direct product, metabelian, supersoluble, monomial
Aliases: C9×C4○D12, D12⋊5C18, C36.75D6, Dic6⋊5C18, (C2×C36)⋊7S3, (C4×S3)⋊4C18, (S3×C36)⋊6C2, (C6×C36)⋊13C2, (C2×C12)⋊4C18, C3⋊D4⋊3C18, (C9×D12)⋊11C2, C4.16(S3×C18), (C6×C12).32C6, D6.1(C2×C18), (C2×C18).23D6, (S3×C12).10C6, C12.111(S3×C6), C12.16(C2×C18), (C9×Dic6)⋊11C2, (C3×D12).12C6, C6.4(C22×C18), C22.2(S3×C18), C62.56(C2×C6), (S3×C18).5C22, C18.52(C22×S3), (C3×C18).31C23, (C6×C18).29C22, (C3×C36).54C22, Dic3.2(C2×C18), (C3×Dic6).12C6, (C9×Dic3).14C22, (C2×C4)⋊3(S3×C9), C3⋊1(C9×C4○D4), C2.5(S3×C2×C18), C6.65(S3×C2×C6), (C9×C3⋊D4)⋊7C2, (C3×C4○D12).C3, (C3×C9)⋊11(C4○D4), (S3×C6).6(C2×C6), (C2×C6).31(S3×C6), C3.4(C3×C4○D12), (C3×C3⋊D4).3C6, (C2×C6).14(C2×C18), (C3×C12).82(C2×C6), (C2×C12).44(C3×S3), C32.2(C3×C4○D4), (C3×C6).41(C22×C6), (C3×Dic3).11(C2×C6), SmallGroup(432,347)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4○D12
G = < a,b,c,d | a9=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >
Subgroups: 260 in 136 conjugacy classes, 69 normal (45 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, C18, C18, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, C3×C4○D4, S3×C9, C3×C18, C3×C18, C2×C36, C2×C36, D4×C9, Q8×C9, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C9×Dic3, C3×C36, S3×C18, C6×C18, C9×C4○D4, C3×C4○D12, C9×Dic6, S3×C36, C9×D12, C9×C3⋊D4, C6×C36, C9×C4○D12
Quotients: C1, C2, C3, C22, S3, C6, C23, C9, D6, C2×C6, C4○D4, C18, C3×S3, C22×S3, C22×C6, C2×C18, S3×C6, C4○D12, C3×C4○D4, S3×C9, C22×C18, S3×C2×C6, S3×C18, C9×C4○D4, C3×C4○D12, S3×C2×C18, C9×C4○D12
►On 72 points
(1 31 61 9 27 69 5 35 65)(2 32 62 10 28 70 6 36 66)(3 33 63 11 29 71 7 25 67)(4 34 64 12 30 72 8 26 68)(13 38 55 17 42 59 21 46 51)(14 39 56 18 43 60 22 47 52)(15 40 57 19 44 49 23 48 53)(16 41 58 20 45 50 24 37 54)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 70 67 64)(62 71 68 65)(63 72 69 66)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 72)
G:=sub<Sym(72)| (1,31,61,9,27,69,5,35,65)(2,32,62,10,28,70,6,36,66)(3,33,63,11,29,71,7,25,67)(4,34,64,12,30,72,8,26,68)(13,38,55,17,42,59,21,46,51)(14,39,56,18,43,60,22,47,52)(15,40,57,19,44,49,23,48,53)(16,41,58,20,45,50,24,37,54), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,72)>;
G:=Group( (1,31,61,9,27,69,5,35,65)(2,32,62,10,28,70,6,36,66)(3,33,63,11,29,71,7,25,67)(4,34,64,12,30,72,8,26,68)(13,38,55,17,42,59,21,46,51)(14,39,56,18,43,60,22,47,52)(15,40,57,19,44,49,23,48,53)(16,41,58,20,45,50,24,37,54), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,70,67,64)(62,71,68,65)(63,72,69,66), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(37,71)(38,70)(39,69)(40,68)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,72) );
G=PermutationGroup([[(1,31,61,9,27,69,5,35,65),(2,32,62,10,28,70,6,36,66),(3,33,63,11,29,71,7,25,67),(4,34,64,12,30,72,8,26,68),(13,38,55,17,42,59,21,46,51),(14,39,56,18,43,60,22,47,52),(15,40,57,19,44,49,23,48,53),(16,41,58,20,45,50,24,37,54)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,70,67,64),(62,71,68,65),(63,72,69,66)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,28),(14,27),(15,26),(16,25),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(37,71),(38,70),(39,69),(40,68),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,72)]])
162 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 | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 12S | 12T | 12U | 12V | 18A | ··· | 18F | 18G | ··· | 18AD | 18AE | ··· | 18AP | 36A | ··· | 36L | 36M | ··· | 36AP | 36AQ | ··· | 36BB |
| 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 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
162 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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C18 | C18 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | S3×C9 | S3×C18 | S3×C18 | C9×C4○D4 | C3×C4○D12 | C9×C4○D12 |
| kernel | C9×C4○D12 | C9×Dic6 | S3×C36 | C9×D12 | C9×C3⋊D4 | C6×C36 | C3×C4○D12 | C3×Dic6 | S3×C12 | C3×D12 | C3×C3⋊D4 | C6×C12 | C4○D12 | Dic6 | C4×S3 | D12 | C3⋊D4 | C2×C12 | C2×C36 | C36 | C2×C18 | C3×C9 | C2×C12 | C12 | C2×C6 | C9 | C32 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 6 | 6 | 12 | 6 | 12 | 6 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 6 | 12 | 6 | 12 | 8 | 24 |
Matrix representation of C9×C4○D12 ►in GL2(𝔽37) generated by
| 9 | 0 |
| 0 | 9 |
| 31 | 0 |
| 0 | 31 |
| 14 | 0 |
| 13 | 8 |
| 29 | 35 |
| 13 | 8 |
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,31],[14,13,0,8],[29,13,35,8] >;
C9×C4○D12 in GAP, Magma, Sage, TeX
C_9\times C_4\circ D_{12} % in TeX
G:=Group("C9xC4oD12"); // GroupNames label
G:=SmallGroup(432,347);
// by ID
G=gap.SmallGroup(432,347);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,192,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^5>;
// generators/relations