?
G = C6.462+ (1+4) order 192 = 26·3
46th non-split extension by C6 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.462+ (1+4), C4⋊C4⋊8D6, (C2×D4)⋊10D6, C4⋊D4⋊20S3, C22⋊C4⋊12D6, C23⋊2D6⋊13C2, (D4×Dic3)⋊24C2, (C6×D4)⋊31C22, C2.29(D4○D12), (C2×C6).161C24, C4⋊Dic3⋊12C22, (C2×Dic6)⋊8C22, (C22×C4).244D6, C23.14D6⋊32C2, C12.48D4⋊44C2, C2.48(D4⋊6D6), (C2×C12).626C23, Dic3⋊C4⋊34C22, D6⋊C4.146C22, C3⋊5(C22.32C24), (C4×Dic3)⋊25C22, C23.31(C22×S3), (C22×C6).27C23, C23.11D6⋊22C2, C22.D12⋊11C2, C22.7(D4⋊2S3), C6.D4⋊26C22, (S3×C23).50C22, (C22×S3).68C23, C22.182(S3×C23), (C22×C12).312C22, (C2×Dic3).229C23, (C22×Dic3)⋊22C22, (C2×D6⋊C4)⋊26C2, C4⋊C4⋊S3⋊14C2, C6.85(C2×C4○D4), (C3×C4⋊D4)⋊23C2, (C3×C4⋊C4)⋊15C22, (C2×C6).23(C4○D4), C2.40(C2×D4⋊2S3), (C3×C22⋊C4)⋊17C22, (C2×C4).180(C22×S3), (C2×C3⋊D4).34C22, SmallGroup(192,1176)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 720 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×2], C22 [×18], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×9], Q8, C23, C23 [×2], C23 [×6], Dic3 [×6], C12 [×4], D6 [×10], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C24, Dic6, C2×Dic3 [×6], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×5], C22×S3 [×2], C22×S3 [×4], C22×C6, C22×C6 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×4], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×4], C22×C12, C6×D4, C6×D4 [×2], S3×C23, C22.32C24, C23.11D6 [×2], C22.D12 [×2], C4⋊C4⋊S3 [×2], C12.48D4, C2×D6⋊C4, D4×Dic3 [×2], C23⋊2D6 [×2], C23.14D6 [×2], C3×C4⋊D4, C6.462+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4) [×2], D4⋊2S3 [×2], S3×C23, C22.32C24, C2×D4⋊2S3, D4⋊6D6, D4○D12, C6.462+ (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >
►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 25 7 31)(2 26 8 32)(3 27 9 33)(4 28 10 34)(5 29 11 35)(6 30 12 36)(13 37 19 43)(14 38 20 44)(15 39 21 45)(16 40 22 46)(17 41 23 47)(18 42 24 48)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 19 10 16)(2 24 11 15)(3 23 12 14)(4 22 7 13)(5 21 8 18)(6 20 9 17)(25 43 34 40)(26 48 35 39)(27 47 36 38)(28 46 31 37)(29 45 32 42)(30 44 33 41)
(1 16 4 13)(2 17 5 14)(3 18 6 15)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 37 28 40)(26 38 29 41)(27 39 30 42)(31 43 34 46)(32 44 35 47)(33 45 36 48)
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), (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19,10,16)(2,24,11,15)(3,23,12,14)(4,22,7,13)(5,21,8,18)(6,20,9,17)(25,43,34,40)(26,48,35,39)(27,47,36,38)(28,46,31,37)(29,45,32,42)(30,44,33,41), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,37,28,40)(26,38,29,41)(27,39,30,42)(31,43,34,46)(32,44,35,47)(33,45,36,48)>;
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), (1,25,7,31)(2,26,8,32)(3,27,9,33)(4,28,10,34)(5,29,11,35)(6,30,12,36)(13,37,19,43)(14,38,20,44)(15,39,21,45)(16,40,22,46)(17,41,23,47)(18,42,24,48), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,19,10,16)(2,24,11,15)(3,23,12,14)(4,22,7,13)(5,21,8,18)(6,20,9,17)(25,43,34,40)(26,48,35,39)(27,47,36,38)(28,46,31,37)(29,45,32,42)(30,44,33,41), (1,16,4,13)(2,17,5,14)(3,18,6,15)(7,22,10,19)(8,23,11,20)(9,24,12,21)(25,37,28,40)(26,38,29,41)(27,39,30,42)(31,43,34,46)(32,44,35,47)(33,45,36,48) );
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)], [(1,25,7,31),(2,26,8,32),(3,27,9,33),(4,28,10,34),(5,29,11,35),(6,30,12,36),(13,37,19,43),(14,38,20,44),(15,39,21,45),(16,40,22,46),(17,41,23,47),(18,42,24,48)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,19,10,16),(2,24,11,15),(3,23,12,14),(4,22,7,13),(5,21,8,18),(6,20,9,17),(25,43,34,40),(26,48,35,39),(27,47,36,38),(28,46,31,37),(29,45,32,42),(30,44,33,41)], [(1,16,4,13),(2,17,5,14),(3,18,6,15),(7,22,10,19),(8,23,11,20),(9,24,12,21),(25,37,28,40),(26,38,29,41),(27,39,30,42),(31,43,34,46),(32,44,35,47),(33,45,36,48)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 1 | 1 | 2 | 1 |
| 0 | 0 | 8 | 8 | 12 | 0 |
| 0 | 0 | 9 | 8 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 8 | 12 | 0 |
| 0 | 0 | 8 | 8 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 | 11 | 9 |
| 0 | 0 | 11 | 2 | 11 | 2 |
| 5 | 11 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 9 | 0 | 11 | 9 |
| 0 | 0 | 0 | 4 | 4 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,12,12,0,1,0,0,0,0,12,1,0,0,0,0,12,0],[12,8,0,0,0,0,0,1,0,0,0,0,0,0,0,1,8,9,0,0,0,1,8,8,0,0,1,2,12,12,0,0,12,1,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,8,8,0,0,0,1,8,8,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,4,2,9,11,0,0,11,9,0,2,0,0,0,0,11,11,0,0,0,0,9,2],[5,0,0,0,0,0,11,8,0,0,0,0,0,0,2,4,9,0,0,0,9,11,0,4,0,0,0,0,11,4,0,0,0,0,9,2] >;
36 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 | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| 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 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | D4⋊2S3 | D4⋊6D6 | D4○D12 |
| kernel | C6.462+ (1+4) | C23.11D6 | C22.D12 | C4⋊C4⋊S3 | C12.48D4 | C2×D6⋊C4 | D4×Dic3 | C23⋊2D6 | C23.14D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_6._{46}2_+^{(1+4)} % in TeX
G:=Group("C6.46ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1176);
// by ID
G=gap.SmallGroup(192,1176);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations