metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12:7M4(2), C42.269D6, C42.12Dic3, C12:C8:4C2, (C4xC12).11C4, C4.80(C2xD12), (C2xC4).87D12, C12.31(C4:C4), (C2xC12).59Q8, C12.83(C2xQ8), (C2xC12).396D4, C12.300(C2xD4), (C2xC42).11S3, C4:2(C4.Dic3), (C2xC4).44Dic6, C4.48(C2xDic6), C3:2(C4:M4(2)), (C22xC12).23C4, C4.14(C4:Dic3), (C22xC4).432D6, C6.37(C2xM4(2)), (C2xC12).842C23, (C4xC12).330C22, (C22xC4).18Dic3, C23.30(C2xDic3), C22.12(C4:Dic3), (C22xC12).535C22, C22.34(C22xDic3), C6.23(C2xC4:C4), (C2xC4xC12).18C2, C2.4(C2xC4:Dic3), (C2xC6).43(C4:C4), (C2xC12).295(C2xC4), C2.6(C2xC4.Dic3), (C2xC3:C8).198C22, (C2xC4).58(C2xDic3), (C2xC4.Dic3).3C2, (C22xC6).130(C2xC4), (C2xC4).784(C22xS3), (C2xC6).171(C22xC4), SmallGroup(192,483)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:7M4(2)
G = < a,b,c | a12=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >
Subgroups: 216 in 126 conjugacy classes, 79 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, C23, C12, C12, C12, C2xC6, C2xC6, C2xC6, C42, C42, C2xC8, M4(2), C22xC4, C22xC4, C3:C8, C2xC12, C2xC12, C2xC12, C22xC6, C4:C8, C2xC42, C2xM4(2), C2xC3:C8, C4.Dic3, C4xC12, C4xC12, C22xC12, C22xC12, C4:M4(2), C12:C8, C2xC4.Dic3, C2xC4xC12, C12:7M4(2)
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, C23, Dic3, D6, C4:C4, M4(2), C22xC4, C2xD4, C2xQ8, Dic6, D12, C2xDic3, C22xS3, C2xC4:C4, C2xM4(2), C4.Dic3, C4:Dic3, C2xDic6, C2xD12, C22xDic3, C4:M4(2), C2xC4.Dic3, C2xC4:Dic3, C12:7M4(2)
►On 96 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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 92 22 68 81 52 45 26)(2 91 23 67 82 51 46 25)(3 90 24 66 83 50 47 36)(4 89 13 65 84 49 48 35)(5 88 14 64 73 60 37 34)(6 87 15 63 74 59 38 33)(7 86 16 62 75 58 39 32)(8 85 17 61 76 57 40 31)(9 96 18 72 77 56 41 30)(10 95 19 71 78 55 42 29)(11 94 20 70 79 54 43 28)(12 93 21 69 80 53 44 27)
(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 85)(58 86)(59 87)(60 88)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,92,22,68,81,52,45,26)(2,91,23,67,82,51,46,25)(3,90,24,66,83,50,47,36)(4,89,13,65,84,49,48,35)(5,88,14,64,73,60,37,34)(6,87,15,63,74,59,38,33)(7,86,16,62,75,58,39,32)(8,85,17,61,76,57,40,31)(9,96,18,72,77,56,41,30)(10,95,19,71,78,55,42,29)(11,94,20,70,79,54,43,28)(12,93,21,69,80,53,44,27), (25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,85)(58,86)(59,87)(60,88)>;
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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,92,22,68,81,52,45,26)(2,91,23,67,82,51,46,25)(3,90,24,66,83,50,47,36)(4,89,13,65,84,49,48,35)(5,88,14,64,73,60,37,34)(6,87,15,63,74,59,38,33)(7,86,16,62,75,58,39,32)(8,85,17,61,76,57,40,31)(9,96,18,72,77,56,41,30)(10,95,19,71,78,55,42,29)(11,94,20,70,79,54,43,28)(12,93,21,69,80,53,44,27), (25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,85)(58,86)(59,87)(60,88) );
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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,92,22,68,81,52,45,26),(2,91,23,67,82,51,46,25),(3,90,24,66,83,50,47,36),(4,89,13,65,84,49,48,35),(5,88,14,64,73,60,37,34),(6,87,15,63,74,59,38,33),(7,86,16,62,75,58,39,32),(8,85,17,61,76,57,40,31),(9,96,18,72,77,56,41,30),(10,95,19,71,78,55,42,29),(11,94,20,70,79,54,43,28),(12,93,21,69,80,53,44,27)], [(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,85),(58,86),(59,87),(60,88)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic3 | D6 | M4(2) | Dic6 | D12 | C4.Dic3 |
| kernel | C12:7M4(2) | C12:C8 | C2xC4.Dic3 | C2xC4xC12 | C4xC12 | C22xC12 | C2xC42 | C2xC12 | C2xC12 | C42 | C42 | C22xC4 | C22xC4 | C12 | C2xC4 | C2xC4 | C4 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 8 | 4 | 4 | 16 |
Matrix representation of C12:7M4(2) ►in GL4(F73) generated by
| 9 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 72 | 1 |
| 0 | 1 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 67 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [9,0,0,0,0,65,0,0,0,0,72,72,0,0,2,1],[0,27,0,0,1,0,0,0,0,0,0,67,0,0,61,0],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;
C12:7M4(2) in GAP, Magma, Sage, TeX
C_{12}\rtimes_7M_4(2) % in TeX
G:=Group("C12:7M4(2)"); // GroupNames label
G:=SmallGroup(192,483);
// by ID
G=gap.SmallGroup(192,483);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,253,120,758,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
// generators/relations