metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22.2D36, C23.6D18, (C2×Dic9)⋊C4, (C22×D9)⋊C4, C9⋊1(C23⋊C4), C22⋊C4⋊1D9, (C2×C6).2D12, C6.9(D6⋊C4), (C2×C18).28D4, C22.3(C4×D9), C2.4(D18⋊C4), (C22×C6).35D6, C18.D4⋊1C2, C18.2(C22⋊C4), C22.8(C9⋊D4), C3.(C23.6D6), (C22×C18).5C22, (C2×C6).2(C4×S3), (C9×C22⋊C4)⋊1C2, (C2×C18).1(C2×C4), (C2×C9⋊D4).1C2, (C3×C22⋊C4).1S3, (C2×C6).66(C3⋊D4), SmallGroup(288,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.D36
G = < a,b,c,d | a2=b2=c36=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >
Subgroups: 420 in 78 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, D9, C18, C18, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C23⋊C4, Dic9, C36, D18, C2×C18, C2×C18, C6.D4, C3×C22⋊C4, C2×C3⋊D4, C2×Dic9, C2×Dic9, C9⋊D4, C2×C36, C22×D9, C22×C18, C23.6D6, C18.D4, C9×C22⋊C4, C2×C9⋊D4, C22.D36
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C23⋊C4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C23.6D6, D18⋊C4, C22.D36
►On 72 points
(1 60)(3 62)(5 64)(7 66)(9 68)(11 70)(13 72)(15 38)(17 40)(19 42)(21 44)(23 46)(25 48)(27 50)(29 52)(31 54)(33 56)(35 58)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 37)(15 38)(16 39)(17 40)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)
(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 50 60 27)(2 49)(3 25 62 48)(4 24)(5 46 64 23)(6 45)(7 21 66 44)(8 20)(9 42 68 19)(10 41)(11 17 70 40)(12 16)(13 38 72 15)(14 37)(18 69)(22 65)(26 61)(28 36)(29 58 52 35)(30 57)(31 33 54 56)(34 53)(39 71)(43 67)(47 63)(51 59)
G:=sub<Sym(72)| (1,60)(3,62)(5,64)(7,66)(9,68)(11,70)(13,72)(15,38)(17,40)(19,42)(21,44)(23,46)(25,48)(27,50)(29,52)(31,54)(33,56)(35,58), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59), (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), (1,50,60,27)(2,49)(3,25,62,48)(4,24)(5,46,64,23)(6,45)(7,21,66,44)(8,20)(9,42,68,19)(10,41)(11,17,70,40)(12,16)(13,38,72,15)(14,37)(18,69)(22,65)(26,61)(28,36)(29,58,52,35)(30,57)(31,33,54,56)(34,53)(39,71)(43,67)(47,63)(51,59)>;
G:=Group( (1,60)(3,62)(5,64)(7,66)(9,68)(11,70)(13,72)(15,38)(17,40)(19,42)(21,44)(23,46)(25,48)(27,50)(29,52)(31,54)(33,56)(35,58), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,70)(12,71)(13,72)(14,37)(15,38)(16,39)(17,40)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59), (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), (1,50,60,27)(2,49)(3,25,62,48)(4,24)(5,46,64,23)(6,45)(7,21,66,44)(8,20)(9,42,68,19)(10,41)(11,17,70,40)(12,16)(13,38,72,15)(14,37)(18,69)(22,65)(26,61)(28,36)(29,58,52,35)(30,57)(31,33,54,56)(34,53)(39,71)(43,67)(47,63)(51,59) );
G=PermutationGroup([[(1,60),(3,62),(5,64),(7,66),(9,68),(11,70),(13,72),(15,38),(17,40),(19,42),(21,44),(23,46),(25,48),(27,50),(29,52),(31,54),(33,56),(35,58)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,70),(12,71),(13,72),(14,37),(15,38),(16,39),(17,40),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59)], [(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)], [(1,50,60,27),(2,49),(3,25,62,48),(4,24),(5,46,64,23),(6,45),(7,21,66,44),(8,20),(9,42,68,19),(10,41),(11,17,70,40),(12,16),(13,38,72,15),(14,37),(18,69),(22,65),(26,61),(28,36),(29,58,52,35),(30,57),(31,33,54,56),(34,53),(39,71),(43,67),(47,63),(51,59)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18O | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 2 | 2 | 36 | 2 | 4 | 4 | 36 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D6 | D9 | C4×S3 | D12 | C3⋊D4 | D18 | C4×D9 | D36 | C9⋊D4 | C23⋊C4 | C23.6D6 | C22.D36 |
| kernel | C22.D36 | C18.D4 | C9×C22⋊C4 | C2×C9⋊D4 | C2×Dic9 | C22×D9 | C3×C22⋊C4 | C2×C18 | C22×C6 | C22⋊C4 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 2 | 6 |
Matrix representation of C22.D36 ►in GL4(𝔽37) generated by
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 20 | 31 |
| 0 | 0 | 6 | 26 |
| 2 | 26 | 0 | 0 |
| 11 | 13 | 0 | 0 |
| 24 | 26 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 0 | 0 | 26 | 6 |
| 0 | 0 | 17 | 11 |
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[0,0,2,11,0,0,26,13,20,6,0,0,31,26,0,0],[24,2,0,0,26,13,0,0,0,0,26,17,0,0,6,11] >;
C22.D36 in GAP, Magma, Sage, TeX
C_2^2.D_{36} % in TeX
G:=Group("C2^2.D36"); // GroupNames label
G:=SmallGroup(288,13);
// by ID
G=gap.SmallGroup(288,13);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
// generators/relations