | C9⋊C6 | C2×C9⋊C6 | Dic9⋊C6 |
| kernel | C2×Dic9⋊C6 | C2×C9⋊C12 | Dic9⋊C6 | C22×C9⋊C6 | C23×3- 1+2 | C2×C9⋊D4 | C2×Dic9 | C9⋊D4 | C22×D9 | C22×C18 | C2×C62 | C2×3- 1+2 | C62 | C22×C6 | C18 | C3×C6 | C2×C6 | C6 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 1 | 2 | 3 | 2 | 4 | 4 | 6 | 8 | 1 | 3 | 4 |
Matrix representation of C2×Dic9⋊C6 ►in GL8(𝔽37)
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 27 | 0 | 0 | 0 |
| 0 | 0 | 26 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 18 | 18 | 26 | 1 | 9 |
| 0 | 0 | 13 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 32 | 5 | 10 | 26 | 11 |
| 25 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 23 | 31 | 0 |
| 0 | 0 | 2 | 30 | 33 | 31 | 8 | 35 |
| 0 | 0 | 5 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 24 | 11 | 15 | 34 | 7 | 26 |
| 0 | 0 | 0 | 21 | 27 | 1 | 33 | 34 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 31 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 15 | 0 | 13 | 0 | 11 | 0 |
| 0 | 0 | 30 | 22 | 12 | 11 | 10 | 27 |
G:=sub<GL(8,GF(37))| [36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,11,0,26,10,13,0,0,0,9,26,1,18,0,32,0,0,0,27,0,18,0,5,0,0,0,0,0,26,11,10,0,0,0,0,0,1,0,26,0,0,0,0,0,9,0,11],[25,1,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,34,0,2,5,24,0,0,0,0,0,30,0,11,21,0,0,0,33,33,0,15,27,0,0,35,23,31,3,34,1,0,0,0,31,8,0,7,33,0,0,0,0,35,0,26,34],[27,31,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,10,36,34,15,30,0,0,0,26,0,0,0,22,0,0,0,0,10,0,13,12,0,0,0,0,0,36,0,11,0,0,0,0,0,0,11,10,0,0,0,0,0,0,0,27] >;
C2×Dic9⋊C6 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_9\rtimes C_6 % in TeX
G:=Group("C2xDic9:C6"); // GroupNames label
G:=SmallGroup(432,379);
// by ID
G=gap.SmallGroup(432,379);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,10085,1034,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^18=d^6=1,c^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^9*c>;
// generators/relations