G = C6.502+ 1+4 order 192 = 26·3
50th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.502+ 1+4, C6.752- 1+4, C12⋊Q8⋊23C2, C4⋊C4.94D6, (C2×Dic3)⋊4Q8, (C2×Q8).98D6, C22⋊Q8.8S3, C22.6(S3×Q8), C22⋊C4.53D6, Dic3.4(C2×Q8), C6.33(C22×Q8), Dic3.Q8⋊16C2, (C2×C12).49C23, (C2×C6).167C24, C2.33(Q8○D12), (C22×C4).248D6, Dic3⋊Q8⋊12C2, C2.52(D4⋊6D6), C4⋊Dic3.47C22, (C6×Q8).102C22, C12.48D4.19C2, C22.188(S3×C23), (C22×C6).195C23, C23.195(C22×S3), Dic3.D4.3C2, C23.16D6.2C2, Dic3⋊C4.161C22, (C22×C12).314C22, C3⋊3(C23.41C23), (C2×Dic6).158C22, (C2×Dic3).231C23, (C4×Dic3).101C22, C6.D4.31C22, (C22×Dic3).117C22, C2.16(C2×S3×Q8), (C2×C6).6(C2×Q8), (C3×C22⋊Q8).8C2, (C3×C4⋊C4).153C22, (C2×Dic3⋊C4).23C2, (C2×C4).181(C22×S3), (C3×C22⋊C4).22C22, SmallGroup(192,1182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.502+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a3c, ce=ec, ede=b2d >
Subgroups: 448 in 206 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×16], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×4], C23, Dic3 [×4], Dic3 [×6], C12 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C22×C4 [×2], C2×Q8, C2×Q8 [×3], Dic6 [×3], C2×Dic3 [×12], C2×Dic3, C2×C12 [×2], C2×C12 [×4], C2×C12, C3×Q8, C22×C6, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8, C22⋊Q8 [×3], C42.C2 [×4], C4⋊Q8 [×4], C4×Dic3 [×4], Dic3⋊C4 [×14], C4⋊Dic3, C4⋊Dic3 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×2], C22×Dic3 [×2], C22×C12, C6×Q8, C23.41C23, C23.16D6 [×2], Dic3.D4 [×2], C12⋊Q8 [×2], Dic3.Q8 [×4], C2×Dic3⋊C4, C12.48D4, Dic3⋊Q8 [×2], C3×C22⋊Q8, C6.502+ 1+4
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, C22×S3 [×7], C22×Q8, 2+ 1+4, 2- 1+4, S3×Q8 [×2], S3×C23, C23.41C23, D4⋊6D6, C2×S3×Q8, Q8○D12, C6.502+ 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 64 17 56)(2 63 18 55)(3 62 13 60)(4 61 14 59)(5 66 15 58)(6 65 16 57)(7 54 92 46)(8 53 93 45)(9 52 94 44)(10 51 95 43)(11 50 96 48)(12 49 91 47)(19 73 26 71)(20 78 27 70)(21 77 28 69)(22 76 29 68)(23 75 30 67)(24 74 25 72)(31 85 38 83)(32 90 39 82)(33 89 40 81)(34 88 41 80)(35 87 42 79)(36 86 37 84)
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 83 10 80)(8 84 11 81)(9 79 12 82)(13 28 16 25)(14 29 17 26)(15 30 18 27)(31 43 34 46)(32 44 35 47)(33 45 36 48)(37 50 40 53)(38 51 41 54)(39 52 42 49)(55 78 58 75)(56 73 59 76)(57 74 60 77)(61 68 64 71)(62 69 65 72)(63 70 66 67)(85 95 88 92)(86 96 89 93)(87 91 90 94)
(1 38 14 34)(2 39 15 35)(3 40 16 36)(4 41 17 31)(5 42 18 32)(6 37 13 33)(7 76 95 71)(8 77 96 72)(9 78 91 67)(10 73 92 68)(11 74 93 69)(12 75 94 70)(19 54 29 43)(20 49 30 44)(21 50 25 45)(22 51 26 46)(23 52 27 47)(24 53 28 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
(7 92)(8 93)(9 94)(10 95)(11 96)(12 91)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)(43 51)(44 52)(45 53)(46 54)(47 49)(48 50)(79 87)(80 88)(81 89)(82 90)(83 85)(84 86)
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), (1,64,17,56)(2,63,18,55)(3,62,13,60)(4,61,14,59)(5,66,15,58)(6,65,16,57)(7,54,92,46)(8,53,93,45)(9,52,94,44)(10,51,95,43)(11,50,96,48)(12,49,91,47)(19,73,26,71)(20,78,27,70)(21,77,28,69)(22,76,29,68)(23,75,30,67)(24,74,25,72)(31,85,38,83)(32,90,39,82)(33,89,40,81)(34,88,41,80)(35,87,42,79)(36,86,37,84), (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,83,10,80)(8,84,11,81)(9,79,12,82)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,43,34,46)(32,44,35,47)(33,45,36,48)(37,50,40,53)(38,51,41,54)(39,52,42,49)(55,78,58,75)(56,73,59,76)(57,74,60,77)(61,68,64,71)(62,69,65,72)(63,70,66,67)(85,95,88,92)(86,96,89,93)(87,91,90,94), (1,38,14,34)(2,39,15,35)(3,40,16,36)(4,41,17,31)(5,42,18,32)(6,37,13,33)(7,76,95,71)(8,77,96,72)(9,78,91,67)(10,73,92,68)(11,74,93,69)(12,75,94,70)(19,54,29,43)(20,49,30,44)(21,50,25,45)(22,51,26,46)(23,52,27,47)(24,53,28,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84), (7,92)(8,93)(9,94)(10,95)(11,96)(12,91)(31,38)(32,39)(33,40)(34,41)(35,42)(36,37)(43,51)(44,52)(45,53)(46,54)(47,49)(48,50)(79,87)(80,88)(81,89)(82,90)(83,85)(84,86)>;
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), (1,64,17,56)(2,63,18,55)(3,62,13,60)(4,61,14,59)(5,66,15,58)(6,65,16,57)(7,54,92,46)(8,53,93,45)(9,52,94,44)(10,51,95,43)(11,50,96,48)(12,49,91,47)(19,73,26,71)(20,78,27,70)(21,77,28,69)(22,76,29,68)(23,75,30,67)(24,74,25,72)(31,85,38,83)(32,90,39,82)(33,89,40,81)(34,88,41,80)(35,87,42,79)(36,86,37,84), (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,83,10,80)(8,84,11,81)(9,79,12,82)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,43,34,46)(32,44,35,47)(33,45,36,48)(37,50,40,53)(38,51,41,54)(39,52,42,49)(55,78,58,75)(56,73,59,76)(57,74,60,77)(61,68,64,71)(62,69,65,72)(63,70,66,67)(85,95,88,92)(86,96,89,93)(87,91,90,94), (1,38,14,34)(2,39,15,35)(3,40,16,36)(4,41,17,31)(5,42,18,32)(6,37,13,33)(7,76,95,71)(8,77,96,72)(9,78,91,67)(10,73,92,68)(11,74,93,69)(12,75,94,70)(19,54,29,43)(20,49,30,44)(21,50,25,45)(22,51,26,46)(23,52,27,47)(24,53,28,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84), (7,92)(8,93)(9,94)(10,95)(11,96)(12,91)(31,38)(32,39)(33,40)(34,41)(35,42)(36,37)(43,51)(44,52)(45,53)(46,54)(47,49)(48,50)(79,87)(80,88)(81,89)(82,90)(83,85)(84,86) );
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)], [(1,64,17,56),(2,63,18,55),(3,62,13,60),(4,61,14,59),(5,66,15,58),(6,65,16,57),(7,54,92,46),(8,53,93,45),(9,52,94,44),(10,51,95,43),(11,50,96,48),(12,49,91,47),(19,73,26,71),(20,78,27,70),(21,77,28,69),(22,76,29,68),(23,75,30,67),(24,74,25,72),(31,85,38,83),(32,90,39,82),(33,89,40,81),(34,88,41,80),(35,87,42,79),(36,86,37,84)], [(1,19,4,22),(2,20,5,23),(3,21,6,24),(7,83,10,80),(8,84,11,81),(9,79,12,82),(13,28,16,25),(14,29,17,26),(15,30,18,27),(31,43,34,46),(32,44,35,47),(33,45,36,48),(37,50,40,53),(38,51,41,54),(39,52,42,49),(55,78,58,75),(56,73,59,76),(57,74,60,77),(61,68,64,71),(62,69,65,72),(63,70,66,67),(85,95,88,92),(86,96,89,93),(87,91,90,94)], [(1,38,14,34),(2,39,15,35),(3,40,16,36),(4,41,17,31),(5,42,18,32),(6,37,13,33),(7,76,95,71),(8,77,96,72),(9,78,91,67),(10,73,92,68),(11,74,93,69),(12,75,94,70),(19,54,29,43),(20,49,30,44),(21,50,25,45),(22,51,26,46),(23,52,27,47),(24,53,28,48),(55,90,66,79),(56,85,61,80),(57,86,62,81),(58,87,63,82),(59,88,64,83),(60,89,65,84)], [(7,92),(8,93),(9,94),(10,95),(11,96),(12,91),(31,38),(32,39),(33,40),(34,41),(35,42),(36,37),(43,51),(44,52),(45,53),(46,54),(47,49),(48,50),(79,87),(80,88),(81,89),(82,90),(83,85),(84,86)])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | S3×Q8 | D4⋊6D6 | Q8○D12 |
| kernel | C6.502+ 1+4 | C23.16D6 | Dic3.D4 | C12⋊Q8 | Dic3.Q8 | C2×Dic3⋊C4 | C12.48D4 | Dic3⋊Q8 | C3×C22⋊Q8 | C22⋊Q8 | C2×Dic3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.502+ 1+4 ►in GL8(𝔽13)
| 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 9 |
| 0 | 10 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 8 | 0 | 0 | 0 | 0 |
| 1 | 2 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 1 | 0 |
| 8 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 8 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 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 | 12 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 3 | 9 | 8 | 0 | 0 | 0 | 0 |
| 9 | 8 | 6 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 | 12 | 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 12 |
G:=sub<GL(8,GF(13))| [10,0,0,0,0,0,0,0,0,10,5,6,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,0,10,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[0,0,1,0,0,0,0,0,10,5,2,10,0,0,0,0,1,0,0,0,0,0,0,0,0,8,11,8,0,0,0,0,0,0,0,0,1,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,11,1,12,1,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,0,10,5,2,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,12,0,12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,9,1,9,0,0,0,0,1,4,3,8,0,0,0,0,0,0,9,6,0,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,12,1,0,1,0,0,0,0,11,1,12,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12] >;
C6.502+ 1+4 in GAP, Magma, Sage, TeX
C_6._{50}2_+^{1+4} % in TeX
G:=Group("C6.50ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1182);
// by ID
G=gap.SmallGroup(192,1182);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,100,1123,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=a^3*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations