metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.43D6, C6.242+ (1+4), C22⋊C4.1D6, C22≀C2.3S3, (D4×Dic3)⋊11C2, (C2×D4).149D6, (C2×C6).131C24, (C2×C12).26C23, Dic3⋊C4⋊7C22, C4⋊Dic3⋊24C22, C2.26(D4⋊6D6), C23.12D6⋊11C2, (C2×Dic6)⋊19C22, (C4×Dic3)⋊13C22, (C6×D4).110C22, C23.8D6⋊11C2, C23.16D6⋊2C2, C23.23D6⋊3C2, (C23×C6).67C22, Dic3.D4⋊12C2, C3⋊4(C22.45C24), C6.D4⋊12C22, (C22×C6).180C23, C23.186(C22×S3), C22.152(S3×C23), C22.17(D4⋊2S3), (C2×Dic3).220C23, (C22×Dic3)⋊11C22, C6.76(C2×C4○D4), (C3×C22≀C2).2C2, (C2×C6).43(C4○D4), C2.27(C2×D4⋊2S3), (C2×C4).26(C22×S3), (C2×C6.D4)⋊19C2, (C3×C22⋊C4).2C22, SmallGroup(192,1146)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 576 in 248 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×11], C22, C22 [×4], C22 [×14], C6, C6 [×2], C6 [×6], C2×C4, C2×C4 [×2], C2×C4 [×15], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], Dic3 [×8], C12 [×3], C2×C6, C2×C6 [×4], C2×C6 [×14], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic6, C2×Dic3 [×8], C2×Dic3 [×7], C2×C12, C2×C12 [×2], C3×D4 [×5], C22×C6 [×2], C22×C6 [×2], C22×C6 [×5], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C6.D4, C6.D4 [×10], C3×C22⋊C4, C3×C22⋊C4 [×2], C2×Dic6, C22×Dic3, C22×Dic3 [×4], C6×D4, C6×D4 [×2], C23×C6, C22.45C24, C23.16D6 [×2], Dic3.D4 [×2], C23.8D6 [×2], D4×Dic3 [×2], C23.23D6, C23.23D6 [×2], C23.12D6, C2×C6.D4 [×2], C3×C22≀C2, C24.43D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), D4⋊2S3 [×4], S3×C23, C22.45C24, C2×D4⋊2S3 [×2], D4⋊6D6, C24.43D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
►On 48 points
(2 7)(4 9)(6 11)(14 48)(16 44)(18 46)(20 25)(22 27)(24 29)(32 40)(34 42)(36 38)
(1 19)(3 21)(5 23)(8 26)(10 28)(12 30)(13 47)(14 40)(15 43)(16 42)(17 45)(18 38)(31 39)(32 48)(33 41)(34 44)(35 37)(36 46)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 47)(14 48)(15 43)(16 44)(17 45)(18 46)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)
(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 46 19 38)(2 45 20 37)(3 44 21 42)(4 43 22 41)(5 48 23 40)(6 47 24 39)(7 17 25 35)(8 16 26 34)(9 15 27 33)(10 14 28 32)(11 13 29 31)(12 18 30 36)
G:=sub<Sym(48)| (2,7)(4,9)(6,11)(14,48)(16,44)(18,46)(20,25)(22,27)(24,29)(32,40)(34,42)(36,38), (1,19)(3,21)(5,23)(8,26)(10,28)(12,30)(13,47)(14,40)(15,43)(16,42)(17,45)(18,38)(31,39)(32,48)(33,41)(34,44)(35,37)(36,46), (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(13,47)(14,48)(15,43)(16,44)(17,45)(18,46)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (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), (1,46,19,38)(2,45,20,37)(3,44,21,42)(4,43,22,41)(5,48,23,40)(6,47,24,39)(7,17,25,35)(8,16,26,34)(9,15,27,33)(10,14,28,32)(11,13,29,31)(12,18,30,36)>;
G:=Group( (2,7)(4,9)(6,11)(14,48)(16,44)(18,46)(20,25)(22,27)(24,29)(32,40)(34,42)(36,38), (1,19)(3,21)(5,23)(8,26)(10,28)(12,30)(13,47)(14,40)(15,43)(16,42)(17,45)(18,38)(31,39)(32,48)(33,41)(34,44)(35,37)(36,46), (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(13,47)(14,48)(15,43)(16,44)(17,45)(18,46)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (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), (1,46,19,38)(2,45,20,37)(3,44,21,42)(4,43,22,41)(5,48,23,40)(6,47,24,39)(7,17,25,35)(8,16,26,34)(9,15,27,33)(10,14,28,32)(11,13,29,31)(12,18,30,36) );
G=PermutationGroup([(2,7),(4,9),(6,11),(14,48),(16,44),(18,46),(20,25),(22,27),(24,29),(32,40),(34,42),(36,38)], [(1,19),(3,21),(5,23),(8,26),(10,28),(12,30),(13,47),(14,40),(15,43),(16,42),(17,45),(18,38),(31,39),(32,48),(33,41),(34,44),(35,37),(36,46)], [(1,12),(2,7),(3,8),(4,9),(5,10),(6,11),(13,47),(14,48),(15,43),(16,44),(17,45),(18,46),(19,30),(20,25),(21,26),(22,27),(23,28),(24,29),(31,39),(32,40),(33,41),(34,42),(35,37),(36,38)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44)], [(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)], [(1,46,19,38),(2,45,20,37),(3,44,21,42),(4,43,22,41),(5,48,23,40),(6,47,24,39),(7,17,25,35),(8,16,26,34),(9,15,27,33),(10,14,28,32),(11,13,29,31),(12,18,30,36)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 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 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,9,3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,10,3,0,0,0,0,6,3,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | ··· | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | ··· | 6I | 6J | 12A | 12B | 12C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | D4⋊2S3 | D4⋊6D6 |
| kernel | C24.43D6 | C23.16D6 | Dic3.D4 | C23.8D6 | D4×Dic3 | C23.23D6 | C23.12D6 | C2×C6.D4 | C3×C22≀C2 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 1 | 3 | 3 | 1 | 8 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
C_2^4._{43}D_6 % in TeX
G:=Group("C2^4.43D6"); // GroupNames label
G:=SmallGroup(192,1146);
// by ID
G=gap.SmallGroup(192,1146);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,1571,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*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^-1>;
// generators/relations