G = C6.682- 1+4 order 192 = 26·3
23rd non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.682- 1+4, C6.322+ 1+4, C4⋊D4⋊5S3, C4⋊C4.89D6, (C2×Dic3)⋊9D4, (C2×D4).89D6, C22⋊C4.4D6, Dic3⋊D4⋊16C2, C12⋊7D4⋊43C2, C22.2(S3×D4), D6⋊Q8⋊13C2, C6.60(C22×D4), C23.14D6⋊8C2, (C2×C6).141C24, D6⋊C4.69C22, C2.26(Q8○D12), Dic3.45(C2×D4), (C22×C4).233D6, C2.34(D4⋊6D6), (C2×C12).172C23, (C2×D12).30C22, (C6×D4).115C22, C4⋊Dic3.44C22, C23.18(C22×S3), (C22×C6).12C23, Dic3.D4⋊15C2, (C22×S3).60C23, C22.162(S3×C23), Dic3⋊C4.158C22, (C22×C12).310C22, C3⋊2(C22.31C24), (C2×Dic3).224C23, (C2×Dic6).151C22, C6.D4.19C22, (C22×Dic3).102C22, C2.33(C2×S3×D4), (C2×C6).4(C2×D4), (C3×C4⋊D4)⋊6C2, (C2×D4⋊2S3)⋊9C2, (S3×C2×C4).80C22, (C2×Dic3⋊C4)⋊28C2, (C2×C4).35(C22×S3), (C3×C4⋊C4).137C22, (C2×C3⋊D4).24C22, (C3×C22⋊C4).6C22, SmallGroup(192,1156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.682- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ece-1=a3c, ede-1=b2d >
Subgroups: 784 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×12], C22, C22 [×2], C22 [×14], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×4], D12, C2×Dic3 [×10], C2×Dic3 [×5], C3⋊D4 [×10], C2×C12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×5], C22×S3 [×2], C22×C6, C22×C6 [×2], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, D4⋊2S3 [×8], C22×Dic3 [×2], C22×Dic3 [×2], C2×C3⋊D4 [×6], C22×C12, C6×D4, C6×D4 [×2], C22.31C24, Dic3.D4 [×2], Dic3⋊D4 [×2], D6⋊Q8 [×2], C2×Dic3⋊C4, C12⋊7D4, C23.14D6 [×4], C3×C4⋊D4, C2×D4⋊2S3 [×2], C6.682- 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2+ 1+4, 2- 1+4, S3×D4 [×2], S3×C23, C22.31C24, C2×S3×D4, D4⋊6D6, Q8○D12, C6.682- 1+4
►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 26 15 19)(2 25 16 24)(3 30 17 23)(4 29 18 22)(5 28 13 21)(6 27 14 20)(7 81 92 88)(8 80 93 87)(9 79 94 86)(10 84 95 85)(11 83 96 90)(12 82 91 89)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 48 42 49)(36 47 37 54)(55 67 62 74)(56 72 63 73)(57 71 64 78)(58 70 65 77)(59 69 66 76)(60 68 61 75)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 43 18 53)(2 48 13 52)(3 47 14 51)(4 46 15 50)(5 45 16 49)(6 44 17 54)(7 60 95 64)(8 59 96 63)(9 58 91 62)(10 57 92 61)(11 56 93 66)(12 55 94 65)(19 31 29 41)(20 36 30 40)(21 35 25 39)(22 34 26 38)(23 33 27 37)(24 32 28 42)(67 89 77 79)(68 88 78 84)(69 87 73 83)(70 86 74 82)(71 85 75 81)(72 90 76 80)
(1 86 18 82)(2 87 13 83)(3 88 14 84)(4 89 15 79)(5 90 16 80)(6 85 17 81)(7 23 95 27)(8 24 96 28)(9 19 91 29)(10 20 92 30)(11 21 93 25)(12 22 94 26)(31 55 41 65)(32 56 42 66)(33 57 37 61)(34 58 38 62)(35 59 39 63)(36 60 40 64)(43 67 53 77)(44 68 54 78)(45 69 49 73)(46 70 50 74)(47 71 51 75)(48 72 52 76)
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), (1,26,15,19)(2,25,16,24)(3,30,17,23)(4,29,18,22)(5,28,13,21)(6,27,14,20)(7,81,92,88)(8,80,93,87)(9,79,94,86)(10,84,95,85)(11,83,96,90)(12,82,91,89)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,48,42,49)(36,47,37,54)(55,67,62,74)(56,72,63,73)(57,71,64,78)(58,70,65,77)(59,69,66,76)(60,68,61,75), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,43,18,53)(2,48,13,52)(3,47,14,51)(4,46,15,50)(5,45,16,49)(6,44,17,54)(7,60,95,64)(8,59,96,63)(9,58,91,62)(10,57,92,61)(11,56,93,66)(12,55,94,65)(19,31,29,41)(20,36,30,40)(21,35,25,39)(22,34,26,38)(23,33,27,37)(24,32,28,42)(67,89,77,79)(68,88,78,84)(69,87,73,83)(70,86,74,82)(71,85,75,81)(72,90,76,80), (1,86,18,82)(2,87,13,83)(3,88,14,84)(4,89,15,79)(5,90,16,80)(6,85,17,81)(7,23,95,27)(8,24,96,28)(9,19,91,29)(10,20,92,30)(11,21,93,25)(12,22,94,26)(31,55,41,65)(32,56,42,66)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64)(43,67,53,77)(44,68,54,78)(45,69,49,73)(46,70,50,74)(47,71,51,75)(48,72,52,76)>;
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), (1,26,15,19)(2,25,16,24)(3,30,17,23)(4,29,18,22)(5,28,13,21)(6,27,14,20)(7,81,92,88)(8,80,93,87)(9,79,94,86)(10,84,95,85)(11,83,96,90)(12,82,91,89)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,48,42,49)(36,47,37,54)(55,67,62,74)(56,72,63,73)(57,71,64,78)(58,70,65,77)(59,69,66,76)(60,68,61,75), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,43,18,53)(2,48,13,52)(3,47,14,51)(4,46,15,50)(5,45,16,49)(6,44,17,54)(7,60,95,64)(8,59,96,63)(9,58,91,62)(10,57,92,61)(11,56,93,66)(12,55,94,65)(19,31,29,41)(20,36,30,40)(21,35,25,39)(22,34,26,38)(23,33,27,37)(24,32,28,42)(67,89,77,79)(68,88,78,84)(69,87,73,83)(70,86,74,82)(71,85,75,81)(72,90,76,80), (1,86,18,82)(2,87,13,83)(3,88,14,84)(4,89,15,79)(5,90,16,80)(6,85,17,81)(7,23,95,27)(8,24,96,28)(9,19,91,29)(10,20,92,30)(11,21,93,25)(12,22,94,26)(31,55,41,65)(32,56,42,66)(33,57,37,61)(34,58,38,62)(35,59,39,63)(36,60,40,64)(43,67,53,77)(44,68,54,78)(45,69,49,73)(46,70,50,74)(47,71,51,75)(48,72,52,76) );
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)], [(1,26,15,19),(2,25,16,24),(3,30,17,23),(4,29,18,22),(5,28,13,21),(6,27,14,20),(7,81,92,88),(8,80,93,87),(9,79,94,86),(10,84,95,85),(11,83,96,90),(12,82,91,89),(31,46,38,53),(32,45,39,52),(33,44,40,51),(34,43,41,50),(35,48,42,49),(36,47,37,54),(55,67,62,74),(56,72,63,73),(57,71,64,78),(58,70,65,77),(59,69,66,76),(60,68,61,75)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,25),(7,90),(8,85),(9,86),(10,87),(11,88),(12,89),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(31,50),(32,51),(33,52),(34,53),(35,54),(36,49),(37,48),(38,43),(39,44),(40,45),(41,46),(42,47),(55,77),(56,78),(57,73),(58,74),(59,75),(60,76),(61,69),(62,70),(63,71),(64,72),(65,67),(66,68),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)], [(1,43,18,53),(2,48,13,52),(3,47,14,51),(4,46,15,50),(5,45,16,49),(6,44,17,54),(7,60,95,64),(8,59,96,63),(9,58,91,62),(10,57,92,61),(11,56,93,66),(12,55,94,65),(19,31,29,41),(20,36,30,40),(21,35,25,39),(22,34,26,38),(23,33,27,37),(24,32,28,42),(67,89,77,79),(68,88,78,84),(69,87,73,83),(70,86,74,82),(71,85,75,81),(72,90,76,80)], [(1,86,18,82),(2,87,13,83),(3,88,14,84),(4,89,15,79),(5,90,16,80),(6,85,17,81),(7,23,95,27),(8,24,96,28),(9,19,91,29),(10,20,92,30),(11,21,93,25),(12,22,94,26),(31,55,41,65),(32,56,42,66),(33,57,37,61),(34,58,38,62),(35,59,39,63),(36,60,40,64),(43,67,53,77),(44,68,54,78),(45,69,49,73),(46,70,50,74),(47,71,51,75),(48,72,52,76)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | S3×D4 | D4⋊6D6 | Q8○D12 |
| kernel | C6.682- 1+4 | Dic3.D4 | Dic3⋊D4 | D6⋊Q8 | C2×Dic3⋊C4 | C12⋊7D4 | C23.14D6 | C3×C4⋊D4 | C2×D4⋊2S3 | C4⋊D4 | C2×Dic3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 2 | 1 | 1 | 3 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.682- 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 | 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 | 12 |
| 1 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 12 | 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 | 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 |
| 12 | 3 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 1 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 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 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 0 | 10 | 6 | 3 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,5,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,2,0,1,0,0,0,0,2,0,12,0],[12,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,7,10,0,0,0,0,0,6,10,0,10,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3] >;
C6.682- 1+4 in GAP, Magma, Sage, TeX
C_6._{68}2_-^{1+4} % in TeX
G:=Group("C6.68ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1156);
// by ID
G=gap.SmallGroup(192,1156);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,1123,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=e^2=a^3*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^3*b,c*d=d*c,e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;
// generators/relations