G = C6.672+ 1+4 order 192 = 26·3
67th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.672+ 1+4, C4⋊C4.201D6, (C2×D4).104D6, C12⋊3D4.8C2, C22⋊C4.31D6, (C22×C4).67D6, D6.D4⋊31C2, (C2×C6).208C24, (C2×C12).75C23, Dic3⋊4D4⋊25C2, Dic6⋊C4⋊33C2, C2.69(D4⋊6D6), D6⋊C4.133C22, (C6×D4).146C22, C22.D4⋊13S3, C23.42(C22×S3), (C22×C6).40C23, Dic3.38(C4○D4), C23.11D6⋊35C2, (C2×D12).159C22, C23.23D6⋊16C2, (C22×S3).89C23, C22.229(S3×C23), Dic3⋊C4.140C22, (C22×C12).370C22, C3⋊3(C22.53C24), (C2×Dic6).170C22, (C4×Dic3).127C22, (C2×Dic3).248C23, C6.D4.46C22, (C22×Dic3).134C22, (C4×C3⋊D4)⋊50C2, C2.70(S3×C4○D4), C6.182(C2×C4○D4), (S3×C2×C4).213C22, (C2×C4).70(C22×S3), (C3×C4⋊C4).181C22, (C2×C3⋊D4).52C22, (C3×C22.D4)⋊16C2, (C3×C22⋊C4).56C22, SmallGroup(192,1223)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.672+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, ebe=a3b, cd=dc, ce=ec, ede=a3b2d >
Subgroups: 608 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C4×Q8, C22.D4, C22.D4, C4.4D4, C4⋊1D4, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C22.53C24, Dic3⋊4D4, C23.11D6, Dic6⋊C4, D6.D4, C4×C3⋊D4, C23.23D6, C12⋊3D4, C3×C22.D4, C6.672+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, S3×C23, C22.53C24, D4⋊6D6, S3×C4○D4, C6.672+ 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 82 7 76)(2 83 8 77)(3 84 9 78)(4 79 10 73)(5 80 11 74)(6 81 12 75)(13 94 19 88)(14 95 20 89)(15 96 21 90)(16 91 22 85)(17 92 23 86)(18 93 24 87)(25 55 31 49)(26 56 32 50)(27 57 33 51)(28 58 34 52)(29 59 35 53)(30 60 36 54)(37 67 43 61)(38 68 44 62)(39 69 45 63)(40 70 46 64)(41 71 47 65)(42 72 48 66)
(1 52 4 49)(2 53 5 50)(3 54 6 51)(7 58 10 55)(8 59 11 56)(9 60 12 57)(13 64 16 61)(14 65 17 62)(15 66 18 63)(19 70 22 67)(20 71 23 68)(21 72 24 69)(25 76 28 73)(26 77 29 74)(27 78 30 75)(31 82 34 79)(32 83 35 80)(33 84 36 81)(37 88 40 85)(38 89 41 86)(39 90 42 87)(43 94 46 91)(44 95 47 92)(45 96 48 93)
(1 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 38)(49 70 58 61)(50 69 59 66)(51 68 60 65)(52 67 55 64)(53 72 56 63)(54 71 57 62)(73 94 82 85)(74 93 83 90)(75 92 84 89)(76 91 79 88)(77 96 80 87)(78 95 81 86)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 88)(74 89)(75 90)(76 85)(77 86)(78 87)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
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), (1,82,7,76)(2,83,8,77)(3,84,9,78)(4,79,10,73)(5,80,11,74)(6,81,12,75)(13,94,19,88)(14,95,20,89)(15,96,21,90)(16,91,22,85)(17,92,23,86)(18,93,24,87)(25,55,31,49)(26,56,32,50)(27,57,33,51)(28,58,34,52)(29,59,35,53)(30,60,36,54)(37,67,43,61)(38,68,44,62)(39,69,45,63)(40,70,46,64)(41,71,47,65)(42,72,48,66), (1,52,4,49)(2,53,5,50)(3,54,6,51)(7,58,10,55)(8,59,11,56)(9,60,12,57)(13,64,16,61)(14,65,17,62)(15,66,18,63)(19,70,22,67)(20,71,23,68)(21,72,24,69)(25,76,28,73)(26,77,29,74)(27,78,30,75)(31,82,34,79)(32,83,35,80)(33,84,36,81)(37,88,40,85)(38,89,41,86)(39,90,42,87)(43,94,46,91)(44,95,47,92)(45,96,48,93), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38)(49,70,58,61)(50,69,59,66)(51,68,60,65)(52,67,55,64)(53,72,56,63)(54,71,57,62)(73,94,82,85)(74,93,83,90)(75,92,84,89)(76,91,79,88)(77,96,80,87)(78,95,81,86), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,88)(74,89)(75,90)(76,85)(77,86)(78,87)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93)>;
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), (1,82,7,76)(2,83,8,77)(3,84,9,78)(4,79,10,73)(5,80,11,74)(6,81,12,75)(13,94,19,88)(14,95,20,89)(15,96,21,90)(16,91,22,85)(17,92,23,86)(18,93,24,87)(25,55,31,49)(26,56,32,50)(27,57,33,51)(28,58,34,52)(29,59,35,53)(30,60,36,54)(37,67,43,61)(38,68,44,62)(39,69,45,63)(40,70,46,64)(41,71,47,65)(42,72,48,66), (1,52,4,49)(2,53,5,50)(3,54,6,51)(7,58,10,55)(8,59,11,56)(9,60,12,57)(13,64,16,61)(14,65,17,62)(15,66,18,63)(19,70,22,67)(20,71,23,68)(21,72,24,69)(25,76,28,73)(26,77,29,74)(27,78,30,75)(31,82,34,79)(32,83,35,80)(33,84,36,81)(37,88,40,85)(38,89,41,86)(39,90,42,87)(43,94,46,91)(44,95,47,92)(45,96,48,93), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38)(49,70,58,61)(50,69,59,66)(51,68,60,65)(52,67,55,64)(53,72,56,63)(54,71,57,62)(73,94,82,85)(74,93,83,90)(75,92,84,89)(76,91,79,88)(77,96,80,87)(78,95,81,86), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,88)(74,89)(75,90)(76,85)(77,86)(78,87)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93) );
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)], [(1,82,7,76),(2,83,8,77),(3,84,9,78),(4,79,10,73),(5,80,11,74),(6,81,12,75),(13,94,19,88),(14,95,20,89),(15,96,21,90),(16,91,22,85),(17,92,23,86),(18,93,24,87),(25,55,31,49),(26,56,32,50),(27,57,33,51),(28,58,34,52),(29,59,35,53),(30,60,36,54),(37,67,43,61),(38,68,44,62),(39,69,45,63),(40,70,46,64),(41,71,47,65),(42,72,48,66)], [(1,52,4,49),(2,53,5,50),(3,54,6,51),(7,58,10,55),(8,59,11,56),(9,60,12,57),(13,64,16,61),(14,65,17,62),(15,66,18,63),(19,70,22,67),(20,71,23,68),(21,72,24,69),(25,76,28,73),(26,77,29,74),(27,78,30,75),(31,82,34,79),(32,83,35,80),(33,84,36,81),(37,88,40,85),(38,89,41,86),(39,90,42,87),(43,94,46,91),(44,95,47,92),(45,96,48,93)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(25,46,34,37),(26,45,35,42),(27,44,36,41),(28,43,31,40),(29,48,32,39),(30,47,33,38),(49,70,58,61),(50,69,59,66),(51,68,60,65),(52,67,55,64),(53,72,56,63),(54,71,57,62),(73,94,82,85),(74,93,83,90),(75,92,84,89),(76,91,79,88),(77,96,80,87),(78,95,81,86)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,88),(74,89),(75,90),(76,85),(77,86),(78,87),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | D4⋊6D6 | S3×C4○D4 |
| kernel | C6.672+ 1+4 | Dic3⋊4D4 | C23.11D6 | Dic6⋊C4 | D6.D4 | C4×C3⋊D4 | C23.23D6 | C12⋊3D4 | C3×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C6 | C2 | C2 |
| # reps | 1 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 8 | 1 | 2 | 4 |
Matrix representation of C6.672+ 1+4 ►in GL6(𝔽13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,8,0,0,0,0,10,8,0,0,0,0,0,0,8,0,0,0,0,0,3,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,8,0,0,0,0,0,8],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
C6.672+ 1+4 in GAP, Magma, Sage, TeX
C_6._{67}2_+^{1+4} % in TeX
G:=Group("C6.67ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1223);
// by ID
G=gap.SmallGroup(192,1223);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,1571,297,6278]);
// 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,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^3*b^2*d>;
// generators/relations