G = C6.1052- 1+4 order 192 = 26·3
60th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1052- 1+4, (C2×D4).235D6, C12.429(C2×D4), (C2×C12).220D4, (C2×Q8).217D6, (C2×C6).311C24, C2.69(Q8○D12), (C22×C4).301D6, C6.163(C22×D4), Dic3⋊Q8⋊32C2, C23.12D6⋊30C2, C12.48D4⋊48C2, (C2×C12).650C23, (C22×Dic6)⋊22C2, (C6×D4).314C22, (C6×Q8).240C22, C23.23D6⋊32C2, C23.26D6⋊35C2, Dic3⋊C4.92C22, C4⋊Dic3.320C22, (C22×C6).237C23, C22.322(S3×C23), C23.218(C22×S3), (C22×C12).320C22, C3⋊7(C23.38C23), (C2×Dic6).310C22, (C4×Dic3).174C22, (C2×Dic3).161C23, C6.D4.133C22, (C22×Dic3).166C22, (C2×C6).79(C2×D4), C4.32(C2×C3⋊D4), (C2×C4○D4).17S3, (C6×C4○D4).12C2, (C2×C4).97(C3⋊D4), C2.36(C22×C3⋊D4), C22.22(C2×C3⋊D4), (C2×C4).249(C22×S3), SmallGroup(192,1384)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.1052- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=b2, e2=a3b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a3b-1, dbd-1=ebe-1=a3b, dcd-1=a3c, ce=ec, ede-1=a3b2d >
Subgroups: 584 in 270 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C23.38C23, C12.48D4, C23.26D6, C23.23D6, C23.12D6, Dic3⋊Q8, C22×Dic6, C6×C4○D4, C6.1052- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, 2- 1+4, C2×C3⋊D4, S3×C23, C23.38C23, Q8○D12, C22×C3⋊D4, C6.1052- 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 76 17 68)(2 75 18 67)(3 74 13 72)(4 73 14 71)(5 78 15 70)(6 77 16 69)(7 41 92 34)(8 40 93 33)(9 39 94 32)(10 38 95 31)(11 37 96 36)(12 42 91 35)(19 59 26 61)(20 58 27 66)(21 57 28 65)(22 56 29 64)(23 55 30 63)(24 60 25 62)(43 83 51 85)(44 82 52 90)(45 81 53 89)(46 80 54 88)(47 79 49 87)(48 84 50 86)
(1 80 4 83)(2 79 5 82)(3 84 6 81)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 86 16 89)(14 85 17 88)(15 90 18 87)(19 95 22 92)(20 94 23 91)(21 93 24 96)(31 61 34 64)(32 66 35 63)(33 65 36 62)(37 60 40 57)(38 59 41 56)(39 58 42 55)(43 73 46 76)(44 78 47 75)(45 77 48 74)(49 67 52 70)(50 72 53 69)(51 71 54 68)
(1 31 17 38)(2 32 18 39)(3 33 13 40)(4 34 14 41)(5 35 15 42)(6 36 16 37)(7 68 92 76)(8 69 93 77)(9 70 94 78)(10 71 95 73)(11 72 96 74)(12 67 91 75)(19 46 26 54)(20 47 27 49)(21 48 28 50)(22 43 29 51)(23 44 30 52)(24 45 25 53)(55 79 63 87)(56 80 64 88)(57 81 65 89)(58 82 66 90)(59 83 61 85)(60 84 62 86)
(1 76 14 71)(2 75 15 70)(3 74 16 69)(4 73 17 68)(5 78 18 67)(6 77 13 72)(7 31 95 41)(8 36 96 40)(9 35 91 39)(10 34 92 38)(11 33 93 37)(12 32 94 42)(19 59 29 64)(20 58 30 63)(21 57 25 62)(22 56 26 61)(23 55 27 66)(24 60 28 65)(43 85 54 80)(44 90 49 79)(45 89 50 84)(46 88 51 83)(47 87 52 82)(48 86 53 81)
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), (1,76,17,68)(2,75,18,67)(3,74,13,72)(4,73,14,71)(5,78,15,70)(6,77,16,69)(7,41,92,34)(8,40,93,33)(9,39,94,32)(10,38,95,31)(11,37,96,36)(12,42,91,35)(19,59,26,61)(20,58,27,66)(21,57,28,65)(22,56,29,64)(23,55,30,63)(24,60,25,62)(43,83,51,85)(44,82,52,90)(45,81,53,89)(46,80,54,88)(47,79,49,87)(48,84,50,86), (1,80,4,83)(2,79,5,82)(3,84,6,81)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,86,16,89)(14,85,17,88)(15,90,18,87)(19,95,22,92)(20,94,23,91)(21,93,24,96)(31,61,34,64)(32,66,35,63)(33,65,36,62)(37,60,40,57)(38,59,41,56)(39,58,42,55)(43,73,46,76)(44,78,47,75)(45,77,48,74)(49,67,52,70)(50,72,53,69)(51,71,54,68), (1,31,17,38)(2,32,18,39)(3,33,13,40)(4,34,14,41)(5,35,15,42)(6,36,16,37)(7,68,92,76)(8,69,93,77)(9,70,94,78)(10,71,95,73)(11,72,96,74)(12,67,91,75)(19,46,26,54)(20,47,27,49)(21,48,28,50)(22,43,29,51)(23,44,30,52)(24,45,25,53)(55,79,63,87)(56,80,64,88)(57,81,65,89)(58,82,66,90)(59,83,61,85)(60,84,62,86), (1,76,14,71)(2,75,15,70)(3,74,16,69)(4,73,17,68)(5,78,18,67)(6,77,13,72)(7,31,95,41)(8,36,96,40)(9,35,91,39)(10,34,92,38)(11,33,93,37)(12,32,94,42)(19,59,29,64)(20,58,30,63)(21,57,25,62)(22,56,26,61)(23,55,27,66)(24,60,28,65)(43,85,54,80)(44,90,49,79)(45,89,50,84)(46,88,51,83)(47,87,52,82)(48,86,53,81)>;
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), (1,76,17,68)(2,75,18,67)(3,74,13,72)(4,73,14,71)(5,78,15,70)(6,77,16,69)(7,41,92,34)(8,40,93,33)(9,39,94,32)(10,38,95,31)(11,37,96,36)(12,42,91,35)(19,59,26,61)(20,58,27,66)(21,57,28,65)(22,56,29,64)(23,55,30,63)(24,60,25,62)(43,83,51,85)(44,82,52,90)(45,81,53,89)(46,80,54,88)(47,79,49,87)(48,84,50,86), (1,80,4,83)(2,79,5,82)(3,84,6,81)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,86,16,89)(14,85,17,88)(15,90,18,87)(19,95,22,92)(20,94,23,91)(21,93,24,96)(31,61,34,64)(32,66,35,63)(33,65,36,62)(37,60,40,57)(38,59,41,56)(39,58,42,55)(43,73,46,76)(44,78,47,75)(45,77,48,74)(49,67,52,70)(50,72,53,69)(51,71,54,68), (1,31,17,38)(2,32,18,39)(3,33,13,40)(4,34,14,41)(5,35,15,42)(6,36,16,37)(7,68,92,76)(8,69,93,77)(9,70,94,78)(10,71,95,73)(11,72,96,74)(12,67,91,75)(19,46,26,54)(20,47,27,49)(21,48,28,50)(22,43,29,51)(23,44,30,52)(24,45,25,53)(55,79,63,87)(56,80,64,88)(57,81,65,89)(58,82,66,90)(59,83,61,85)(60,84,62,86), (1,76,14,71)(2,75,15,70)(3,74,16,69)(4,73,17,68)(5,78,18,67)(6,77,13,72)(7,31,95,41)(8,36,96,40)(9,35,91,39)(10,34,92,38)(11,33,93,37)(12,32,94,42)(19,59,29,64)(20,58,30,63)(21,57,25,62)(22,56,26,61)(23,55,27,66)(24,60,28,65)(43,85,54,80)(44,90,49,79)(45,89,50,84)(46,88,51,83)(47,87,52,82)(48,86,53,81) );
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)], [(1,76,17,68),(2,75,18,67),(3,74,13,72),(4,73,14,71),(5,78,15,70),(6,77,16,69),(7,41,92,34),(8,40,93,33),(9,39,94,32),(10,38,95,31),(11,37,96,36),(12,42,91,35),(19,59,26,61),(20,58,27,66),(21,57,28,65),(22,56,29,64),(23,55,30,63),(24,60,25,62),(43,83,51,85),(44,82,52,90),(45,81,53,89),(46,80,54,88),(47,79,49,87),(48,84,50,86)], [(1,80,4,83),(2,79,5,82),(3,84,6,81),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,86,16,89),(14,85,17,88),(15,90,18,87),(19,95,22,92),(20,94,23,91),(21,93,24,96),(31,61,34,64),(32,66,35,63),(33,65,36,62),(37,60,40,57),(38,59,41,56),(39,58,42,55),(43,73,46,76),(44,78,47,75),(45,77,48,74),(49,67,52,70),(50,72,53,69),(51,71,54,68)], [(1,31,17,38),(2,32,18,39),(3,33,13,40),(4,34,14,41),(5,35,15,42),(6,36,16,37),(7,68,92,76),(8,69,93,77),(9,70,94,78),(10,71,95,73),(11,72,96,74),(12,67,91,75),(19,46,26,54),(20,47,27,49),(21,48,28,50),(22,43,29,51),(23,44,30,52),(24,45,25,53),(55,79,63,87),(56,80,64,88),(57,81,65,89),(58,82,66,90),(59,83,61,85),(60,84,62,86)], [(1,76,14,71),(2,75,15,70),(3,74,16,69),(4,73,17,68),(5,78,18,67),(6,77,13,72),(7,31,95,41),(8,36,96,40),(9,35,91,39),(10,34,92,38),(11,33,93,37),(12,32,94,42),(19,59,29,64),(20,58,30,63),(21,57,25,62),(22,56,26,61),(23,55,27,66),(24,60,28,65),(43,85,54,80),(44,90,49,79),(45,89,50,84),(46,88,51,83),(47,87,52,82),(48,86,53,81)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C3⋊D4 | 2- 1+4 | Q8○D12 |
| kernel | C6.1052- 1+4 | C12.48D4 | C23.26D6 | C23.23D6 | C23.12D6 | Dic3⋊Q8 | C22×Dic6 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C6 | C2 |
| # reps | 1 | 4 | 1 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 8 | 2 | 4 |
Matrix representation of C6.1052- 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 4 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 7 | 12 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 |
| 0 | 0 | 7 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 7 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 4 | 5 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 4 | 5 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,8,0,0,0,0,0,0,8,0,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,9,12,0,0,12,6,0,0,0,0,4,1,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,7,0,0,0,0,9,12,0,0,0,0,0,0,1,7,0,0,0,0,9,12],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,4,0,0,0,0,0,5,0,0,8,4,0,0,0,0,0,5,0,0] >;
C6.1052- 1+4 in GAP, Magma, Sage, TeX
C_6._{105}2_-^{1+4} % in TeX
G:=Group("C6.105ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1384);
// by ID
G=gap.SmallGroup(192,1384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,675,570,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=a^3,d^2=b^2,e^2=a^3*b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b^-1,d*b*d^-1=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