G = C6.632+ 1+4 order 192 = 26·3
63rd non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.632+ 1+4, C6.832- 1+4, C4⋊C4.107D6, D6⋊Q8⋊31C2, D6.7(C4○D4), (C2×D4).100D6, C22⋊C4.28D6, D6⋊3D4.11C2, C23.9D6⋊33C2, Dic3.Q8⋊27C2, (C2×C6).204C24, C2.44(Q8○D12), C4.Dic6⋊29C2, (C22×C4).275D6, C22.D4⋊9S3, Dic3⋊4D4⋊22C2, C12.48D4⋊22C2, C2.65(D4⋊6D6), (C2×C12).180C23, D6⋊C4.108C22, (C6×D4).142C22, C23.8D6⋊32C2, (C22×C6).36C23, C23.38(C22×S3), Dic3.D4⋊33C2, C23.23D6⋊15C2, Dic3⋊C4.42C22, C4⋊Dic3.228C22, C22.225(S3×C23), (C2×Dic6).37C22, (C22×S3).210C23, (C22×C12).115C22, C3⋊6(C22.33C24), (C4×Dic3).124C22, (C2×Dic3).106C23, C6.D4.44C22, (C22×Dic3).130C22, (S3×C4⋊C4)⋊33C2, (C4×C3⋊D4)⋊7C2, C2.66(S3×C4○D4), C6.178(C2×C4○D4), (S3×C2×C4).113C22, (C2×C4).66(C22×S3), (C3×C4⋊C4).177C22, (C2×C3⋊D4).48C22, (C3×C22.D4)⋊12C2, (C3×C22⋊C4).52C22, SmallGroup(192,1219)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.632+ 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=b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=b2d >
Subgroups: 528 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, 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, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C42.C2, C42⋊2C2, 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, C22.33C24, Dic3.D4, C23.8D6, Dic3⋊4D4, C23.9D6, Dic3.Q8, C4.Dic6, S3×C4⋊C4, D6⋊Q8, C12.48D4, C4×C3⋊D4, C23.23D6, D6⋊3D4, C3×C22.D4, C6.632+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, S3×C23, C22.33C24, D4⋊6D6, S3×C4○D4, Q8○D12, C6.632+ 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 7 73)(2 80 8 74)(3 81 9 75)(4 82 10 76)(5 83 11 77)(6 84 12 78)(13 91 19 85)(14 92 20 86)(15 93 21 87)(16 94 22 88)(17 95 23 89)(18 96 24 90)(25 58 31 52)(26 59 32 53)(27 60 33 54)(28 55 34 49)(29 56 35 50)(30 57 36 51)(37 70 43 64)(38 71 44 65)(39 72 45 66)(40 67 46 61)(41 68 47 62)(42 69 48 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 19 7 13)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(25 46 31 40)(26 45 32 39)(27 44 33 38)(28 43 34 37)(29 48 35 42)(30 47 36 41)(49 67 55 61)(50 72 56 66)(51 71 57 65)(52 70 58 64)(53 69 59 63)(54 68 60 62)(73 94 79 88)(74 93 80 87)(75 92 81 86)(76 91 82 85)(77 96 83 90)(78 95 84 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 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
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), (1,79,7,73)(2,80,8,74)(3,81,9,75)(4,82,10,76)(5,83,11,77)(6,84,12,78)(13,91,19,85)(14,92,20,86)(15,93,21,87)(16,94,22,88)(17,95,23,89)(18,96,24,90)(25,58,31,52)(26,59,32,53)(27,60,33,54)(28,55,34,49)(29,56,35,50)(30,57,36,51)(37,70,43,64)(38,71,44,65)(39,72,45,66)(40,67,46,61)(41,68,47,62)(42,69,48,63), (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), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,46,31,40)(26,45,32,39)(27,44,33,38)(28,43,34,37)(29,48,35,42)(30,47,36,41)(49,67,55,61)(50,72,56,66)(51,71,57,65)(52,70,58,64)(53,69,59,63)(54,68,60,62)(73,94,79,88)(74,93,80,87)(75,92,81,86)(76,91,82,85)(77,96,83,90)(78,95,84,89), (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)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,88)(74,89)(75,90)(76,85)(77,86)(78,87)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93)>;
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), (1,79,7,73)(2,80,8,74)(3,81,9,75)(4,82,10,76)(5,83,11,77)(6,84,12,78)(13,91,19,85)(14,92,20,86)(15,93,21,87)(16,94,22,88)(17,95,23,89)(18,96,24,90)(25,58,31,52)(26,59,32,53)(27,60,33,54)(28,55,34,49)(29,56,35,50)(30,57,36,51)(37,70,43,64)(38,71,44,65)(39,72,45,66)(40,67,46,61)(41,68,47,62)(42,69,48,63), (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), (1,19,7,13)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,46,31,40)(26,45,32,39)(27,44,33,38)(28,43,34,37)(29,48,35,42)(30,47,36,41)(49,67,55,61)(50,72,56,66)(51,71,57,65)(52,70,58,64)(53,69,59,63)(54,68,60,62)(73,94,79,88)(74,93,80,87)(75,92,81,86)(76,91,82,85)(77,96,83,90)(78,95,84,89), (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)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,88)(74,89)(75,90)(76,85)(77,86)(78,87)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93) );
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)], [(1,79,7,73),(2,80,8,74),(3,81,9,75),(4,82,10,76),(5,83,11,77),(6,84,12,78),(13,91,19,85),(14,92,20,86),(15,93,21,87),(16,94,22,88),(17,95,23,89),(18,96,24,90),(25,58,31,52),(26,59,32,53),(27,60,33,54),(28,55,34,49),(29,56,35,50),(30,57,36,51),(37,70,43,64),(38,71,44,65),(39,72,45,66),(40,67,46,61),(41,68,47,62),(42,69,48,63)], [(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)], [(1,19,7,13),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14),(25,46,31,40),(26,45,32,39),(27,44,33,38),(28,43,34,37),(29,48,35,42),(30,47,36,41),(49,67,55,61),(50,72,56,66),(51,71,57,65),(52,70,58,64),(53,69,59,63),(54,68,60,62),(73,94,79,88),(74,93,80,87),(75,92,81,86),(76,91,82,85),(77,96,83,90),(78,95,84,89)], [(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),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,88),(74,89),(75,90),(76,85),(77,86),(78,87),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | S3×C4○D4 | Q8○D12 |
| kernel | C6.632+ 1+4 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | Dic3.Q8 | C4.Dic6 | S3×C4⋊C4 | D6⋊Q8 | C12.48D4 | C4×C3⋊D4 | C23.23D6 | D6⋊3D4 | C3×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D6 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.632+ 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 12 | 0 |
| 7 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 3 | 0 | 0 | 0 | 0 |
| 10 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 | 12 | 12 |
| 0 | 0 | 0 | 0 | 9 | 5 | 0 | 12 |
| 0 | 0 | 0 | 0 | 6 | 11 | 8 | 8 |
| 0 | 0 | 0 | 0 | 2 | 6 | 4 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 | 4 | 4 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 4 |
| 0 | 0 | 0 | 0 | 6 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 3 | 6 | 10 | 7 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 11 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 5 | 5 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 | 4 | 12 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 11 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 12 | 0 |
G:=sub<GL(8,GF(13))| [1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,1,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0],[7,0,10,0,0,0,0,0,0,7,0,10,0,0,0,0,3,0,6,0,0,0,0,0,0,3,0,6,0,0,0,0,0,0,0,0,1,9,6,2,0,0,0,0,5,5,11,6,0,0,0,0,12,0,8,4,0,0,0,0,12,12,8,12],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,6,3,6,3,0,0,0,0,6,3,0,6,0,0,0,0,4,0,10,10,0,0,0,0,4,4,7,7],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,5,5,0,0,0,0,1,0,5,4,0,0,0,0,11,12,12,12,0,0,0,0,12,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,12,11,12,12,0,0,0,0,1,12,0,0] >;
C6.632+ 1+4 in GAP, Magma, Sage, TeX
C_6._{63}2_+^{1+4} % in TeX
G:=Group("C6.63ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1219);
// by ID
G=gap.SmallGroup(192,1219);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,184,675,297,6278]);
// 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=b^-1,d*b*d^-1=e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations