direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C9×C4≀C2, D4⋊2C36, Q8⋊3C36, C42⋊6C18, C36.66D4, M4(2)⋊4C18, (D4×C9)⋊5C4, (Q8×C9)⋊5C4, (C4×C36)⋊10C2, C4.3(C2×C36), C4.17(D4×C9), C36.32(C2×C4), (C4×C12).14C6, C4○D4.3C18, (C3×D4).3C12, C12.83(C3×D4), (C2×C18).23D4, (C3×Q8).7C12, C22.3(D4×C9), C12.32(C2×C12), (C9×M4(2))⋊10C2, (C3×M4(2)).5C6, C18.26(C22⋊C4), (C2×C36).115C22, C3.(C3×C4≀C2), (C3×C4≀C2).C3, (C9×C4○D4).4C2, (C2×C6).26(C3×D4), C2.8(C9×C22⋊C4), (C2×C4).14(C2×C18), (C3×C4○D4).12C6, C6.26(C3×C22⋊C4), (C2×C12).135(C2×C6), SmallGroup(288,54)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4≀C2
G = < a,b,c,d | a9=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Subgroups: 102 in 66 conjugacy classes, 36 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C9, C12, C12, C2×C6, C2×C6, C42, M4(2), C4○D4, C18, C18, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4≀C2, C36, C36, C2×C18, C2×C18, C4×C12, C3×M4(2), C3×C4○D4, C72, C2×C36, C2×C36, D4×C9, D4×C9, Q8×C9, C3×C4≀C2, C4×C36, C9×M4(2), C9×C4○D4, C9×C4≀C2
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C18, C2×C12, C3×D4, C4≀C2, C36, C2×C18, C3×C22⋊C4, C2×C36, D4×C9, C3×C4≀C2, C9×C22⋊C4, C9×C4≀C2
►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 59 46 37)(2 60 47 38)(3 61 48 39)(4 62 49 40)(5 63 50 41)(6 55 51 42)(7 56 52 43)(8 57 53 44)(9 58 54 45)(10 67 24 29)(11 68 25 30)(12 69 26 31)(13 70 27 32)(14 71 19 33)(15 72 20 34)(16 64 21 35)(17 65 22 36)(18 66 23 28)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 37)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 64)(54 65)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 67 24 29)(11 68 25 30)(12 69 26 31)(13 70 27 32)(14 71 19 33)(15 72 20 34)(16 64 21 35)(17 65 22 36)(18 66 23 28)(37 59)(38 60)(39 61)(40 62)(41 63)(42 55)(43 56)(44 57)(45 58)
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), (1,59,46,37)(2,60,47,38)(3,61,48,39)(4,62,49,40)(5,63,50,41)(6,55,51,42)(7,56,52,43)(8,57,53,44)(9,58,54,45)(10,67,24,29)(11,68,25,30)(12,69,26,31)(13,70,27,32)(14,71,19,33)(15,72,20,34)(16,64,21,35)(17,65,22,36)(18,66,23,28), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,37)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,64)(54,65), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,67,24,29)(11,68,25,30)(12,69,26,31)(13,70,27,32)(14,71,19,33)(15,72,20,34)(16,64,21,35)(17,65,22,36)(18,66,23,28)(37,59)(38,60)(39,61)(40,62)(41,63)(42,55)(43,56)(44,57)(45,58)>;
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), (1,59,46,37)(2,60,47,38)(3,61,48,39)(4,62,49,40)(5,63,50,41)(6,55,51,42)(7,56,52,43)(8,57,53,44)(9,58,54,45)(10,67,24,29)(11,68,25,30)(12,69,26,31)(13,70,27,32)(14,71,19,33)(15,72,20,34)(16,64,21,35)(17,65,22,36)(18,66,23,28), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,37)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,64)(54,65), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,67,24,29)(11,68,25,30)(12,69,26,31)(13,70,27,32)(14,71,19,33)(15,72,20,34)(16,64,21,35)(17,65,22,36)(18,66,23,28)(37,59)(38,60)(39,61)(40,62)(41,63)(42,55)(43,56)(44,57)(45,58) );
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)], [(1,59,46,37),(2,60,47,38),(3,61,48,39),(4,62,49,40),(5,63,50,41),(6,55,51,42),(7,56,52,43),(8,57,53,44),(9,58,54,45),(10,67,24,29),(11,68,25,30),(12,69,26,31),(13,70,27,32),(14,71,19,33),(15,72,20,34),(16,64,21,35),(17,65,22,36),(18,66,23,28)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,37),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,64),(54,65)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,67,24,29),(11,68,25,30),(12,69,26,31),(13,70,27,32),(14,71,19,33),(15,72,20,34),(16,64,21,35),(17,65,22,36),(18,66,23,28),(37,59),(38,60),(39,61),(40,62),(41,63),(42,55),(43,56),(44,57),(45,58)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18R | 24A | 24B | 24C | 24D | 36A | ··· | 36L | 36M | ··· | 36AP | 36AQ | ··· | 36AV | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C18 | C36 | C36 | D4 | D4 | C3×D4 | C3×D4 | C4≀C2 | D4×C9 | D4×C9 | C3×C4≀C2 | C9×C4≀C2 |
| kernel | C9×C4≀C2 | C4×C36 | C9×M4(2) | C9×C4○D4 | C3×C4≀C2 | D4×C9 | Q8×C9 | C4×C12 | C3×M4(2) | C3×C4○D4 | C4≀C2 | C3×D4 | C3×Q8 | C42 | M4(2) | C4○D4 | D4 | Q8 | C36 | C2×C18 | C12 | C2×C6 | C9 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 8 | 24 |
Matrix representation of C9×C4≀C2 ►in GL2(𝔽37) generated by
| 12 | 0 |
| 0 | 12 |
| 31 | 0 |
| 0 | 6 |
| 0 | 33 |
| 9 | 0 |
| 6 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(37))| [12,0,0,12],[31,0,0,6],[0,9,33,0],[6,0,0,1] >;
C9×C4≀C2 in GAP, Magma, Sage, TeX
C_9\times C_4\wr C_2
% in TeX
G:=Group("C9xC4wrC2"); // GroupNames label
G:=SmallGroup(288,54);
// by ID
G=gap.SmallGroup(288,54);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2194,360,242]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=c^2=d^4=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^-1=b^-1*c>;
// generators/relations