metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊23D6, C6.732+ (1+4), (C2×Q8)⋊12D6, C22⋊C4⋊35D6, (C4×C12)⋊33C22, D6⋊3Q8⋊31C2, (C2×D4).111D6, C23⋊2D6.6C2, C4.4D4⋊13S3, (C6×Q8)⋊15C22, C42⋊2S3⋊36C2, D6.23(C4○D4), C23.9D6⋊44C2, (C2×C6).223C24, C4⋊Dic3⋊42C22, Dic3⋊4D4⋊32C2, C2.76(D4⋊6D6), (C2×C12).632C23, Dic3⋊C4⋊67C22, D6⋊C4.136C22, (C4×Dic3)⋊57C22, (C6×D4).211C22, C23.8D6⋊40C2, (C22×C6).53C23, C23.55(C22×S3), C3⋊8(C22.45C24), C23.23D6⋊25C2, C6.D4⋊34C22, (S3×C23).66C22, C22.244(S3×C23), (C22×S3).217C23, (C2×Dic3).255C23, (C22×Dic3)⋊28C22, C2.79(S3×C4○D4), (S3×C22⋊C4)⋊19C2, C6.190(C2×C4○D4), (C3×C4.4D4)⋊15C2, (S3×C2×C4).215C22, (C2×C4).74(C22×S3), (C3×C22⋊C4)⋊31C22, (C2×C3⋊D4).61C22, SmallGroup(192,1238)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 672 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×11], C22, C22 [×18], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], Dic3 [×6], C12 [×5], D6 [×4], D6 [×8], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, C4×S3 [×6], C2×Dic3 [×6], C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12 [×4], C3×D4, C3×Q8, C22×S3 [×2], C22×S3 [×5], C22×C6 [×2], 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 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×6], C6.D4 [×2], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C6×D4, C6×Q8, S3×C23, C22.45C24, C42⋊2S3 [×2], C23.8D6 [×2], S3×C22⋊C4 [×2], Dic3⋊4D4 [×2], C23.9D6 [×2], C23.23D6, C23⋊2D6, D6⋊3Q8 [×2], C3×C4.4D4, C42⋊23D6
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), S3×C23, C22.45C24, D4⋊6D6, S3×C4○D4 [×2], C42⋊23D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 48 points
(1 31 7 25)(2 35 8 29)(3 33 9 27)(4 34 10 28)(5 32 11 26)(6 36 12 30)(13 38 16 43)(14 47 17 42)(15 40 18 45)(19 44 22 39)(20 37 23 48)(21 46 24 41)
(1 21 4 13)(2 19 5 17)(3 23 6 15)(7 24 10 16)(8 22 11 14)(9 20 12 18)(25 41 28 43)(26 47 29 39)(27 37 30 45)(31 46 34 38)(32 42 35 44)(33 48 36 40)
(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 9)(2 8)(3 7)(4 12)(5 11)(6 10)(13 15)(16 18)(20 24)(21 23)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 41)(38 40)(43 45)(46 48)
G:=sub<Sym(48)| (1,31,7,25)(2,35,8,29)(3,33,9,27)(4,34,10,28)(5,32,11,26)(6,36,12,30)(13,38,16,43)(14,47,17,42)(15,40,18,45)(19,44,22,39)(20,37,23,48)(21,46,24,41), (1,21,4,13)(2,19,5,17)(3,23,6,15)(7,24,10,16)(8,22,11,14)(9,20,12,18)(25,41,28,43)(26,47,29,39)(27,37,30,45)(31,46,34,38)(32,42,35,44)(33,48,36,40), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,15)(16,18)(20,24)(21,23)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,41)(38,40)(43,45)(46,48)>;
G:=Group( (1,31,7,25)(2,35,8,29)(3,33,9,27)(4,34,10,28)(5,32,11,26)(6,36,12,30)(13,38,16,43)(14,47,17,42)(15,40,18,45)(19,44,22,39)(20,37,23,48)(21,46,24,41), (1,21,4,13)(2,19,5,17)(3,23,6,15)(7,24,10,16)(8,22,11,14)(9,20,12,18)(25,41,28,43)(26,47,29,39)(27,37,30,45)(31,46,34,38)(32,42,35,44)(33,48,36,40), (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), (1,9)(2,8)(3,7)(4,12)(5,11)(6,10)(13,15)(16,18)(20,24)(21,23)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)(37,41)(38,40)(43,45)(46,48) );
G=PermutationGroup([(1,31,7,25),(2,35,8,29),(3,33,9,27),(4,34,10,28),(5,32,11,26),(6,36,12,30),(13,38,16,43),(14,47,17,42),(15,40,18,45),(19,44,22,39),(20,37,23,48),(21,46,24,41)], [(1,21,4,13),(2,19,5,17),(3,23,6,15),(7,24,10,16),(8,22,11,14),(9,20,12,18),(25,41,28,43),(26,47,29,39),(27,37,30,45),(31,46,34,38),(32,42,35,44),(33,48,36,40)], [(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)], [(1,9),(2,8),(3,7),(4,12),(5,11),(6,10),(13,15),(16,18),(20,24),(21,23),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34),(37,41),(38,40),(43,45),(46,48)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,11,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;
39 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 | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 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 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | D4⋊6D6 | S3×C4○D4 |
| kernel | C42⋊23D6 | C42⋊2S3 | C23.8D6 | S3×C22⋊C4 | Dic3⋊4D4 | C23.9D6 | C23.23D6 | C23⋊2D6 | D6⋊3Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | D6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{23}D_6 % in TeX
G:=Group("C4^2:23D6"); // GroupNames label
G:=SmallGroup(192,1238);
// by ID
G=gap.SmallGroup(192,1238);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,387,100,346,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations