G = C6.492+ 1+4 order 192 = 26·3
49th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.492+ 1+4, C6.742- 1+4, C4⋊C4.93D6, C4⋊D4⋊24S3, (C2×D4).96D6, D6⋊3D4⋊29C2, D6⋊Q8⋊16C2, C22⋊C4.12D6, C23.9D6⋊23C2, Dic3.Q8⋊14C2, (C2×C12).47C23, (C2×C6).165C24, D6⋊C4.18C22, C2.32(Q8○D12), (C22×C4).247D6, C23.14D6⋊19C2, C23.12D6⋊20C2, C12.48D4⋊45C2, C2.51(D4⋊6D6), (C6×D4).129C22, C4⋊Dic3.46C22, Dic3.D4⋊21C2, C23.11D6⋊23C2, C23.28D6⋊13C2, C23.23D6⋊23C2, Dic3⋊C4.79C22, (C22×S3).72C23, C22.186(S3×C23), C23.125(C22×S3), (C22×C6).193C23, (C2×Dic3).82C23, (C2×Dic6).34C22, (C22×C12).313C22, C3⋊1(C22.56C24), (C4×Dic3).100C22, C6.D4.29C22, (C22×Dic3).116C22, (C3×C4⋊D4)⋊27C2, (S3×C2×C4).90C22, (C2×C4).43(C22×S3), (C3×C4⋊C4).151C22, (C2×C3⋊D4).36C22, (C3×C22⋊C4).20C22, SmallGroup(192,1180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.492+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a3b-1, dbd-1=a3b, be=eb, dcd-1=ece=a3c, ede=b2d >
Subgroups: 576 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.56C24, Dic3.D4, C23.9D6, C23.11D6, Dic3.Q8, D6⋊Q8, C12.48D4, C23.28D6, C23.23D6, C23.12D6, D6⋊3D4, C23.14D6, C3×C4⋊D4, C6.492+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.56C24, D4⋊6D6, Q8○D12, C6.492+ 1+4
►On 96 points
Generators in S96
(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 79 10 76)(2 80 11 77)(3 81 12 78)(4 82 7 73)(5 83 8 74)(6 84 9 75)(13 91 22 88)(14 92 23 89)(15 93 24 90)(16 94 19 85)(17 95 20 86)(18 96 21 87)(25 55 34 52)(26 56 35 53)(27 57 36 54)(28 58 31 49)(29 59 32 50)(30 60 33 51)(37 67 46 64)(38 68 47 65)(39 69 48 66)(40 70 43 61)(41 71 44 62)(42 72 45 63)
(1 52 4 49)(2 53 5 50)(3 54 6 51)(7 58 10 55)(8 59 11 56)(9 60 12 57)(13 64 16 61)(14 65 17 62)(15 66 18 63)(19 70 22 67)(20 71 23 68)(21 72 24 69)(25 76 28 73)(26 77 29 74)(27 78 30 75)(31 82 34 79)(32 83 35 80)(33 84 36 81)(37 88 40 85)(38 89 41 86)(39 90 42 87)(43 94 46 91)(44 95 47 92)(45 96 48 93)
(1 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 38)(49 67 58 64)(50 72 59 63)(51 71 60 62)(52 70 55 61)(53 69 56 66)(54 68 57 65)(73 91 82 88)(74 96 83 87)(75 95 84 86)(76 94 79 85)(77 93 80 90)(78 92 81 89)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 64)(50 65)(51 66)(52 61)(53 62)(54 63)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
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), (1,79,10,76)(2,80,11,77)(3,81,12,78)(4,82,7,73)(5,83,8,74)(6,84,9,75)(13,91,22,88)(14,92,23,89)(15,93,24,90)(16,94,19,85)(17,95,20,86)(18,96,21,87)(25,55,34,52)(26,56,35,53)(27,57,36,54)(28,58,31,49)(29,59,32,50)(30,60,33,51)(37,67,46,64)(38,68,47,65)(39,69,48,66)(40,70,43,61)(41,71,44,62)(42,72,45,63), (1,52,4,49)(2,53,5,50)(3,54,6,51)(7,58,10,55)(8,59,11,56)(9,60,12,57)(13,64,16,61)(14,65,17,62)(15,66,18,63)(19,70,22,67)(20,71,23,68)(21,72,24,69)(25,76,28,73)(26,77,29,74)(27,78,30,75)(31,82,34,79)(32,83,35,80)(33,84,36,81)(37,88,40,85)(38,89,41,86)(39,90,42,87)(43,94,46,91)(44,95,47,92)(45,96,48,93), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38)(49,67,58,64)(50,72,59,63)(51,71,60,62)(52,70,55,61)(53,69,56,66)(54,68,57,65)(73,91,82,88)(74,96,83,87)(75,95,84,86)(76,94,79,85)(77,93,80,90)(78,92,81,89), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,64)(50,65)(51,66)(52,61)(53,62)(54,63)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)>;
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), (1,79,10,76)(2,80,11,77)(3,81,12,78)(4,82,7,73)(5,83,8,74)(6,84,9,75)(13,91,22,88)(14,92,23,89)(15,93,24,90)(16,94,19,85)(17,95,20,86)(18,96,21,87)(25,55,34,52)(26,56,35,53)(27,57,36,54)(28,58,31,49)(29,59,32,50)(30,60,33,51)(37,67,46,64)(38,68,47,65)(39,69,48,66)(40,70,43,61)(41,71,44,62)(42,72,45,63), (1,52,4,49)(2,53,5,50)(3,54,6,51)(7,58,10,55)(8,59,11,56)(9,60,12,57)(13,64,16,61)(14,65,17,62)(15,66,18,63)(19,70,22,67)(20,71,23,68)(21,72,24,69)(25,76,28,73)(26,77,29,74)(27,78,30,75)(31,82,34,79)(32,83,35,80)(33,84,36,81)(37,88,40,85)(38,89,41,86)(39,90,42,87)(43,94,46,91)(44,95,47,92)(45,96,48,93), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38)(49,67,58,64)(50,72,59,63)(51,71,60,62)(52,70,55,61)(53,69,56,66)(54,68,57,65)(73,91,82,88)(74,96,83,87)(75,95,84,86)(76,94,79,85)(77,93,80,90)(78,92,81,89), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,64)(50,65)(51,66)(52,61)(53,62)(54,63)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96) );
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)], [(1,79,10,76),(2,80,11,77),(3,81,12,78),(4,82,7,73),(5,83,8,74),(6,84,9,75),(13,91,22,88),(14,92,23,89),(15,93,24,90),(16,94,19,85),(17,95,20,86),(18,96,21,87),(25,55,34,52),(26,56,35,53),(27,57,36,54),(28,58,31,49),(29,59,32,50),(30,60,33,51),(37,67,46,64),(38,68,47,65),(39,69,48,66),(40,70,43,61),(41,71,44,62),(42,72,45,63)], [(1,52,4,49),(2,53,5,50),(3,54,6,51),(7,58,10,55),(8,59,11,56),(9,60,12,57),(13,64,16,61),(14,65,17,62),(15,66,18,63),(19,70,22,67),(20,71,23,68),(21,72,24,69),(25,76,28,73),(26,77,29,74),(27,78,30,75),(31,82,34,79),(32,83,35,80),(33,84,36,81),(37,88,40,85),(38,89,41,86),(39,90,42,87),(43,94,46,91),(44,95,47,92),(45,96,48,93)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(25,46,34,37),(26,45,35,42),(27,44,36,41),(28,43,31,40),(29,48,32,39),(30,47,33,38),(49,67,58,64),(50,72,59,63),(51,71,60,62),(52,70,55,61),(53,69,56,66),(54,68,57,65),(73,91,82,88),(74,96,83,87),(75,95,84,86),(76,94,79,85),(77,93,80,90),(78,92,81,89)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(49,64),(50,65),(51,66),(52,61),(53,62),(54,63),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 12 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | Q8○D12 |
| kernel | C6.492+ 1+4 | Dic3.D4 | C23.9D6 | C23.11D6 | Dic3.Q8 | D6⋊Q8 | C12.48D4 | C23.28D6 | C23.23D6 | C23.12D6 | D6⋊3D4 | C23.14D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 1 | 4 | 2 |
Matrix representation of C6.492+ 1+4 ►in GL8(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 11 | 9 | 9 | 5 | 0 | 0 | 0 | 0 |
| 4 | 2 | 8 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 | 2 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 | 0 | 2 |
| 0 | 0 | 0 | 0 | 2 | 12 | 5 | 11 |
| 0 | 0 | 0 | 0 | 1 | 3 | 2 | 7 |
| 3 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 12 | 0 | 0 | 0 | 0 |
| 10 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 | 11 | 0 |
| 0 | 0 | 0 | 0 | 2 | 7 | 0 | 11 |
| 0 | 0 | 0 | 0 | 10 | 1 | 8 | 2 |
| 0 | 0 | 0 | 0 | 12 | 9 | 11 | 6 |
| 12 | 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 7 | 7 | 11 | 2 |
| 12 | 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 |
G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0],[11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,9,8,2,9,0,0,0,0,5,4,4,11,0,0,0,0,0,0,0,0,8,11,2,1,0,0,0,0,2,6,12,3,0,0,0,0,2,0,5,2,0,0,0,0,0,2,11,7],[3,0,10,0,0,0,0,0,0,3,0,10,0,0,0,0,12,0,10,0,0,0,0,0,0,12,0,10,0,0,0,0,0,0,0,0,5,2,10,12,0,0,0,0,11,7,1,9,0,0,0,0,11,0,8,11,0,0,0,0,0,11,2,6],[12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,2,1,12,0,0,0,0,0,2,0,12,0,0,0,0,0,0,0,0,11,11,6,7,0,0,0,0,9,2,0,7,0,0,0,0,0,0,11,11,0,0,0,0,0,0,9,2],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,0,1,0,0,0,0,0,0,11,0,1,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2] >;
C6.492+ 1+4 in GAP, Magma, Sage, TeX
C_6._{49}2_+^{1+4} % in TeX
G:=Group("C6.49ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1180);
// by ID
G=gap.SmallGroup(192,1180);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=b^2*d>;
// generators/relations