metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.42D6, C6.62+ (1+4), C24⋊4S3.C2, D6⋊C4⋊49C22, C23.9D6⋊3C2, (C2×C6).39C24, C4⋊Dic3⋊7C22, C22⋊C4.88D6, (C2×Dic6)⋊4C22, C23.8D6⋊2C2, (C22×C4).188D6, Dic3⋊4D4⋊41C2, C12.48D4⋊17C2, C2.10(D4⋊6D6), (C2×C12).575C23, Dic3⋊C4⋊50C22, C23.11D6⋊3C2, Dic3.D4⋊3C2, (C4×Dic3)⋊49C22, C23.26D6⋊3C2, C22.D12⋊3C2, (C23×C6).65C22, C22.78(S3×C23), C3⋊1(C22.45C24), C22.17(C4○D12), C23.28D6⋊10C2, C23.16D6⋊25C2, (C22×S3).11C23, (C22×C6).129C23, C23.232(C22×S3), (C2×Dic3).12C23, C22.23(D4⋊2S3), (C22×C12).101C22, C6.D4.140C22, (C22×Dic3).80C22, (C4×C3⋊D4)⋊2C2, (S3×C2×C4)⋊42C22, C6.17(C2×C4○D4), (C2×C22⋊C4)⋊18S3, (C6×C22⋊C4)⋊21C2, C2.19(C2×C4○D12), (C2×C6).40(C4○D4), C2.12(C2×D4⋊2S3), (C2×C3⋊D4).8C22, (C2×C6.D4)⋊18C2, (C2×C4).262(C22×S3), (C3×C22⋊C4).110C22, SmallGroup(192,1054)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 600 in 248 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×11], C22, C22 [×4], C22 [×14], S3, C6 [×3], C6 [×5], C2×C4 [×4], C2×C4 [×14], D4 [×5], Q8, C23 [×3], C23 [×6], Dic3 [×7], C12 [×4], D6 [×3], C2×C6, C2×C6 [×4], C2×C6 [×11], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C24, Dic6, C4×S3, C2×Dic3 [×7], C2×Dic3 [×3], C3⋊D4 [×5], C2×C12 [×4], C2×C12 [×3], C22×S3, C22×C6 [×3], C22×C6 [×5], C2×C22⋊C4, C2×C22⋊C4, C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], C4×Dic3 [×3], Dic3⋊C4 [×5], C4⋊Dic3 [×3], D6⋊C4 [×3], C6.D4 [×7], C3×C22⋊C4 [×4], C2×Dic6, S3×C2×C4, C22×Dic3 [×2], C2×C3⋊D4 [×3], C22×C12 [×2], C23×C6, C22.45C24, C23.16D6, Dic3.D4, C23.8D6 [×2], Dic3⋊4D4, C23.9D6, C23.11D6, C22.D12, C12.48D4, C23.26D6, C4×C3⋊D4, C23.28D6, C2×C6.D4, C24⋊4S3, C6×C22⋊C4, C24.42D6
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), C4○D12 [×2], D4⋊2S3 [×2], S3×C23, C22.45C24, C2×C4○D12, C2×D4⋊2S3, D4⋊6D6, C24.42D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
►On 48 points
(2 31)(4 33)(6 35)(8 25)(10 27)(12 29)(13 43)(14 20)(15 45)(16 22)(17 47)(18 24)(19 37)(21 39)(23 41)(38 44)(40 46)(42 48)
(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(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 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 45 7 39)(2 38 8 44)(3 43 9 37)(4 48 10 42)(5 41 11 47)(6 46 12 40)(13 32 19 26)(14 25 20 31)(15 30 21 36)(16 35 22 29)(17 28 23 34)(18 33 24 27)
G:=sub<Sym(48)| (2,31)(4,33)(6,35)(8,25)(10,27)(12,29)(13,43)(14,20)(15,45)(16,22)(17,47)(18,24)(19,37)(21,39)(23,41)(38,44)(40,46)(42,48), (13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,45,7,39)(2,38,8,44)(3,43,9,37)(4,48,10,42)(5,41,11,47)(6,46,12,40)(13,32,19,26)(14,25,20,31)(15,30,21,36)(16,35,22,29)(17,28,23,34)(18,33,24,27)>;
G:=Group( (2,31)(4,33)(6,35)(8,25)(10,27)(12,29)(13,43)(14,20)(15,45)(16,22)(17,47)(18,24)(19,37)(21,39)(23,41)(38,44)(40,46)(42,48), (13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,45,7,39)(2,38,8,44)(3,43,9,37)(4,48,10,42)(5,41,11,47)(6,46,12,40)(13,32,19,26)(14,25,20,31)(15,30,21,36)(16,35,22,29)(17,28,23,34)(18,33,24,27) );
G=PermutationGroup([(2,31),(4,33),(6,35),(8,25),(10,27),(12,29),(13,43),(14,20),(15,45),(16,22),(17,47),(18,24),(19,37),(21,39),(23,41),(38,44),(40,46),(42,48)], [(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,45,7,39),(2,38,8,44),(3,43,9,37),(4,48,10,42),(5,41,11,47),(6,46,12,40),(13,32,19,26),(14,25,20,31),(15,30,21,36),(16,35,22,29),(17,28,23,34),(18,33,24,27)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 12 |
| 1 | 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 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 8 | 11 |
| 0 | 0 | 0 | 5 |
| 0 | 6 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,1,8,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[11,0,0,0,0,7,0,0,0,0,8,0,0,0,11,5],[0,2,0,0,6,0,0,0,0,0,8,0,0,0,0,8] >;
45 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 | ··· | 4O | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 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 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ (1+4) | D4⋊2S3 | D4⋊6D6 |
| kernel | C24.42D6 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | C23.11D6 | C22.D12 | C12.48D4 | C23.26D6 | C4×C3⋊D4 | C23.28D6 | C2×C6.D4 | C24⋊4S3 | C6×C22⋊C4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C22 | C6 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 8 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_2^4._{42}D_6 % in TeX
G:=Group("C2^4.42D6"); // GroupNames label
G:=SmallGroup(192,1054);
// by ID
G=gap.SmallGroup(192,1054);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,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