G = C6.172- 1+4 order 192 = 26·3
17th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.172- 1+4, C6.1162+ 1+4, (C4×S3)⋊3D4, C4⋊C4.96D6, D6.4(C2×D4), C22⋊Q8⋊6S3, C4.188(S3×D4), Dic3⋊D4⋊24C2, C12⋊D4⋊24C2, C12⋊7D4⋊36C2, D6⋊3Q8⋊15C2, D6⋊Q8⋊18C2, C12.233(C2×D4), (C2×Q8).149D6, C22⋊C4.15D6, C6.75(C22×D4), C2.33(D4○D12), (C2×C6).173C24, (C2×C12).53C23, D6⋊C4.22C22, Dic3.48(C2×D4), (C22×C4).251D6, (C6×Q8).106C22, (C2×D12).148C22, Dic3⋊C4.26C22, C4⋊Dic3.214C22, C22.194(S3×C23), (C22×C6).201C23, C23.128(C22×S3), (C2×Dic3).88C23, (C22×S3).195C23, (C22×C12).253C22, C2.18(Q8.15D6), C3⋊4(C22.31C24), (C2×Dic6).247C22, (S3×C4⋊C4)⋊25C2, C2.48(C2×S3×D4), (C3×C22⋊Q8)⋊9C2, (C2×C4○D12)⋊23C2, (C2×Q8⋊3S3)⋊6C2, (S3×C2×C4).94C22, (C3×C4⋊C4).157C22, (C2×C4).591(C22×S3), (C2×C3⋊D4).121C22, (C3×C22⋊C4).28C22, SmallGroup(192,1188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.172- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=b2d >
Subgroups: 832 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C22⋊Q8, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C4○D12, Q8⋊3S3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×Q8, C22.31C24, Dic3⋊D4, S3×C4⋊C4, C12⋊D4, C12⋊D4, D6⋊Q8, C12⋊7D4, D6⋊3Q8, C3×C22⋊Q8, C2×C4○D12, C2×Q8⋊3S3, C6.172- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, 2- 1+4, S3×D4, S3×C23, C22.31C24, C2×S3×D4, Q8.15D6, D4○D12, C6.172- 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 67 7 61)(2 68 8 62)(3 69 9 63)(4 70 10 64)(5 71 11 65)(6 72 12 66)(13 55 19 49)(14 56 20 50)(15 57 21 51)(16 58 22 52)(17 59 23 53)(18 60 24 54)(25 94 31 88)(26 95 32 89)(27 96 33 90)(28 91 34 85)(29 92 35 86)(30 93 36 87)(37 82 43 76)(38 83 44 77)(39 84 45 78)(40 79 46 73)(41 80 47 74)(42 81 48 75)
(1 76)(2 75)(3 74)(4 73)(5 78)(6 77)(7 82)(8 81)(9 80)(10 79)(11 84)(12 83)(13 91)(14 96)(15 95)(16 94)(17 93)(18 92)(19 85)(20 90)(21 89)(22 88)(23 87)(24 86)(25 58)(26 57)(27 56)(28 55)(29 60)(30 59)(31 52)(32 51)(33 50)(34 49)(35 54)(36 53)(37 61)(38 66)(39 65)(40 64)(41 63)(42 62)(43 67)(44 72)(45 71)(46 70)(47 69)(48 68)
(1 19 10 16)(2 24 11 15)(3 23 12 14)(4 22 7 13)(5 21 8 18)(6 20 9 17)(25 43 34 40)(26 48 35 39)(27 47 36 38)(28 46 31 37)(29 45 32 42)(30 44 33 41)(49 61 58 70)(50 66 59 69)(51 65 60 68)(52 64 55 67)(53 63 56 72)(54 62 57 71)(73 85 82 94)(74 90 83 93)(75 89 84 92)(76 88 79 91)(77 87 80 96)(78 86 81 95)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 91 43 85)(38 92 44 86)(39 93 45 87)(40 94 46 88)(41 95 47 89)(42 96 48 90)
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), (1,67,7,61)(2,68,8,62)(3,69,9,63)(4,70,10,64)(5,71,11,65)(6,72,12,66)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,94,31,88)(26,95,32,89)(27,96,33,90)(28,91,34,85)(29,92,35,86)(30,93,36,87)(37,82,43,76)(38,83,44,77)(39,84,45,78)(40,79,46,73)(41,80,47,74)(42,81,48,75), (1,76)(2,75)(3,74)(4,73)(5,78)(6,77)(7,82)(8,81)(9,80)(10,79)(11,84)(12,83)(13,91)(14,96)(15,95)(16,94)(17,93)(18,92)(19,85)(20,90)(21,89)(22,88)(23,87)(24,86)(25,58)(26,57)(27,56)(28,55)(29,60)(30,59)(31,52)(32,51)(33,50)(34,49)(35,54)(36,53)(37,61)(38,66)(39,65)(40,64)(41,63)(42,62)(43,67)(44,72)(45,71)(46,70)(47,69)(48,68), (1,19,10,16)(2,24,11,15)(3,23,12,14)(4,22,7,13)(5,21,8,18)(6,20,9,17)(25,43,34,40)(26,48,35,39)(27,47,36,38)(28,46,31,37)(29,45,32,42)(30,44,33,41)(49,61,58,70)(50,66,59,69)(51,65,60,68)(52,64,55,67)(53,63,56,72)(54,62,57,71)(73,85,82,94)(74,90,83,93)(75,89,84,92)(76,88,79,91)(77,87,80,96)(78,86,81,95), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,79,31,73)(26,80,32,74)(27,81,33,75)(28,82,34,76)(29,83,35,77)(30,84,36,78)(37,91,43,85)(38,92,44,86)(39,93,45,87)(40,94,46,88)(41,95,47,89)(42,96,48,90)>;
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), (1,67,7,61)(2,68,8,62)(3,69,9,63)(4,70,10,64)(5,71,11,65)(6,72,12,66)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,94,31,88)(26,95,32,89)(27,96,33,90)(28,91,34,85)(29,92,35,86)(30,93,36,87)(37,82,43,76)(38,83,44,77)(39,84,45,78)(40,79,46,73)(41,80,47,74)(42,81,48,75), (1,76)(2,75)(3,74)(4,73)(5,78)(6,77)(7,82)(8,81)(9,80)(10,79)(11,84)(12,83)(13,91)(14,96)(15,95)(16,94)(17,93)(18,92)(19,85)(20,90)(21,89)(22,88)(23,87)(24,86)(25,58)(26,57)(27,56)(28,55)(29,60)(30,59)(31,52)(32,51)(33,50)(34,49)(35,54)(36,53)(37,61)(38,66)(39,65)(40,64)(41,63)(42,62)(43,67)(44,72)(45,71)(46,70)(47,69)(48,68), (1,19,10,16)(2,24,11,15)(3,23,12,14)(4,22,7,13)(5,21,8,18)(6,20,9,17)(25,43,34,40)(26,48,35,39)(27,47,36,38)(28,46,31,37)(29,45,32,42)(30,44,33,41)(49,61,58,70)(50,66,59,69)(51,65,60,68)(52,64,55,67)(53,63,56,72)(54,62,57,71)(73,85,82,94)(74,90,83,93)(75,89,84,92)(76,88,79,91)(77,87,80,96)(78,86,81,95), (1,55,7,49)(2,56,8,50)(3,57,9,51)(4,58,10,52)(5,59,11,53)(6,60,12,54)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,79,31,73)(26,80,32,74)(27,81,33,75)(28,82,34,76)(29,83,35,77)(30,84,36,78)(37,91,43,85)(38,92,44,86)(39,93,45,87)(40,94,46,88)(41,95,47,89)(42,96,48,90) );
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)], [(1,67,7,61),(2,68,8,62),(3,69,9,63),(4,70,10,64),(5,71,11,65),(6,72,12,66),(13,55,19,49),(14,56,20,50),(15,57,21,51),(16,58,22,52),(17,59,23,53),(18,60,24,54),(25,94,31,88),(26,95,32,89),(27,96,33,90),(28,91,34,85),(29,92,35,86),(30,93,36,87),(37,82,43,76),(38,83,44,77),(39,84,45,78),(40,79,46,73),(41,80,47,74),(42,81,48,75)], [(1,76),(2,75),(3,74),(4,73),(5,78),(6,77),(7,82),(8,81),(9,80),(10,79),(11,84),(12,83),(13,91),(14,96),(15,95),(16,94),(17,93),(18,92),(19,85),(20,90),(21,89),(22,88),(23,87),(24,86),(25,58),(26,57),(27,56),(28,55),(29,60),(30,59),(31,52),(32,51),(33,50),(34,49),(35,54),(36,53),(37,61),(38,66),(39,65),(40,64),(41,63),(42,62),(43,67),(44,72),(45,71),(46,70),(47,69),(48,68)], [(1,19,10,16),(2,24,11,15),(3,23,12,14),(4,22,7,13),(5,21,8,18),(6,20,9,17),(25,43,34,40),(26,48,35,39),(27,47,36,38),(28,46,31,37),(29,45,32,42),(30,44,33,41),(49,61,58,70),(50,66,59,69),(51,65,60,68),(52,64,55,67),(53,63,56,72),(54,62,57,71),(73,85,82,94),(74,90,83,93),(75,89,84,92),(76,88,79,91),(77,87,80,96),(78,86,81,95)], [(1,55,7,49),(2,56,8,50),(3,57,9,51),(4,58,10,52),(5,59,11,53),(6,60,12,54),(13,67,19,61),(14,68,20,62),(15,69,21,63),(16,70,22,64),(17,71,23,65),(18,72,24,66),(25,79,31,73),(26,80,32,74),(27,81,33,75),(28,82,34,76),(29,83,35,77),(30,84,36,78),(37,91,43,85),(38,92,44,86),(39,93,45,87),(40,94,46,88),(41,95,47,89),(42,96,48,90)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | S3×D4 | Q8.15D6 | D4○D12 |
| kernel | C6.172- 1+4 | Dic3⋊D4 | S3×C4⋊C4 | C12⋊D4 | D6⋊Q8 | C12⋊7D4 | D6⋊3Q8 | C3×C22⋊Q8 | C2×C4○D12 | C2×Q8⋊3S3 | C22⋊Q8 | C4×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 4 | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.172- 1+4 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 | 2 | 11 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 10 | 8 | 3 | 3 |
| 0 | 0 | 0 | 0 | 5 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 | 0 | 5 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 11 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 12 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,12,12,12,0,0,0,0,0,0,2,0,12,0,0,0,0,11,11,12,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,5,10,5,8,0,0,0,0,0,8,0,5,0,0,0,0,3,3,8,0,0,0,0,0,0,3,0,5],[0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,2,1,12,0,0,0,0,12,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,12,0,0,0,0,11,11,12,0,0,0,0,0,0,2,0,12] >;
C6.172- 1+4 in GAP, Magma, Sage, TeX
C_6._{17}2_-^{1+4} % in TeX
G:=Group("C6.17ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1188);
// by ID
G=gap.SmallGroup(192,1188);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,e^2=b^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations