G = C6.1462+ 1+4 order 192 = 26·3
55th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.1462+ 1+4, (C2×D4)⋊45D6, (C2×C12)⋊16D4, (C2×Q8)⋊37D6, (C22×C4)⋊34D6, C23⋊2D6⋊32C2, C12⋊3D4⋊30C2, C12⋊7D4⋊48C2, D6⋊C4⋊38C22, C12.430(C2×D4), (C6×D4)⋊48C22, (C6×Q8)⋊41C22, C2.70(D4○D12), (C22×D12)⋊21C2, (C2×C6).316C24, C4⋊Dic3⋊66C22, C6.166(C22×D4), C12.23D4⋊32C2, (C2×C12).653C23, (C22×C12)⋊32C22, C3⋊7(C22.29C24), (C4×Dic3)⋊45C22, (C2×D12).281C22, C23.26D6⋊39C2, (S3×C23).80C22, C23.222(C22×S3), (C22×C6).242C23, C22.325(S3×C23), (C22×S3).138C23, (C2×Dic3).163C23, C6.D4.136C22, (C6×C4○D4)⋊8C2, (C2×C4○D4)⋊12S3, (C2×C4)⋊7(C3⋊D4), (C2×C6).82(C2×D4), C4.33(C2×C3⋊D4), (C2×C3⋊D4)⋊31C22, C2.39(C22×C3⋊D4), C22.24(C2×C3⋊D4), (C2×C4).251(C22×S3), SmallGroup(192,1389)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.1462+ 1+4
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=a3b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a3b-1, dbd-1=a3b, be=eb, cd=dc, ece=a3c, ede=a3b2d >
Subgroups: 1032 in 334 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22×C6, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C2×D12, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, S3×C23, C22.29C24, C23.26D6, C12⋊7D4, C23⋊2D6, C12⋊3D4, C12.23D4, C22×D12, C6×C4○D4, C6.1462+ 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, C22.29C24, D4○D12, C22×C3⋊D4, C6.1462+ 1+4
►On 48 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)
(1 12 17 43)(2 7 18 44)(3 8 13 45)(4 9 14 46)(5 10 15 47)(6 11 16 48)(19 41 26 34)(20 42 27 35)(21 37 28 36)(22 38 29 31)(23 39 30 32)(24 40 25 33)
(1 43 4 46)(2 48 5 45)(3 47 6 44)(7 13 10 16)(8 18 11 15)(9 17 12 14)(19 31 22 34)(20 36 23 33)(21 35 24 32)(25 39 28 42)(26 38 29 41)(27 37 30 40)
(1 29 14 19)(2 28 15 24)(3 27 16 23)(4 26 17 22)(5 25 18 21)(6 30 13 20)(7 33 47 37)(8 32 48 42)(9 31 43 41)(10 36 44 40)(11 35 45 39)(12 34 46 38)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 28)(14 29)(15 30)(16 25)(17 26)(18 27)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
G:=sub<Sym(48)| (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), (1,12,17,43)(2,7,18,44)(3,8,13,45)(4,9,14,46)(5,10,15,47)(6,11,16,48)(19,41,26,34)(20,42,27,35)(21,37,28,36)(22,38,29,31)(23,39,30,32)(24,40,25,33), (1,43,4,46)(2,48,5,45)(3,47,6,44)(7,13,10,16)(8,18,11,15)(9,17,12,14)(19,31,22,34)(20,36,23,33)(21,35,24,32)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,29,14,19)(2,28,15,24)(3,27,16,23)(4,26,17,22)(5,25,18,21)(6,30,13,20)(7,33,47,37)(8,32,48,42)(9,31,43,41)(10,36,44,40)(11,35,45,39)(12,34,46,38), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)>;
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), (1,12,17,43)(2,7,18,44)(3,8,13,45)(4,9,14,46)(5,10,15,47)(6,11,16,48)(19,41,26,34)(20,42,27,35)(21,37,28,36)(22,38,29,31)(23,39,30,32)(24,40,25,33), (1,43,4,46)(2,48,5,45)(3,47,6,44)(7,13,10,16)(8,18,11,15)(9,17,12,14)(19,31,22,34)(20,36,23,33)(21,35,24,32)(25,39,28,42)(26,38,29,41)(27,37,30,40), (1,29,14,19)(2,28,15,24)(3,27,16,23)(4,26,17,22)(5,25,18,21)(6,30,13,20)(7,33,47,37)(8,32,48,42)(9,31,43,41)(10,36,44,40)(11,35,45,39)(12,34,46,38), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,28)(14,29)(15,30)(16,25)(17,26)(18,27)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45) );
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)], [(1,12,17,43),(2,7,18,44),(3,8,13,45),(4,9,14,46),(5,10,15,47),(6,11,16,48),(19,41,26,34),(20,42,27,35),(21,37,28,36),(22,38,29,31),(23,39,30,32),(24,40,25,33)], [(1,43,4,46),(2,48,5,45),(3,47,6,44),(7,13,10,16),(8,18,11,15),(9,17,12,14),(19,31,22,34),(20,36,23,33),(21,35,24,32),(25,39,28,42),(26,38,29,41),(27,37,30,40)], [(1,29,14,19),(2,28,15,24),(3,27,16,23),(4,26,17,22),(5,25,18,21),(6,30,13,20),(7,33,47,37),(8,32,48,42),(9,31,43,41),(10,36,44,40),(11,35,45,39),(12,34,46,38)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,42),(8,37),(9,38),(10,39),(11,40),(12,41),(13,28),(14,29),(15,30),(16,25),(17,26),(18,27),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | ··· | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 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 | D4○D12 |
| kernel | C6.1462+ 1+4 | C23.26D6 | C12⋊7D4 | C23⋊2D6 | C12⋊3D4 | C12.23D4 | C22×D12 | C6×C4○D4 | C2×C4○D4 | C2×C12 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 8 | 2 | 4 |
Matrix representation of C6.1462+ 1+4 ►in GL6(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 6 | 1 |
| 0 | 0 | 7 | 3 | 12 | 7 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 2 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 10 | 1 | 6 |
| 0 | 0 | 3 | 7 | 7 | 12 |
| 0 | 0 | 0 | 0 | 7 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 2 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 2 |
| 0 | 0 | 12 | 0 | 2 | 0 |
| 0 | 0 | 0 | 12 | 0 | 1 |
| 0 | 0 | 12 | 0 | 1 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 9 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 | 0 |
| 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,6,12,3,6,0,0,1,7,7,10],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,6,3,0,0,0,0,10,7,0,0,0,0,1,7,7,10,0,0,6,12,3,6],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,0,12,0,12,0,0,12,0,12,0,0,0,0,2,0,1,0,0,2,0,1,0],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12] >;
C6.1462+ 1+4 in GAP, Magma, Sage, TeX
C_6._{146}2_+^{1+4} % in TeX
G:=Group("C6.146ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1389);
// by ID
G=gap.SmallGroup(192,1389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,6278]);
// 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,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^3*b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
// generators/relations