direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C2×C4×3- 1+2, C36⋊4C6, C18⋊2C12, C62.5C6, C6.7C62, (C2×C36)⋊C3, (C6×C12).C3, C9⋊3(C2×C12), C18.6(C2×C6), C6.5(C3×C12), C3.2(C6×C12), (C3×C6).5C12, (C3×C12).8C6, (C2×C18).3C6, C32.(C2×C12), C12.13(C3×C6), (C2×C12).2C32, C22.(C2×3- 1+2), C2.1(C22×3- 1+2), (C2×3- 1+2).6C22, (C22×3- 1+2).2C2, (C2×C6).11(C3×C6), (C3×C6).12(C2×C6), SmallGroup(216,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4×3- 1+2
G = < a,b,c,d | a2=b4=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 80 in 64 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C18, C3×C6, C3×C6, C2×C12, C2×C12, 3- 1+2, C36, C2×C18, C3×C12, C62, C2×3- 1+2, C2×3- 1+2, C2×C36, C6×C12, C4×3- 1+2, C22×3- 1+2, C2×C4×3- 1+2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, C3×C6, C2×C12, 3- 1+2, C3×C12, C62, C2×3- 1+2, C6×C12, C4×3- 1+2, C22×3- 1+2, C2×C4×3- 1+2
►On 72 points
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 53)(11 54)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 34)(20 35)(21 36)(22 28)(23 29)(24 30)(25 31)(26 32)(27 33)(55 70)(56 71)(57 72)(58 64)(59 65)(60 66)(61 67)(62 68)(63 69)
(1 18 19 70)(2 10 20 71)(3 11 21 72)(4 12 22 64)(5 13 23 65)(6 14 24 66)(7 15 25 67)(8 16 26 68)(9 17 27 69)(28 58 40 46)(29 59 41 47)(30 60 42 48)(31 61 43 49)(32 62 44 50)(33 63 45 51)(34 55 37 52)(35 56 38 53)(36 57 39 54)
(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(20 26 23)(21 24 27)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(47 53 50)(48 51 54)(56 62 59)(57 60 63)(65 71 68)(66 69 72)
G:=sub<Sym(72)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,53)(11,54)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,34)(20,35)(21,36)(22,28)(23,29)(24,30)(25,31)(26,32)(27,33)(55,70)(56,71)(57,72)(58,64)(59,65)(60,66)(61,67)(62,68)(63,69), (1,18,19,70)(2,10,20,71)(3,11,21,72)(4,12,22,64)(5,13,23,65)(6,14,24,66)(7,15,25,67)(8,16,26,68)(9,17,27,69)(28,58,40,46)(29,59,41,47)(30,60,42,48)(31,61,43,49)(32,62,44,50)(33,63,45,51)(34,55,37,52)(35,56,38,53)(36,57,39,54), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(56,62,59)(57,60,63)(65,71,68)(66,69,72)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,53)(11,54)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,34)(20,35)(21,36)(22,28)(23,29)(24,30)(25,31)(26,32)(27,33)(55,70)(56,71)(57,72)(58,64)(59,65)(60,66)(61,67)(62,68)(63,69), (1,18,19,70)(2,10,20,71)(3,11,21,72)(4,12,22,64)(5,13,23,65)(6,14,24,66)(7,15,25,67)(8,16,26,68)(9,17,27,69)(28,58,40,46)(29,59,41,47)(30,60,42,48)(31,61,43,49)(32,62,44,50)(33,63,45,51)(34,55,37,52)(35,56,38,53)(36,57,39,54), (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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(20,26,23)(21,24,27)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(47,53,50)(48,51,54)(56,62,59)(57,60,63)(65,71,68)(66,69,72) );
G=PermutationGroup([[(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,53),(11,54),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,34),(20,35),(21,36),(22,28),(23,29),(24,30),(25,31),(26,32),(27,33),(55,70),(56,71),(57,72),(58,64),(59,65),(60,66),(61,67),(62,68),(63,69)], [(1,18,19,70),(2,10,20,71),(3,11,21,72),(4,12,22,64),(5,13,23,65),(6,14,24,66),(7,15,25,67),(8,16,26,68),(9,17,27,69),(28,58,40,46),(29,59,41,47),(30,60,42,48),(31,61,43,49),(32,62,44,50),(33,63,45,51),(34,55,37,52),(35,56,38,53),(36,57,39,54)], [(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)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(20,26,23),(21,24,27),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(47,53,50),(48,51,54),(56,62,59),(57,60,63),(65,71,68),(66,69,72)]])
C2×C4×3- 1+2 is a maximal subgroup of
C36.C12 Dic9⋊C12 C36⋊C12 D18⋊C12 D36⋊6C6
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6L | 9A | ··· | 9F | 12A | ··· | 12H | 12I | ··· | 12P | 18A | ··· | 18R | 36A | ··· | 36X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C4×3- 1+2 |
| kernel | C2×C4×3- 1+2 | C4×3- 1+2 | C22×3- 1+2 | C2×C36 | C6×C12 | C2×3- 1+2 | C36 | C2×C18 | C3×C12 | C62 | C18 | C3×C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 6 | 2 | 4 | 12 | 6 | 4 | 2 | 24 | 8 | 2 | 4 | 2 | 8 |
Matrix representation of C2×C4×3- 1+2 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 9 | 0 |
| 0 | 1 | 36 | 10 |
| 0 | 0 | 36 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 10 | 0 |
| 0 | 11 | 0 | 26 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[1,0,0,0,0,1,1,0,0,9,36,36,0,0,10,0],[26,0,0,0,0,1,1,11,0,0,10,0,0,0,0,26] >;
C2×C4×3- 1+2 in GAP, Magma, Sage, TeX
C_2\times C_4\times 3_-^{1+2} % in TeX
G:=Group("C2xC4xES-(3,1)"); // GroupNames label
G:=SmallGroup(216,75);
// by ID
G=gap.SmallGroup(216,75);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,338,519]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations