G = C6.572+ 1+4 order 192 = 26·3
57th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.572+ 1+4, C6.782- 1+4, C4⋊C4.100D6, C22⋊Q8⋊22S3, D6⋊3Q8⋊24C2, (C2×Q8).102D6, C22⋊C4.65D6, D6.D4⋊23C2, C12⋊7D4.19C2, (C2×C6).189C24, (C2×C12).65C23, D6⋊C4.72C22, C2.38(Q8○D12), C12.3Q8⋊26C2, (C22×C4).267D6, Dic3⋊4D4⋊16C2, C2.59(D4⋊6D6), (C2×D12).31C22, (C6×Q8).118C22, C22.D12⋊17C2, (C22×S3).80C23, C4⋊Dic3.221C22, (C22×C6).217C23, C22.210(S3×C23), C23.207(C22×S3), C22.5(Q8⋊3S3), (C2×Dic3).95C23, Dic3⋊C4.119C22, (C22×C12).317C22, C3⋊5(C22.33C24), (C4×Dic3).116C22, (C22×Dic3).125C22, C4⋊C4⋊S3⋊24C2, C6.117(C2×C4○D4), (C3×C22⋊Q8)⋊25C2, (C2×Dic3⋊C4)⋊30C2, (C2×C6).29(C4○D4), (S3×C2×C4).105C22, C2.21(C2×Q8⋊3S3), (C3×C4⋊C4).169C22, (C2×C4).186(C22×S3), (C2×C3⋊D4).41C22, (C3×C22⋊C4).44C22, SmallGroup(192,1204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.572+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=cac=dad-1=a-1, ae=ea, cbc=a3b-1, bd=db, ebe-1=a3b, dcd-1=a3c, ce=ec, ede-1=a3b2d >
Subgroups: 544 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×12], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], Dic3 [×6], C12 [×6], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×11], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C4×S3 [×2], D12, C2×Dic3 [×6], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12, C3×Q8, C22×S3 [×2], C22×C6, C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×8], C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C6×Q8, C22.33C24, Dic3⋊4D4 [×2], C22.D12 [×2], C12.3Q8 [×2], D6.D4 [×2], C4⋊C4⋊S3 [×2], C2×Dic3⋊C4, C12⋊7D4, D6⋊3Q8 [×2], C3×C22⋊Q8, C6.572+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, Q8⋊3S3 [×2], S3×C23, C22.33C24, D4⋊6D6, C2×Q8⋊3S3, Q8○D12, C6.572+ 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 94 17 9)(2 93 18 8)(3 92 13 7)(4 91 14 12)(5 96 15 11)(6 95 16 10)(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 62 50 55)(44 61 51 60)(45 66 52 59)(46 65 53 58)(47 64 54 57)(48 63 49 56)
(2 6)(3 5)(7 93)(8 92)(9 91)(10 96)(11 95)(12 94)(13 15)(16 18)(19 22)(20 21)(23 24)(25 30)(26 29)(27 28)(32 36)(33 35)(37 39)(40 42)(43 46)(44 45)(47 48)(49 54)(50 53)(51 52)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(67 77)(68 76)(69 75)(70 74)(71 73)(72 78)(79 86)(80 85)(81 90)(82 89)(83 88)(84 87)
(1 53 17 46)(2 52 18 45)(3 51 13 44)(4 50 14 43)(5 49 15 48)(6 54 16 47)(7 61 92 60)(8 66 93 59)(9 65 94 58)(10 64 95 57)(11 63 96 56)(12 62 91 55)(19 38 26 31)(20 37 27 36)(21 42 28 35)(22 41 29 34)(23 40 30 33)(24 39 25 32)(67 86 74 79)(68 85 75 84)(69 90 76 83)(70 89 77 82)(71 88 78 81)(72 87 73 80)
(1 31 4 34)(2 32 5 35)(3 33 6 36)(7 68 10 71)(8 69 11 72)(9 70 12 67)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)(25 52 28 49)(26 53 29 50)(27 54 30 51)(55 89 58 86)(56 90 59 87)(57 85 60 88)(61 81 64 84)(62 82 65 79)(63 83 66 80)(73 93 76 96)(74 94 77 91)(75 95 78 92)
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), (1,94,17,9)(2,93,18,8)(3,92,13,7)(4,91,14,12)(5,96,15,11)(6,95,16,10)(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)(43,62,50,55)(44,61,51,60)(45,66,52,59)(46,65,53,58)(47,64,54,57)(48,63,49,56), (2,6)(3,5)(7,93)(8,92)(9,91)(10,96)(11,95)(12,94)(13,15)(16,18)(19,22)(20,21)(23,24)(25,30)(26,29)(27,28)(32,36)(33,35)(37,39)(40,42)(43,46)(44,45)(47,48)(49,54)(50,53)(51,52)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(67,77)(68,76)(69,75)(70,74)(71,73)(72,78)(79,86)(80,85)(81,90)(82,89)(83,88)(84,87), (1,53,17,46)(2,52,18,45)(3,51,13,44)(4,50,14,43)(5,49,15,48)(6,54,16,47)(7,61,92,60)(8,66,93,59)(9,65,94,58)(10,64,95,57)(11,63,96,56)(12,62,91,55)(19,38,26,31)(20,37,27,36)(21,42,28,35)(22,41,29,34)(23,40,30,33)(24,39,25,32)(67,86,74,79)(68,85,75,84)(69,90,76,83)(70,89,77,82)(71,88,78,81)(72,87,73,80), (1,31,4,34)(2,32,5,35)(3,33,6,36)(7,68,10,71)(8,69,11,72)(9,70,12,67)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45)(25,52,28,49)(26,53,29,50)(27,54,30,51)(55,89,58,86)(56,90,59,87)(57,85,60,88)(61,81,64,84)(62,82,65,79)(63,83,66,80)(73,93,76,96)(74,94,77,91)(75,95,78,92)>;
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), (1,94,17,9)(2,93,18,8)(3,92,13,7)(4,91,14,12)(5,96,15,11)(6,95,16,10)(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)(43,62,50,55)(44,61,51,60)(45,66,52,59)(46,65,53,58)(47,64,54,57)(48,63,49,56), (2,6)(3,5)(7,93)(8,92)(9,91)(10,96)(11,95)(12,94)(13,15)(16,18)(19,22)(20,21)(23,24)(25,30)(26,29)(27,28)(32,36)(33,35)(37,39)(40,42)(43,46)(44,45)(47,48)(49,54)(50,53)(51,52)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(67,77)(68,76)(69,75)(70,74)(71,73)(72,78)(79,86)(80,85)(81,90)(82,89)(83,88)(84,87), (1,53,17,46)(2,52,18,45)(3,51,13,44)(4,50,14,43)(5,49,15,48)(6,54,16,47)(7,61,92,60)(8,66,93,59)(9,65,94,58)(10,64,95,57)(11,63,96,56)(12,62,91,55)(19,38,26,31)(20,37,27,36)(21,42,28,35)(22,41,29,34)(23,40,30,33)(24,39,25,32)(67,86,74,79)(68,85,75,84)(69,90,76,83)(70,89,77,82)(71,88,78,81)(72,87,73,80), (1,31,4,34)(2,32,5,35)(3,33,6,36)(7,68,10,71)(8,69,11,72)(9,70,12,67)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45)(25,52,28,49)(26,53,29,50)(27,54,30,51)(55,89,58,86)(56,90,59,87)(57,85,60,88)(61,81,64,84)(62,82,65,79)(63,83,66,80)(73,93,76,96)(74,94,77,91)(75,95,78,92) );
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)], [(1,94,17,9),(2,93,18,8),(3,92,13,7),(4,91,14,12),(5,96,15,11),(6,95,16,10),(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),(43,62,50,55),(44,61,51,60),(45,66,52,59),(46,65,53,58),(47,64,54,57),(48,63,49,56)], [(2,6),(3,5),(7,93),(8,92),(9,91),(10,96),(11,95),(12,94),(13,15),(16,18),(19,22),(20,21),(23,24),(25,30),(26,29),(27,28),(32,36),(33,35),(37,39),(40,42),(43,46),(44,45),(47,48),(49,54),(50,53),(51,52),(55,62),(56,61),(57,66),(58,65),(59,64),(60,63),(67,77),(68,76),(69,75),(70,74),(71,73),(72,78),(79,86),(80,85),(81,90),(82,89),(83,88),(84,87)], [(1,53,17,46),(2,52,18,45),(3,51,13,44),(4,50,14,43),(5,49,15,48),(6,54,16,47),(7,61,92,60),(8,66,93,59),(9,65,94,58),(10,64,95,57),(11,63,96,56),(12,62,91,55),(19,38,26,31),(20,37,27,36),(21,42,28,35),(22,41,29,34),(23,40,30,33),(24,39,25,32),(67,86,74,79),(68,85,75,84),(69,90,76,83),(70,89,77,82),(71,88,78,81),(72,87,73,80)], [(1,31,4,34),(2,32,5,35),(3,33,6,36),(7,68,10,71),(8,69,11,72),(9,70,12,67),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45),(25,52,28,49),(26,53,29,50),(27,54,30,51),(55,89,58,86),(56,90,59,87),(57,85,60,88),(61,81,64,84),(62,82,65,79),(63,83,66,80),(73,93,76,96),(74,94,77,91),(75,95,78,92)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| 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 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 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 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | Q8⋊3S3 | D4⋊6D6 | Q8○D12 |
| kernel | C6.572+ 1+4 | Dic3⋊4D4 | C22.D12 | C12.3Q8 | D6.D4 | C4⋊C4⋊S3 | C2×Dic3⋊C4 | C12⋊7D4 | D6⋊3Q8 | C3×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 3 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.572+ 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 |
| 8 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 5 | 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 | 7 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 | 4 | 4 |
| 0 | 0 | 0 | 0 | 10 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 | 9 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 12 |
| 8 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 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 | 12 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 8 | 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 | 12 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 1 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[8,3,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,7,10,9,0,0,0,0,0,10,0,4,0,0,0,0,8,4,6,9,0,0,0,0,0,4,0,3],[1,2,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[8,3,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,11,12,1,1],[5,10,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,11,12,1,1,0,0,0,0,0,1,0,0] >;
C6.572+ 1+4 in GAP, Magma, Sage, TeX
C_6._{57}2_+^{1+4} % in TeX
G:=Group("C6.57ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1204);
// by ID
G=gap.SmallGroup(192,1204);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,387,100,675,409,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=b^2,e^2=a^3,b*a*b^-1=c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^3*b^-1,b*d=d*b,e*b*e^-1=a^3*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=a^3*b^2*d>;
// generators/relations