direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C9×C22≀C2, C24⋊5C18, (C2×C18)⋊7D4, (C2×D4)⋊1C18, (C6×D4).9C6, C2.4(D4×C18), C6.67(C6×D4), C22⋊3(D4×C9), (D4×C18)⋊10C2, C22⋊C4⋊2C18, C23⋊1(C2×C18), (C23×C18)⋊1C2, (C2×C36)⋊8C22, C18.67(C2×D4), (C23×C6).9C6, (C2×C18).75C23, (C22×C18)⋊1C22, C22.10(C22×C18), (C2×C4)⋊1(C2×C18), C3.(C3×C22≀C2), (C2×C6).27(C3×D4), (C9×C22⋊C4)⋊10C2, (C2×C12).62(C2×C6), (C3×C22⋊C4).5C6, (C3×C22≀C2).2C3, (C2×C6).80(C22×C6), (C22×C6).45(C2×C6), SmallGroup(288,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C22≀C2
G = < a,b,c,d,e,f | a9=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 318 in 195 conjugacy classes, 78 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, D4, C23, C23, C23, C9, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C18, C18, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22≀C2, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C6×D4, C23×C6, C2×C36, D4×C9, C22×C18, C22×C18, C22×C18, C3×C22≀C2, C9×C22⋊C4, D4×C18, C23×C18, C9×C22≀C2
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C18, C3×D4, C22×C6, C22≀C2, C2×C18, C6×D4, D4×C9, C22×C18, C3×C22≀C2, D4×C18, C9×C22≀C2
►On 72 points
Generators in S72
(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 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(55 68)(56 69)(57 70)(58 71)(59 72)(60 64)(61 65)(62 66)(63 67)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 46)(10 62)(11 63)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 41)(29 42)(30 43)(31 44)(32 45)(33 37)(34 38)(35 39)(36 40)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 37)(10 62)(11 63)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 72)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 50)(29 51)(30 52)(31 53)(32 54)(33 46)(34 47)(35 48)(36 49)
(1 34)(2 35)(3 36)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 64)(18 65)(19 59)(20 60)(21 61)(22 62)(23 63)(24 55)(25 56)(26 57)(27 58)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 46)(26 47)(27 48)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 55)
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), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,68)(56,69)(57,70)(58,71)(59,72)(60,64)(61,65)(62,66)(63,67), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,46)(10,62)(11,63)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,41)(29,42)(30,43)(31,44)(32,45)(33,37)(34,38)(35,39)(36,40), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,37)(10,62)(11,63)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,50)(29,51)(30,52)(31,53)(32,54)(33,46)(34,47)(35,48)(36,49), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,64)(18,65)(19,59)(20,60)(21,61)(22,62)(23,63)(24,55)(25,56)(26,57)(27,58)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,46)(26,47)(27,48)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,55)>;
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), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(55,68)(56,69)(57,70)(58,71)(59,72)(60,64)(61,65)(62,66)(63,67), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,46)(10,62)(11,63)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,41)(29,42)(30,43)(31,44)(32,45)(33,37)(34,38)(35,39)(36,40), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,37)(10,62)(11,63)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,72)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,50)(29,51)(30,52)(31,53)(32,54)(33,46)(34,47)(35,48)(36,49), (1,34)(2,35)(3,36)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,64)(18,65)(19,59)(20,60)(21,61)(22,62)(23,63)(24,55)(25,56)(26,57)(27,58)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,46)(26,47)(27,48)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,55) );
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)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,19),(17,20),(18,21),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(55,68),(56,69),(57,70),(58,71),(59,72),(60,64),(61,65),(62,66),(63,67)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,46),(10,62),(11,63),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,72),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,41),(29,42),(30,43),(31,44),(32,45),(33,37),(34,38),(35,39),(36,40)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,37),(10,62),(11,63),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,72),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,50),(29,51),(30,52),(31,53),(32,54),(33,46),(34,47),(35,48),(36,49)], [(1,34),(2,35),(3,36),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,64),(18,65),(19,59),(20,60),(21,61),(22,62),(23,63),(24,55),(25,56),(26,57),(27,58),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,46),(26,47),(27,48),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,55)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 6A | ··· | 6F | 6G | ··· | 6R | 6S | 6T | 9A | ··· | 9F | 12A | ··· | 12F | 18A | ··· | 18R | 18S | ··· | 18BB | 18BC | ··· | 18BH | 36A | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | D4 | C3×D4 | D4×C9 |
| kernel | C9×C22≀C2 | C9×C22⋊C4 | D4×C18 | C23×C18 | C3×C22≀C2 | C3×C22⋊C4 | C6×D4 | C23×C6 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C18 | C2×C6 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 18 | 18 | 6 | 6 | 12 | 36 |
Matrix representation of C9×C22≀C2 ►in GL4(𝔽37) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(37))| [12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,1,0,0,0,0,1,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C9×C22≀C2 in GAP, Magma, Sage, TeX
C_9\times C_2^2\wr C_2
% in TeX
G:=Group("C9xC2^2wrC2"); // GroupNames label
G:=SmallGroup(288,170);
// by ID
G=gap.SmallGroup(288,170);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,365,1094,360]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^9=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations