metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.42- 1+4, C6.92+ 1+4, (C2×C4)⋊4D12, (C2×C12)⋊9D4, C12⋊D4⋊9C2, C12⋊7D4⋊5C2, C4⋊C4.263D6, C4.D12⋊8C2, C4.69(C2×D12), C12.222(C2×D4), (C2×C6).54C24, D6⋊C4.1C22, C22.7(C2×D12), C6.11(C22×D4), C2.13(C22×D12), (C22×C4).195D6, C2.12(D4⋊6D6), (C2×C12).137C23, C4⋊Dic3.29C22, C22.88(S3×C23), (C2×D12).254C22, (C22×S3).13C23, C23.236(C22×S3), (C22×C12).75C22, (C22×C6).403C23, C2.7(Q8.15D6), (C2×Dic3).16C23, C3⋊1(C22.31C24), (C2×Dic6).282C22, (C6×C4⋊C4)⋊16C2, (C2×C4⋊C4)⋊19S3, (C2×C4○D12)⋊16C2, (C2×C6).176(C2×D4), (S3×C2×C4).55C22, (C3×C4⋊C4).296C22, (C2×C4).572(C22×S3), (C2×C3⋊D4).93C22, SmallGroup(192,1069)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.2+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, bab-1=dad-1=eae=a-1, ac=ca, cbc-1=a3b-1, dbd-1=a3b, be=eb, dcd-1=ece=a3c, ede=a3b2d >
Subgroups: 824 in 294 conjugacy classes, 111 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×C4○D4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C22.31C24, C12⋊D4, C4.D12, C12⋊7D4, C6×C4⋊C4, C2×C4○D12, C6.2+ 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, 2+ 1+4, 2- 1+4, C2×D12, S3×C23, C22.31C24, C22×D12, D4⋊6D6, Q8.15D6, C6.2+ 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 41 17 36)(2 40 18 35)(3 39 13 34)(4 38 14 33)(5 37 15 32)(6 42 16 31)(7 76 94 71)(8 75 95 70)(9 74 96 69)(10 73 91 68)(11 78 92 67)(12 77 93 72)(19 44 30 49)(20 43 25 54)(21 48 26 53)(22 47 27 52)(23 46 28 51)(24 45 29 50)(55 83 66 88)(56 82 61 87)(57 81 62 86)(58 80 63 85)(59 79 64 90)(60 84 65 89)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 82 10 79)(8 83 11 80)(9 84 12 81)(13 28 16 25)(14 29 17 26)(15 30 18 27)(31 46 34 43)(32 47 35 44)(33 48 36 45)(37 52 40 49)(38 53 41 50)(39 54 42 51)(55 75 58 78)(56 76 59 73)(57 77 60 74)(61 71 64 68)(62 72 65 69)(63 67 66 70)(85 95 88 92)(86 96 89 93)(87 91 90 94)
(1 9 17 96)(2 8 18 95)(3 7 13 94)(4 12 14 93)(5 11 15 92)(6 10 16 91)(19 83 30 88)(20 82 25 87)(21 81 26 86)(22 80 27 85)(23 79 28 90)(24 84 29 89)(31 71 42 76)(32 70 37 75)(33 69 38 74)(34 68 39 73)(35 67 40 78)(36 72 41 77)(43 64 54 59)(44 63 49 58)(45 62 50 57)(46 61 51 56)(47 66 52 55)(48 65 53 60)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 66)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(13 58)(14 57)(15 56)(16 55)(17 60)(18 59)(19 68)(20 67)(21 72)(22 71)(23 70)(24 69)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 83)(32 82)(33 81)(34 80)(35 79)(36 84)(37 87)(38 86)(39 85)(40 90)(41 89)(42 88)(49 91)(50 96)(51 95)(52 94)(53 93)(54 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), (1,41,17,36)(2,40,18,35)(3,39,13,34)(4,38,14,33)(5,37,15,32)(6,42,16,31)(7,76,94,71)(8,75,95,70)(9,74,96,69)(10,73,91,68)(11,78,92,67)(12,77,93,72)(19,44,30,49)(20,43,25,54)(21,48,26,53)(22,47,27,52)(23,46,28,51)(24,45,29,50)(55,83,66,88)(56,82,61,87)(57,81,62,86)(58,80,63,85)(59,79,64,90)(60,84,65,89), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,82,10,79)(8,83,11,80)(9,84,12,81)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,46,34,43)(32,47,35,44)(33,48,36,45)(37,52,40,49)(38,53,41,50)(39,54,42,51)(55,75,58,78)(56,76,59,73)(57,77,60,74)(61,71,64,68)(62,72,65,69)(63,67,66,70)(85,95,88,92)(86,96,89,93)(87,91,90,94), (1,9,17,96)(2,8,18,95)(3,7,13,94)(4,12,14,93)(5,11,15,92)(6,10,16,91)(19,83,30,88)(20,82,25,87)(21,81,26,86)(22,80,27,85)(23,79,28,90)(24,84,29,89)(31,71,42,76)(32,70,37,75)(33,69,38,74)(34,68,39,73)(35,67,40,78)(36,72,41,77)(43,64,54,59)(44,63,49,58)(45,62,50,57)(46,61,51,56)(47,66,52,55)(48,65,53,60), (1,65)(2,64)(3,63)(4,62)(5,61)(6,66)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(13,58)(14,57)(15,56)(16,55)(17,60)(18,59)(19,68)(20,67)(21,72)(22,71)(23,70)(24,69)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,83)(32,82)(33,81)(34,80)(35,79)(36,84)(37,87)(38,86)(39,85)(40,90)(41,89)(42,88)(49,91)(50,96)(51,95)(52,94)(53,93)(54,92)>;
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), (1,41,17,36)(2,40,18,35)(3,39,13,34)(4,38,14,33)(5,37,15,32)(6,42,16,31)(7,76,94,71)(8,75,95,70)(9,74,96,69)(10,73,91,68)(11,78,92,67)(12,77,93,72)(19,44,30,49)(20,43,25,54)(21,48,26,53)(22,47,27,52)(23,46,28,51)(24,45,29,50)(55,83,66,88)(56,82,61,87)(57,81,62,86)(58,80,63,85)(59,79,64,90)(60,84,65,89), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,82,10,79)(8,83,11,80)(9,84,12,81)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,46,34,43)(32,47,35,44)(33,48,36,45)(37,52,40,49)(38,53,41,50)(39,54,42,51)(55,75,58,78)(56,76,59,73)(57,77,60,74)(61,71,64,68)(62,72,65,69)(63,67,66,70)(85,95,88,92)(86,96,89,93)(87,91,90,94), (1,9,17,96)(2,8,18,95)(3,7,13,94)(4,12,14,93)(5,11,15,92)(6,10,16,91)(19,83,30,88)(20,82,25,87)(21,81,26,86)(22,80,27,85)(23,79,28,90)(24,84,29,89)(31,71,42,76)(32,70,37,75)(33,69,38,74)(34,68,39,73)(35,67,40,78)(36,72,41,77)(43,64,54,59)(44,63,49,58)(45,62,50,57)(46,61,51,56)(47,66,52,55)(48,65,53,60), (1,65)(2,64)(3,63)(4,62)(5,61)(6,66)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(13,58)(14,57)(15,56)(16,55)(17,60)(18,59)(19,68)(20,67)(21,72)(22,71)(23,70)(24,69)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,83)(32,82)(33,81)(34,80)(35,79)(36,84)(37,87)(38,86)(39,85)(40,90)(41,89)(42,88)(49,91)(50,96)(51,95)(52,94)(53,93)(54,92) );
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)], [(1,41,17,36),(2,40,18,35),(3,39,13,34),(4,38,14,33),(5,37,15,32),(6,42,16,31),(7,76,94,71),(8,75,95,70),(9,74,96,69),(10,73,91,68),(11,78,92,67),(12,77,93,72),(19,44,30,49),(20,43,25,54),(21,48,26,53),(22,47,27,52),(23,46,28,51),(24,45,29,50),(55,83,66,88),(56,82,61,87),(57,81,62,86),(58,80,63,85),(59,79,64,90),(60,84,65,89)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(7,82,10,79),(8,83,11,80),(9,84,12,81),(13,28,16,25),(14,29,17,26),(15,30,18,27),(31,46,34,43),(32,47,35,44),(33,48,36,45),(37,52,40,49),(38,53,41,50),(39,54,42,51),(55,75,58,78),(56,76,59,73),(57,77,60,74),(61,71,64,68),(62,72,65,69),(63,67,66,70),(85,95,88,92),(86,96,89,93),(87,91,90,94)], [(1,9,17,96),(2,8,18,95),(3,7,13,94),(4,12,14,93),(5,11,15,92),(6,10,16,91),(19,83,30,88),(20,82,25,87),(21,81,26,86),(22,80,27,85),(23,79,28,90),(24,84,29,89),(31,71,42,76),(32,70,37,75),(33,69,38,74),(34,68,39,73),(35,67,40,78),(36,72,41,77),(43,64,54,59),(44,63,49,58),(45,62,50,57),(46,61,51,56),(47,66,52,55),(48,65,53,60)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,66),(7,47),(8,46),(9,45),(10,44),(11,43),(12,48),(13,58),(14,57),(15,56),(16,55),(17,60),(18,59),(19,68),(20,67),(21,72),(22,71),(23,70),(24,69),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,83),(32,82),(33,81),(34,80),(35,79),(36,84),(37,87),(38,86),(39,85),(40,90),(41,89),(42,88),(49,91),(50,96),(51,95),(52,94),(53,93),(54,92)]])
42 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 | ··· | 6G | 12A | ··· | 12L |
| 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 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D12 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | Q8.15D6 |
| kernel | C6.2+ 1+4 | C12⋊D4 | C4.D12 | C12⋊7D4 | C6×C4⋊C4 | C2×C4○D12 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 3 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C6.2+ 1+4 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 4 | 10 | 3 |
| 0 | 0 | 0 | 4 | 0 | 3 |
| 0 | 0 | 10 | 3 | 4 | 9 |
| 0 | 0 | 0 | 3 | 0 | 9 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 5 | 5 |
| 0 | 0 | 12 | 7 | 10 | 8 |
| 0 | 0 | 5 | 5 | 7 | 7 |
| 0 | 0 | 10 | 8 | 1 | 6 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,11,9,0,0,0,0,11,2,0,0,0,0,0,0,11,9,0,0,0,0,11,2],[10,7,0,0,0,0,6,3,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,9,0,10,0,0,0,4,4,3,3,0,0,10,0,4,0,0,0,3,3,9,9],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,6,12,5,10,0,0,6,7,5,8,0,0,5,10,7,1,0,0,5,8,7,6] >;
C6.2+ 1+4 in GAP, Magma, Sage, TeX
C_6.2_+^{1+4} % in TeX
G:=Group("C6.ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1069);
// by ID
G=gap.SmallGroup(192,1069);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
// generators/relations