direct product, metacyclic, supersoluble, monomial
Aliases: C3×C4.Dic9, C36.7C12, C12.78D18, C12.9Dic9, C62.16Dic3, C9⋊C8⋊12C6, C4.(C3×Dic9), (C3×C36).3C4, (C6×C18).5C4, C4.15(C6×D9), (C3×C9)⋊8M4(2), C12.91(S3×C6), C36.38(C2×C6), (C6×C36).11C2, (C2×C36).18C6, (C6×C12).38S3, (C2×C12).18D9, C9⋊5(C3×M4(2)), (C2×C6).2Dic9, C6.7(C6×Dic3), C2.3(C6×Dic9), (C2×C18).10C12, C18.21(C2×C12), C22.(C3×Dic9), (C3×C12).215D6, C6.18(C2×Dic9), C12.1(C3×Dic3), (C3×C36).68C22, (C3×C12).18Dic3, C32.3(C4.Dic3), (C3×C9⋊C8)⋊12C2, (C2×C4).2(C3×D9), (C2×C12).22(C3×S3), (C3×C18).33(C2×C4), C3.1(C3×C4.Dic3), (C2×C6).13(C3×Dic3), (C3×C6).55(C2×Dic3), SmallGroup(432,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.Dic9
G = < a,b,c,d | a3=b4=1, c18=b2, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c17 >
Subgroups: 158 in 76 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C4.Dic3, C3×M4(2), C3×C18, C3×C18, C9⋊C8, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C4.Dic9, C3×C4.Dic3, C3×C9⋊C8, C6×C36, C3×C4.Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), D9, C3×S3, C2×Dic3, C2×C12, Dic9, D18, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), C3×D9, C2×Dic9, C6×Dic3, C3×Dic9, C6×D9, C4.Dic9, C3×C4.Dic3, C6×Dic9, C3×C4.Dic9
►On 72 points
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)
(1 10 19 28)(2 11 20 29)(3 12 21 30)(4 13 22 31)(5 14 23 32)(6 15 24 33)(7 16 25 34)(8 17 26 35)(9 18 27 36)(37 64 55 46)(38 65 56 47)(39 66 57 48)(40 67 58 49)(41 68 59 50)(42 69 60 51)(43 70 61 52)(44 71 62 53)(45 72 63 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)
(1 55 10 64 19 37 28 46)(2 72 11 45 20 54 29 63)(3 53 12 62 21 71 30 44)(4 70 13 43 22 52 31 61)(5 51 14 60 23 69 32 42)(6 68 15 41 24 50 33 59)(7 49 16 58 25 67 34 40)(8 66 17 39 26 48 35 57)(9 47 18 56 27 65 36 38)
G:=sub<Sym(72)| (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72), (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,64,55,46)(38,65,56,47)(39,66,57,48)(40,67,58,49)(41,68,59,50)(42,69,60,51)(43,70,61,52)(44,71,62,53)(45,72,63,54), (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), (1,55,10,64,19,37,28,46)(2,72,11,45,20,54,29,63)(3,53,12,62,21,71,30,44)(4,70,13,43,22,52,31,61)(5,51,14,60,23,69,32,42)(6,68,15,41,24,50,33,59)(7,49,16,58,25,67,34,40)(8,66,17,39,26,48,35,57)(9,47,18,56,27,65,36,38)>;
G:=Group( (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72), (1,10,19,28)(2,11,20,29)(3,12,21,30)(4,13,22,31)(5,14,23,32)(6,15,24,33)(7,16,25,34)(8,17,26,35)(9,18,27,36)(37,64,55,46)(38,65,56,47)(39,66,57,48)(40,67,58,49)(41,68,59,50)(42,69,60,51)(43,70,61,52)(44,71,62,53)(45,72,63,54), (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), (1,55,10,64,19,37,28,46)(2,72,11,45,20,54,29,63)(3,53,12,62,21,71,30,44)(4,70,13,43,22,52,31,61)(5,51,14,60,23,69,32,42)(6,68,15,41,24,50,33,59)(7,49,16,58,25,67,34,40)(8,66,17,39,26,48,35,57)(9,47,18,56,27,65,36,38) );
G=PermutationGroup([[(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72)], [(1,10,19,28),(2,11,20,29),(3,12,21,30),(4,13,22,31),(5,14,23,32),(6,15,24,33),(7,16,25,34),(8,17,26,35),(9,18,27,36),(37,64,55,46),(38,65,56,47),(39,66,57,48),(40,67,58,49),(41,68,59,50),(42,69,60,51),(43,70,61,52),(44,71,62,53),(45,72,63,54)], [(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)], [(1,55,10,64,19,37,28,46),(2,72,11,45,20,54,29,63),(3,53,12,62,21,71,30,44),(4,70,13,43,22,52,31,61),(5,51,14,60,23,69,32,42),(6,68,15,41,24,50,33,59),(7,49,16,58,25,67,34,40),(8,66,17,39,26,48,35,57),(9,47,18,56,27,65,36,38)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6M | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 18A | ··· | 18AA | 24A | ··· | 24H | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 |
126 irreducible representations
| dim | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | Dic3 | M4(2) | D9 | C3×S3 | Dic9 | D18 | C3×Dic3 | S3×C6 | Dic9 | C3×Dic3 | C3×M4(2) | C4.Dic3 | C3×D9 | C3×Dic9 | C6×D9 | C3×Dic9 | C4.Dic9 | C3×C4.Dic3 | C3×C4.Dic9 |
| kernel | C3×C4.Dic9 | C3×C9⋊C8 | C6×C36 | C4.Dic9 | C3×C36 | C6×C18 | C9⋊C8 | C2×C36 | C36 | C2×C18 | C6×C12 | C3×C12 | C3×C12 | C62 | C3×C9 | C2×C12 | C2×C12 | C12 | C12 | C12 | C12 | C2×C6 | C2×C6 | C9 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 3 | 2 | 2 | 3 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 8 | 24 |
Matrix representation of C3×C4.Dic9 ►in GL2(𝔽37) generated by
| 26 | 0 |
| 0 | 26 |
| 31 | 0 |
| 0 | 6 |
| 13 | 0 |
| 0 | 17 |
| 0 | 6 |
| 1 | 0 |
G:=sub<GL(2,GF(37))| [26,0,0,26],[31,0,0,6],[13,0,0,17],[0,1,6,0] >;
C3×C4.Dic9 in GAP, Magma, Sage, TeX
C_3\times C_4.{\rm Dic}_9 % in TeX
G:=Group("C3xC4.Dic9"); // GroupNames label
G:=SmallGroup(432,125);
// by ID
G=gap.SmallGroup(432,125);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^18=b^2,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^17>;
// generators/relations