direct product, metabelian, supersoluble, monomial
Aliases: C3×D36⋊5C2, D36⋊11C6, C12.79D18, Dic18⋊11C6, C62.131D6, (C6×C36)⋊8C2, (C4×D9)⋊9C6, (C2×C12)⋊7D9, C9⋊D4⋊7C6, (C2×C36)⋊12C6, (C12×D9)⋊9C2, C4.16(C6×D9), (C3×D36)⋊11C2, C12.75(S3×C6), C36.39(C2×C6), (C6×C12).39S3, D18.6(C2×C6), (C2×C6).23D18, C22.2(C6×D9), (C3×C12).221D6, Dic9.7(C2×C6), C6.52(C22×D9), (C3×Dic18)⋊11C2, (C6×C18).45C22, (C3×C18).41C23, (C3×C36).62C22, C18.18(C22×C6), (C6×D9).12C22, C32.6(C4○D12), (C3×Dic9).14C22, C2.5(C2×C6×D9), C9⋊5(C3×C4○D4), (C2×C4)⋊3(C3×D9), C6.22(S3×C2×C6), (C3×C9⋊D4)⋊7C2, (C3×C9)⋊12(C4○D4), (C2×C6).52(S3×C6), C3.1(C3×C4○D12), (C2×C12).25(C3×S3), (C2×C18).31(C2×C6), (C3×C6).155(C22×S3), SmallGroup(432,344)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D36⋊5C2
G = < a,b,c,d | a3=b36=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b18c >
Subgroups: 470 in 136 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, C3×D9, C3×C18, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C2×C36, C2×C36, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C3×Dic9, C3×C36, C6×D9, C6×C18, D36⋊5C2, C3×C4○D12, C3×Dic18, C12×D9, C3×D36, C3×C9⋊D4, C6×C36, C3×D36⋊5C2
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, D9, C3×S3, C22×S3, C22×C6, D18, S3×C6, C4○D12, C3×C4○D4, C3×D9, C22×D9, S3×C2×C6, C6×D9, D36⋊5C2, C3×C4○D12, C2×C6×D9, C3×D36⋊5C2
►On 72 points
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 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 40)(2 39)(3 38)(4 37)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)
(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)
G:=sub<Sym(72)| (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72), (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), (1,40)(2,39)(3,38)(4,37)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41), (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)>;
G:=Group( (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72), (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), (1,40)(2,39)(3,38)(4,37)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41), (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) );
G=PermutationGroup([[(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41)], [(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)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6M | 6N | 6O | 6P | 6Q | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 12S | 12T | 12U | 12V | 18A | ··· | 18AA | 36A | ··· | 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 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 18 | 18 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
126 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 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | D9 | C3×S3 | D18 | S3×C6 | D18 | S3×C6 | C3×C4○D4 | C4○D12 | C3×D9 | C6×D9 | C6×D9 | D36⋊5C2 | C3×C4○D12 | C3×D36⋊5C2 |
| kernel | C3×D36⋊5C2 | C3×Dic18 | C12×D9 | C3×D36 | C3×C9⋊D4 | C6×C36 | D36⋊5C2 | Dic18 | C4×D9 | D36 | C9⋊D4 | C2×C36 | C6×C12 | C3×C12 | C62 | C3×C9 | C2×C12 | C2×C12 | C12 | C12 | C2×C6 | C2×C6 | C9 | C32 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | 3 | 2 | 6 | 4 | 3 | 2 | 4 | 4 | 6 | 12 | 6 | 12 | 8 | 24 |
Matrix representation of C3×D36⋊5C2 ►in GL2(𝔽37) generated by
| 10 | 0 |
| 0 | 10 |
| 22 | 0 |
| 0 | 32 |
| 0 | 32 |
| 22 | 0 |
| 1 | 0 |
| 0 | 36 |
G:=sub<GL(2,GF(37))| [10,0,0,10],[22,0,0,32],[0,22,32,0],[1,0,0,36] >;
C3×D36⋊5C2 in GAP, Magma, Sage, TeX
C_3\times D_{36}\rtimes_5C_2 % in TeX
G:=Group("C3xD36:5C2"); // GroupNames label
G:=SmallGroup(432,344);
// by ID
G=gap.SmallGroup(432,344);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^36=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d=b^18*c>;
// generators/relations