G = C6.1072- 1+4 order 192 = 26·3
62nd non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1072- 1+4, (C3×Q8)⋊18D4, C3⋊7(Q8⋊5D4), D6⋊16(C4○D4), D6⋊3D4⋊42C2, (Q8×Dic3)⋊29C2, Q8⋊10(C3⋊D4), (C2×D4).236D6, C12.267(C2×D4), (C2×Q8).234D6, (C2×C6).317C24, C2.70(Q8○D12), (C22×C4).303D6, C6.167(C22×D4), C12.48D4⋊40C2, C23.12D6⋊31C2, (C2×C12).561C23, D6⋊C4.164C22, (C6×D4).276C22, (C6×Q8).243C22, C4⋊Dic3.260C22, C23.148(C22×S3), (C22×C6).243C23, C22.326(S3×C23), Dic3⋊C4.101C22, (C22×S3).244C23, (C22×C12).296C22, (C2×Dic3).164C23, (C2×Dic6).204C22, (C4×Dic3).176C22, C6.D4.76C22, (C2×S3×Q8)⋊19C2, (C6×C4○D4)⋊9C2, (C2×C4○D4)⋊13S3, (C4×C3⋊D4)⋊30C2, C4.73(C2×C3⋊D4), C6.217(C2×C4○D4), C2.105(S3×C4○D4), (S3×C2×C4).169C22, C2.40(C22×C3⋊D4), (C2×C4).640(C22×S3), (C2×C3⋊D4).141C22, SmallGroup(192,1390)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.1072- 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=a3b, be=eb, dcd-1=a3c, ce=ec, ede-1=a3b2d >
Subgroups: 664 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, S3×Q8, C2×C3⋊D4, C22×C12, C6×D4, C6×Q8, C3×C4○D4, Q8⋊5D4, C12.48D4, C4×C3⋊D4, C23.12D6, D6⋊3D4, Q8×Dic3, C2×S3×Q8, C6×C4○D4, C6.1072- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, C2×C3⋊D4, S3×C23, Q8⋊5D4, S3×C4○D4, Q8○D12, C22×C3⋊D4, C6.1072- 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 74 15 67)(2 73 16 72)(3 78 17 71)(4 77 18 70)(5 76 13 69)(6 75 14 68)(7 36 92 37)(8 35 93 42)(9 34 94 41)(10 33 95 40)(11 32 96 39)(12 31 91 38)(19 58 26 65)(20 57 27 64)(21 56 28 63)(22 55 29 62)(23 60 30 61)(24 59 25 66)(43 82 50 89)(44 81 51 88)(45 80 52 87)(46 79 53 86)(47 84 54 85)(48 83 49 90)
(1 70)(2 71)(3 72)(4 67)(5 68)(6 69)(7 35)(8 36)(9 31)(10 32)(11 33)(12 34)(13 75)(14 76)(15 77)(16 78)(17 73)(18 74)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(37 93)(38 94)(39 95)(40 96)(41 91)(42 92)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)
(1 38 18 34)(2 37 13 33)(3 42 14 32)(4 41 15 31)(5 40 16 36)(6 39 17 35)(7 72 95 76)(8 71 96 75)(9 70 91 74)(10 69 92 73)(11 68 93 78)(12 67 94 77)(19 50 29 46)(20 49 30 45)(21 54 25 44)(22 53 26 43)(23 52 27 48)(24 51 28 47)(55 89 65 79)(56 88 66 84)(57 87 61 83)(58 86 62 82)(59 85 63 81)(60 90 64 80)
(1 58 18 62)(2 59 13 63)(3 60 14 64)(4 55 15 65)(5 56 16 66)(6 57 17 61)(7 51 95 47)(8 52 96 48)(9 53 91 43)(10 54 92 44)(11 49 93 45)(12 50 94 46)(19 77 29 67)(20 78 30 68)(21 73 25 69)(22 74 26 70)(23 75 27 71)(24 76 28 72)(31 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)
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), (1,74,15,67)(2,73,16,72)(3,78,17,71)(4,77,18,70)(5,76,13,69)(6,75,14,68)(7,36,92,37)(8,35,93,42)(9,34,94,41)(10,33,95,40)(11,32,96,39)(12,31,91,38)(19,58,26,65)(20,57,27,64)(21,56,28,63)(22,55,29,62)(23,60,30,61)(24,59,25,66)(43,82,50,89)(44,81,51,88)(45,80,52,87)(46,79,53,86)(47,84,54,85)(48,83,49,90), (1,70)(2,71)(3,72)(4,67)(5,68)(6,69)(7,35)(8,36)(9,31)(10,32)(11,33)(12,34)(13,75)(14,76)(15,77)(16,78)(17,73)(18,74)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(37,93)(38,94)(39,95)(40,96)(41,91)(42,92)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90), (1,38,18,34)(2,37,13,33)(3,42,14,32)(4,41,15,31)(5,40,16,36)(6,39,17,35)(7,72,95,76)(8,71,96,75)(9,70,91,74)(10,69,92,73)(11,68,93,78)(12,67,94,77)(19,50,29,46)(20,49,30,45)(21,54,25,44)(22,53,26,43)(23,52,27,48)(24,51,28,47)(55,89,65,79)(56,88,66,84)(57,87,61,83)(58,86,62,82)(59,85,63,81)(60,90,64,80), (1,58,18,62)(2,59,13,63)(3,60,14,64)(4,55,15,65)(5,56,16,66)(6,57,17,61)(7,51,95,47)(8,52,96,48)(9,53,91,43)(10,54,92,44)(11,49,93,45)(12,50,94,46)(19,77,29,67)(20,78,30,68)(21,73,25,69)(22,74,26,70)(23,75,27,71)(24,76,28,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)>;
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), (1,74,15,67)(2,73,16,72)(3,78,17,71)(4,77,18,70)(5,76,13,69)(6,75,14,68)(7,36,92,37)(8,35,93,42)(9,34,94,41)(10,33,95,40)(11,32,96,39)(12,31,91,38)(19,58,26,65)(20,57,27,64)(21,56,28,63)(22,55,29,62)(23,60,30,61)(24,59,25,66)(43,82,50,89)(44,81,51,88)(45,80,52,87)(46,79,53,86)(47,84,54,85)(48,83,49,90), (1,70)(2,71)(3,72)(4,67)(5,68)(6,69)(7,35)(8,36)(9,31)(10,32)(11,33)(12,34)(13,75)(14,76)(15,77)(16,78)(17,73)(18,74)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(37,93)(38,94)(39,95)(40,96)(41,91)(42,92)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90), (1,38,18,34)(2,37,13,33)(3,42,14,32)(4,41,15,31)(5,40,16,36)(6,39,17,35)(7,72,95,76)(8,71,96,75)(9,70,91,74)(10,69,92,73)(11,68,93,78)(12,67,94,77)(19,50,29,46)(20,49,30,45)(21,54,25,44)(22,53,26,43)(23,52,27,48)(24,51,28,47)(55,89,65,79)(56,88,66,84)(57,87,61,83)(58,86,62,82)(59,85,63,81)(60,90,64,80), (1,58,18,62)(2,59,13,63)(3,60,14,64)(4,55,15,65)(5,56,16,66)(6,57,17,61)(7,51,95,47)(8,52,96,48)(9,53,91,43)(10,54,92,44)(11,49,93,45)(12,50,94,46)(19,77,29,67)(20,78,30,68)(21,73,25,69)(22,74,26,70)(23,75,27,71)(24,76,28,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84) );
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)], [(1,74,15,67),(2,73,16,72),(3,78,17,71),(4,77,18,70),(5,76,13,69),(6,75,14,68),(7,36,92,37),(8,35,93,42),(9,34,94,41),(10,33,95,40),(11,32,96,39),(12,31,91,38),(19,58,26,65),(20,57,27,64),(21,56,28,63),(22,55,29,62),(23,60,30,61),(24,59,25,66),(43,82,50,89),(44,81,51,88),(45,80,52,87),(46,79,53,86),(47,84,54,85),(48,83,49,90)], [(1,70),(2,71),(3,72),(4,67),(5,68),(6,69),(7,35),(8,36),(9,31),(10,32),(11,33),(12,34),(13,75),(14,76),(15,77),(16,78),(17,73),(18,74),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(37,93),(38,94),(39,95),(40,96),(41,91),(42,92),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90)], [(1,38,18,34),(2,37,13,33),(3,42,14,32),(4,41,15,31),(5,40,16,36),(6,39,17,35),(7,72,95,76),(8,71,96,75),(9,70,91,74),(10,69,92,73),(11,68,93,78),(12,67,94,77),(19,50,29,46),(20,49,30,45),(21,54,25,44),(22,53,26,43),(23,52,27,48),(24,51,28,47),(55,89,65,79),(56,88,66,84),(57,87,61,83),(58,86,62,82),(59,85,63,81),(60,90,64,80)], [(1,58,18,62),(2,59,13,63),(3,60,14,64),(4,55,15,65),(5,56,16,66),(6,57,17,61),(7,51,95,47),(8,52,96,48),(9,53,91,43),(10,54,92,44),(11,49,93,45),(12,50,94,46),(19,77,29,67),(20,78,30,68),(21,73,25,69),(22,74,26,70),(23,75,27,71),(24,76,28,72),(31,89,41,79),(32,90,42,80),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | ··· | 2 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C6.1072- 1+4 | C12.48D4 | C4×C3⋊D4 | C23.12D6 | D6⋊3D4 | Q8×Dic3 | C2×S3×Q8 | C6×C4○D4 | C2×C4○D4 | C3×Q8 | C22×C4 | C2×D4 | C2×Q8 | D6 | Q8 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C6.1072- 1+4 ►in GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 6 | 2 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 9 |
| 0 | 0 | 0 | 0 | 9 | 3 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,11,6,0,0,0,0,6,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,11,6,0,0,0,0,6,2],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,4,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,9,0,0,0,0,9,3] >;
C6.1072- 1+4 in GAP, Magma, Sage, TeX
C_6._{107}2_-^{1+4} % in TeX
G:=Group("C6.107ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1390);
// by ID
G=gap.SmallGroup(192,1390);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,136,6278]);
// 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=a^3*b,b*e=e*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=a^3*b^2*d>;
// generators/relations