metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.38D6, C6.32+ (1+4), C3⋊D4⋊8D4, Dic3⋊D4⋊1C2, D6⋊D4⋊2C2, C3⋊1(D4⋊5D4), C22⋊C4⋊40D6, D6.12(C2×D4), (C22×C4)⋊11D6, C12⋊7D4⋊17C2, C24⋊4S3⋊2C2, D6⋊C4⋊46C22, C23.9D6⋊1C2, (C2×D12)⋊2C22, (C2×C6).34C24, C4⋊Dic3⋊4C22, C22.18(S3×D4), C6.37(C22×D4), C2.7(D4⋊6D6), C22⋊5(C4○D12), Dic3.14(C2×D4), Dic3⋊4D4⋊40C2, (C2×C12).128C23, Dic3⋊C4⋊49C22, (C22×C12)⋊14C22, C23.11D6⋊1C2, Dic3.D4⋊2C2, (C2×Dic6)⋊48C22, (C4×Dic3)⋊47C22, C23.28D6⋊9C2, C6.D4⋊8C22, C22.73(S3×C23), (C23×C6).60C22, (S3×C23).31C22, (C22×C6).387C23, C23.231(C22×S3), (C22×S3).152C23, (C2×Dic3).180C23, (C22×Dic3).78C22, C2.11(C2×S3×D4), (C4×C3⋊D4)⋊1C2, (C2×C4○D12)⋊3C2, (C2×C6)⋊8(C4○D4), (S3×C2×C4)⋊40C22, C6.14(C2×C4○D4), (S3×C22⋊C4)⋊24C2, (C6×C22⋊C4)⋊18C2, (C2×C22⋊C4)⋊13S3, C2.16(C2×C4○D12), (C2×C6).383(C2×D4), (C22×C3⋊D4)⋊5C2, (C2×C3⋊D4)⋊1C22, (C3×C22⋊C4)⋊53C22, (C2×C4).259(C22×S3), SmallGroup(192,1049)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 952 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×4], C22 [×25], S3 [×4], C6 [×3], C6 [×5], C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], Dic3 [×2], Dic3 [×4], C12 [×4], D6 [×2], D6 [×12], C2×C6, C2×C6 [×4], C2×C6 [×11], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24, C24, Dic6 [×2], C4×S3 [×5], D12 [×3], C2×Dic3 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C3⋊D4 [×11], C2×C12 [×4], C2×C12 [×3], C22×S3 [×3], C22×S3 [×5], C22×C6 [×3], C22×C6 [×5], C2×C22⋊C4, C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×3], C4⋊Dic3, D6⋊C4 [×5], C6.D4 [×3], C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4 [×3], C2×D12 [×2], C4○D12 [×4], C22×Dic3, C2×C3⋊D4 [×7], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, D4⋊5D4, Dic3.D4, S3×C22⋊C4, Dic3⋊4D4, D6⋊D4, C23.9D6, Dic3⋊D4 [×2], C23.11D6, C4×C3⋊D4, C23.28D6, C12⋊7D4, C24⋊4S3, C6×C22⋊C4, C2×C4○D12, C22×C3⋊D4, C24.38D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C4○D12 [×2], S3×D4 [×2], S3×C23, D4⋊5D4, C2×C4○D12, C2×S3×D4, D4⋊6D6, C24.38D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
►On 48 points
(1 7)(2 24)(3 9)(4 14)(5 11)(6 16)(8 18)(10 20)(12 22)(13 19)(15 21)(17 23)(25 31)(26 41)(27 33)(28 43)(29 35)(30 45)(32 47)(34 37)(36 39)(38 44)(40 46)(42 48)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 33)(14 34)(15 35)(16 36)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(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 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 14 19 20)(15 24 21 18)(16 17 22 23)(25 36 31 30)(26 29 32 35)(27 34 33 28)(37 48 43 42)(38 41 44 47)(39 46 45 40)
G:=sub<Sym(48)| (1,7)(2,24)(3,9)(4,14)(5,11)(6,16)(8,18)(10,20)(12,22)(13,19)(15,21)(17,23)(25,31)(26,41)(27,33)(28,43)(29,35)(30,45)(32,47)(34,37)(36,39)(38,44)(40,46)(42,48), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,33)(14,34)(15,35)(16,36)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,36,31,30)(26,29,32,35)(27,34,33,28)(37,48,43,42)(38,41,44,47)(39,46,45,40)>;
G:=Group( (1,7)(2,24)(3,9)(4,14)(5,11)(6,16)(8,18)(10,20)(12,22)(13,19)(15,21)(17,23)(25,31)(26,41)(27,33)(28,43)(29,35)(30,45)(32,47)(34,37)(36,39)(38,44)(40,46)(42,48), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,33)(14,34)(15,35)(16,36)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,46)(26,47)(27,48)(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,36,31,30)(26,29,32,35)(27,34,33,28)(37,48,43,42)(38,41,44,47)(39,46,45,40) );
G=PermutationGroup([(1,7),(2,24),(3,9),(4,14),(5,11),(6,16),(8,18),(10,20),(12,22),(13,19),(15,21),(17,23),(25,31),(26,41),(27,33),(28,43),(29,35),(30,45),(32,47),(34,37),(36,39),(38,44),(40,46),(42,48)], [(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,33),(14,34),(15,35),(16,36),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,46),(26,47),(27,48),(28,37),(29,38),(30,39),(31,40),(32,41),(33,42),(34,43),(35,44),(36,45)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,14,19,20),(15,24,21,18),(16,17,22,23),(25,36,31,30),(26,29,32,35),(27,34,33,28),(37,48,43,42),(38,41,44,47),(39,46,45,40)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 12 |
| 2 | 9 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 |
| 8 | 5 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,5,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[0,8,0,0,5,5,0,0,0,0,12,0,0,0,3,1],[5,5,0,0,0,8,0,0,0,0,12,0,0,0,3,1] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 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 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | S3×D4 | D4⋊6D6 |
| kernel | C24.38D6 | Dic3.D4 | S3×C22⋊C4 | Dic3⋊4D4 | D6⋊D4 | C23.9D6 | Dic3⋊D4 | C23.11D6 | C4×C3⋊D4 | C23.28D6 | C12⋊7D4 | C24⋊4S3 | C6×C22⋊C4 | C2×C4○D12 | C22×C3⋊D4 | C2×C22⋊C4 | C3⋊D4 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C22 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2^4._{38}D_6 % in TeX
G:=Group("C2^4.38D6"); // GroupNames label
G:=SmallGroup(192,1049);
// by ID
G=gap.SmallGroup(192,1049);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,100,675,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
// generators/relations