G = C6.852- 1+4 order 192 = 26·3
40th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.852- 1+4, C6.1232+ 1+4, C4⋊C4.108D6, C4.D12⋊33C2, (D4×Dic3)⋊29C2, (C2×D4).167D6, D6⋊C4.9C22, C22⋊C4.69D6, C23.9D6⋊37C2, C2.45(D4○D12), Dic3.Q8⋊30C2, (C2×C6).209C24, C2.46(Q8○D12), C4.Dic6⋊30C2, (C22×C4).278D6, Dic3⋊4D4⋊26C2, (C2×C12).183C23, C23.14D6.3C2, (C6×D4).147C22, C23.28D6⋊8C2, C23.8D6⋊35C2, C22.D4⋊14S3, (C22×C6).41C23, C23.43(C22×S3), Dic3.D4⋊35C2, C23.21D6⋊23C2, Dic3⋊C4.10C22, (C22×S3).90C23, C4⋊Dic3.313C22, C22.230(S3×C23), (C22×C12).87C22, C22.12(D4⋊2S3), C3⋊7(C22.33C24), (C2×Dic6).171C22, (C2×Dic3).108C23, (C4×Dic3).128C22, C6.D4.47C22, (C22×Dic3).135C22, C4⋊C4⋊S3⋊31C2, C6.92(C2×C4○D4), (C2×C4⋊Dic3)⋊27C2, (C2×C6).48(C4○D4), C2.54(C2×D4⋊2S3), (S3×C2×C4).116C22, (C3×C4⋊C4).182C22, (C2×C4).190(C22×S3), (C2×C3⋊D4).53C22, (C3×C22.D4)⋊17C2, (C3×C22⋊C4).57C22, SmallGroup(192,1224)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.852- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 528 in 218 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, 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, C23.21D6, Dic3.Q8, C4.Dic6, C4.D12, C4⋊C4⋊S3, C2×C4⋊Dic3, C23.28D6, D4×Dic3, C23.14D6, C3×C22.D4, C6.852- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, D4⋊2S3, S3×C23, C22.33C24, C2×D4⋊2S3, D4○D12, Q8○D12, C6.852- 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 94 18 12)(2 95 13 7)(3 96 14 8)(4 91 15 9)(5 92 16 10)(6 93 17 11)(19 79 29 89)(20 80 30 90)(21 81 25 85)(22 82 26 86)(23 83 27 87)(24 84 28 88)(31 67 41 77)(32 68 42 78)(33 69 37 73)(34 70 38 74)(35 71 39 75)(36 72 40 76)(43 65 53 55)(44 66 54 56)(45 61 49 57)(46 62 50 58)(47 63 51 59)(48 64 52 60)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 61)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 59)(14 60)(15 55)(16 56)(17 57)(18 58)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 9 15 94)(2 8 16 93)(3 7 17 92)(4 12 18 91)(5 11 13 96)(6 10 14 95)(19 79 26 86)(20 84 27 85)(21 83 28 90)(22 82 29 89)(23 81 30 88)(24 80 25 87)(31 74 38 67)(32 73 39 72)(33 78 40 71)(34 77 41 70)(35 76 42 69)(36 75 37 68)(43 65 50 58)(44 64 51 57)(45 63 52 56)(46 62 53 55)(47 61 54 60)(48 66 49 59)
(1 19 18 29)(2 20 13 30)(3 21 14 25)(4 22 15 26)(5 23 16 27)(6 24 17 28)(7 87 95 83)(8 88 96 84)(9 89 91 79)(10 90 92 80)(11 85 93 81)(12 86 94 82)(31 53 41 43)(32 54 42 44)(33 49 37 45)(34 50 38 46)(35 51 39 47)(36 52 40 48)(55 74 65 70)(56 75 66 71)(57 76 61 72)(58 77 62 67)(59 78 63 68)(60 73 64 69)
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), (1,94,18,12)(2,95,13,7)(3,96,14,8)(4,91,15,9)(5,92,16,10)(6,93,17,11)(19,79,29,89)(20,80,30,90)(21,81,25,85)(22,82,26,86)(23,83,27,87)(24,84,28,88)(31,67,41,77)(32,68,42,78)(33,69,37,73)(34,70,38,74)(35,71,39,75)(36,72,40,76)(43,65,53,55)(44,66,54,56)(45,61,49,57)(46,62,50,58)(47,63,51,59)(48,64,52,60), (1,62)(2,63)(3,64)(4,65)(5,66)(6,61)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,9,15,94)(2,8,16,93)(3,7,17,92)(4,12,18,91)(5,11,13,96)(6,10,14,95)(19,79,26,86)(20,84,27,85)(21,83,28,90)(22,82,29,89)(23,81,30,88)(24,80,25,87)(31,74,38,67)(32,73,39,72)(33,78,40,71)(34,77,41,70)(35,76,42,69)(36,75,37,68)(43,65,50,58)(44,64,51,57)(45,63,52,56)(46,62,53,55)(47,61,54,60)(48,66,49,59), (1,19,18,29)(2,20,13,30)(3,21,14,25)(4,22,15,26)(5,23,16,27)(6,24,17,28)(7,87,95,83)(8,88,96,84)(9,89,91,79)(10,90,92,80)(11,85,93,81)(12,86,94,82)(31,53,41,43)(32,54,42,44)(33,49,37,45)(34,50,38,46)(35,51,39,47)(36,52,40,48)(55,74,65,70)(56,75,66,71)(57,76,61,72)(58,77,62,67)(59,78,63,68)(60,73,64,69)>;
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), (1,94,18,12)(2,95,13,7)(3,96,14,8)(4,91,15,9)(5,92,16,10)(6,93,17,11)(19,79,29,89)(20,80,30,90)(21,81,25,85)(22,82,26,86)(23,83,27,87)(24,84,28,88)(31,67,41,77)(32,68,42,78)(33,69,37,73)(34,70,38,74)(35,71,39,75)(36,72,40,76)(43,65,53,55)(44,66,54,56)(45,61,49,57)(46,62,50,58)(47,63,51,59)(48,64,52,60), (1,62)(2,63)(3,64)(4,65)(5,66)(6,61)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,59)(14,60)(15,55)(16,56)(17,57)(18,58)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,9,15,94)(2,8,16,93)(3,7,17,92)(4,12,18,91)(5,11,13,96)(6,10,14,95)(19,79,26,86)(20,84,27,85)(21,83,28,90)(22,82,29,89)(23,81,30,88)(24,80,25,87)(31,74,38,67)(32,73,39,72)(33,78,40,71)(34,77,41,70)(35,76,42,69)(36,75,37,68)(43,65,50,58)(44,64,51,57)(45,63,52,56)(46,62,53,55)(47,61,54,60)(48,66,49,59), (1,19,18,29)(2,20,13,30)(3,21,14,25)(4,22,15,26)(5,23,16,27)(6,24,17,28)(7,87,95,83)(8,88,96,84)(9,89,91,79)(10,90,92,80)(11,85,93,81)(12,86,94,82)(31,53,41,43)(32,54,42,44)(33,49,37,45)(34,50,38,46)(35,51,39,47)(36,52,40,48)(55,74,65,70)(56,75,66,71)(57,76,61,72)(58,77,62,67)(59,78,63,68)(60,73,64,69) );
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)], [(1,94,18,12),(2,95,13,7),(3,96,14,8),(4,91,15,9),(5,92,16,10),(6,93,17,11),(19,79,29,89),(20,80,30,90),(21,81,25,85),(22,82,26,86),(23,83,27,87),(24,84,28,88),(31,67,41,77),(32,68,42,78),(33,69,37,73),(34,70,38,74),(35,71,39,75),(36,72,40,76),(43,65,53,55),(44,66,54,56),(45,61,49,57),(46,62,50,58),(47,63,51,59),(48,64,52,60)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,61),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,59),(14,60),(15,55),(16,56),(17,57),(18,58),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,9,15,94),(2,8,16,93),(3,7,17,92),(4,12,18,91),(5,11,13,96),(6,10,14,95),(19,79,26,86),(20,84,27,85),(21,83,28,90),(22,82,29,89),(23,81,30,88),(24,80,25,87),(31,74,38,67),(32,73,39,72),(33,78,40,71),(34,77,41,70),(35,76,42,69),(36,75,37,68),(43,65,50,58),(44,64,51,57),(45,63,52,56),(46,62,53,55),(47,61,54,60),(48,66,49,59)], [(1,19,18,29),(2,20,13,30),(3,21,14,25),(4,22,15,26),(5,23,16,27),(6,24,17,28),(7,87,95,83),(8,88,96,84),(9,89,91,79),(10,90,92,80),(11,85,93,81),(12,86,94,82),(31,53,41,43),(32,54,42,44),(33,49,37,45),(34,50,38,46),(35,51,39,47),(36,52,40,48),(55,74,65,70),(56,75,66,71),(57,76,61,72),(58,77,62,67),(59,78,63,68),(60,73,64,69)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 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 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 2 | 4 | ··· | 4 | 6 | 6 | 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 | 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 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | D4⋊2S3 | D4○D12 | Q8○D12 |
| kernel | C6.852- 1+4 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | C23.21D6 | Dic3.Q8 | C4.Dic6 | C4.D12 | C4⋊C4⋊S3 | C2×C4⋊Dic3 | C23.28D6 | D4×Dic3 | C23.14D6 | C3×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.852- 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 |
| 8 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 5 | 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 | 9 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 10 |
| 0 | 0 | 0 | 0 | 10 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 4 |
| 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 | 0 | 10 | 0 | 4 |
| 0 | 0 | 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 3 |
| 0 | 0 | 0 | 0 | 9 | 0 | 10 | 0 |
| 8 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 4 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 10 |
| 0 | 0 | 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 4 |
| 1 | 11 | 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 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 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 | 0 |
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],[8,8,0,0,0,0,0,0,10,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,9,0,10,0,0,0,0,0,0,9,0,10,0,0,0,0,10,0,4,0,0,0,0,0,0,10,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,3,0,9,0,0,0,0,10,0,4,0,0,0,0,0,0,9,0,10,0,0,0,0,4,0,3,0],[8,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,9,0,10,0,0,0,0,3,0,9,0,0,0,0,0,0,10,0,4],[1,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0] >;
C6.852- 1+4 in GAP, Magma, Sage, TeX
C_6._{85}2_-^{1+4} % in TeX
G:=Group("C6.85ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1224);
// by ID
G=gap.SmallGroup(192,1224);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,100,675,409,136,6278]);
// 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,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations