metabelian, supersoluble, monomial
Aliases: C36⋊2D6, D18⋊2D6, D6⋊2D18, D36⋊4S3, D12⋊4D9, C12⋊2D18, C9⋊S3⋊1D4, C9⋊2(S3×D4), C3⋊2(D4×D9), C4⋊3(S3×D9), C12.20S32, (C3×D36)⋊8C2, (C9×D12)⋊6C2, (S3×C6).4D6, D6⋊D9⋊3C2, (C3×C36)⋊3C22, (C3×D12).8S3, (C6×D9)⋊2C22, C32.3(S3×D4), (S3×C18)⋊2C22, (C3×C12).100D6, C9⋊Dic3⋊4C22, C6.14(C22×D9), C3.1(D6⋊D6), C18.14(C22×S3), (C3×C18).14C23, (C2×S3×D9)⋊3C2, (C3×C9)⋊3(C2×D4), (C4×C9⋊S3)⋊3C2, C6.33(C2×S32), C2.17(C2×S3×D9), (C2×C9⋊S3).10C22, (C3×C6).82(C22×S3), SmallGroup(432,293)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C36⋊D6
G = < a,b,c | a36=b6=c2=1, bab-1=a-1, cac=a17, cbc=b-1 >
Subgroups: 1352 in 178 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×D4, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, D12, C3⋊D4, C3×D4, C22×S3, C3×C9, Dic9, C36, C36, D18, D18, C2×C18, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, S3×D4, C3×D9, S3×C9, C9⋊S3, C3×C18, C4×D9, D36, C9⋊D4, D4×C9, C22×D9, D6⋊S3, C3×D12, C3×D12, C4×C3⋊S3, C2×S32, C9⋊Dic3, C3×C36, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, D4×D9, D6⋊D6, D6⋊D9, C3×D36, C9×D12, C4×C9⋊S3, C2×S3×D9, C36⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, D18, S32, S3×D4, C22×D9, C2×S32, S3×D9, D4×D9, D6⋊D6, C2×S3×D9, C36⋊D6
►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 49 13 37 25 61)(2 48 14 72 26 60)(3 47 15 71 27 59)(4 46 16 70 28 58)(5 45 17 69 29 57)(6 44 18 68 30 56)(7 43 19 67 31 55)(8 42 20 66 32 54)(9 41 21 65 33 53)(10 40 22 64 34 52)(11 39 23 63 35 51)(12 38 24 62 36 50)
(1 13)(2 30)(3 11)(4 28)(5 9)(6 26)(8 24)(10 22)(12 20)(14 18)(15 35)(17 33)(19 31)(21 29)(23 27)(32 36)(37 61)(38 42)(39 59)(41 57)(43 55)(44 72)(45 53)(46 70)(47 51)(48 68)(50 66)(52 64)(54 62)(56 60)(63 71)(65 69)
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), (1,49,13,37,25,61)(2,48,14,72,26,60)(3,47,15,71,27,59)(4,46,16,70,28,58)(5,45,17,69,29,57)(6,44,18,68,30,56)(7,43,19,67,31,55)(8,42,20,66,32,54)(9,41,21,65,33,53)(10,40,22,64,34,52)(11,39,23,63,35,51)(12,38,24,62,36,50), (1,13)(2,30)(3,11)(4,28)(5,9)(6,26)(8,24)(10,22)(12,20)(14,18)(15,35)(17,33)(19,31)(21,29)(23,27)(32,36)(37,61)(38,42)(39,59)(41,57)(43,55)(44,72)(45,53)(46,70)(47,51)(48,68)(50,66)(52,64)(54,62)(56,60)(63,71)(65,69)>;
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), (1,49,13,37,25,61)(2,48,14,72,26,60)(3,47,15,71,27,59)(4,46,16,70,28,58)(5,45,17,69,29,57)(6,44,18,68,30,56)(7,43,19,67,31,55)(8,42,20,66,32,54)(9,41,21,65,33,53)(10,40,22,64,34,52)(11,39,23,63,35,51)(12,38,24,62,36,50), (1,13)(2,30)(3,11)(4,28)(5,9)(6,26)(8,24)(10,22)(12,20)(14,18)(15,35)(17,33)(19,31)(21,29)(23,27)(32,36)(37,61)(38,42)(39,59)(41,57)(43,55)(44,72)(45,53)(46,70)(47,51)(48,68)(50,66)(52,64)(54,62)(56,60)(63,71)(65,69) );
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)], [(1,49,13,37,25,61),(2,48,14,72,26,60),(3,47,15,71,27,59),(4,46,16,70,28,58),(5,45,17,69,29,57),(6,44,18,68,30,56),(7,43,19,67,31,55),(8,42,20,66,32,54),(9,41,21,65,33,53),(10,40,22,64,34,52),(11,39,23,63,35,51),(12,38,24,62,36,50)], [(1,13),(2,30),(3,11),(4,28),(5,9),(6,26),(8,24),(10,22),(12,20),(14,18),(15,35),(17,33),(19,31),(21,29),(23,27),(32,36),(37,61),(38,42),(39,59),(41,57),(43,55),(44,72),(45,53),(46,70),(47,51),(48,68),(50,66),(52,64),(54,62),(56,60),(63,71),(65,69)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 6 | 18 | 18 | 27 | 27 | 2 | 2 | 4 | 2 | 54 | 2 | 2 | 4 | 12 | 12 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D6 | D9 | D18 | D18 | S32 | S3×D4 | S3×D4 | C2×S32 | S3×D9 | D4×D9 | D6⋊D6 | C2×S3×D9 | C36⋊D6 |
| kernel | C36⋊D6 | D6⋊D9 | C3×D36 | C9×D12 | C4×C9⋊S3 | C2×S3×D9 | D36 | C3×D12 | C9⋊S3 | C36 | D18 | C3×C12 | S3×C6 | D12 | C12 | D6 | C12 | C9 | C32 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 3 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 2 | 3 | 6 |
Matrix representation of C36⋊D6 ►in GL6(𝔽37)
| 1 | 3 | 0 | 0 | 0 | 0 |
| 24 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 11 |
| 0 | 0 | 0 | 0 | 26 | 31 |
| 15 | 32 | 0 | 0 | 0 | 0 |
| 30 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(37))| [1,24,0,0,0,0,3,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,26,0,0,0,0,11,31],[15,30,0,0,0,0,32,22,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C36⋊D6 in GAP, Magma, Sage, TeX
C_{36}\rtimes D_6 % in TeX
G:=Group("C36:D6"); // GroupNames label
G:=SmallGroup(432,293);
// by ID
G=gap.SmallGroup(432,293);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c|a^36=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations