metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.35D6, C6.22+ (1+4), C23⋊4(C4×S3), (C22×C4)⋊6D6, C22⋊C4⋊51D6, C6.9(C23×C4), D6⋊C4⋊57C22, (C2×C6).30C24, D6.2(C22×C4), C2.1(D4⋊6D6), Dic3⋊4D4⋊39C2, (C2×C12).571C23, Dic3⋊C4⋊58C22, (C22×C12)⋊34C22, C3⋊1(C22.11C24), (C4×Dic3)⋊46C22, C22.19(S3×C23), (C23×C6).56C22, Dic3.3(C22×C4), C23.16D6⋊24C2, C6.D4⋊67C22, (S3×C23).30C22, C23.230(C22×S3), (C22×C6).122C23, (C22×Dic3)⋊5C22, (C22×S3).150C23, (C2×Dic3).177C23, (C2×C3⋊D4)⋊9C4, C3⋊D4⋊9(C2×C4), (C4×C3⋊D4)⋊33C2, (C2×C22⋊C4)⋊7S3, (S3×C2×C4)⋊39C22, C2.11(S3×C22×C4), C22.24(S3×C2×C4), (S3×C22⋊C4)⋊23C2, (C6×C22⋊C4)⋊26C2, (C22×S3)⋊6(C2×C4), (C22×C6)⋊10(C2×C4), (C2×Dic3)⋊10(C2×C4), (C2×C6).18(C22×C4), (C22×C3⋊D4).9C2, (C2×C6.D4)⋊15C2, (C3×C22⋊C4)⋊61C22, (C2×C4).257(C22×S3), (C2×C3⋊D4).88C22, SmallGroup(192,1045)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 840 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×12], C22, C22 [×6], C22 [×22], S3 [×4], C6, C6 [×2], C6 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×16], C23 [×3], C23 [×4], C23 [×10], Dic3 [×4], Dic3 [×4], C12 [×4], D6 [×4], D6 [×8], C2×C6, C2×C6 [×6], C2×C6 [×10], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×7], C2×D4 [×12], C24, C24, C4×S3 [×4], C2×Dic3 [×10], C2×Dic3 [×2], C3⋊D4 [×16], C2×C12 [×4], C2×C12 [×2], C22×S3 [×6], C22×S3 [×2], C22×C6 [×3], C22×C6 [×4], C22×C6 [×2], C2×C22⋊C4, C2×C22⋊C4 [×3], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C4×Dic3 [×4], Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×12], C22×C12 [×2], S3×C23, C23×C6, C22.11C24, C23.16D6 [×2], S3×C22⋊C4 [×2], Dic3⋊4D4 [×4], C4×C3⋊D4 [×4], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.35D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ (1+4) [×2], S3×C2×C4 [×6], S3×C23, C22.11C24, S3×C22×C4, D4⋊6D6 [×2], C24.35D6
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, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >
►On 48 points
(2 30)(4 32)(6 34)(8 36)(10 26)(12 28)(14 47)(16 37)(18 39)(20 41)(22 43)(24 45)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)
(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 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 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 43 19 37)(14 48 20 42)(15 41 21 47)(16 46 22 40)(17 39 23 45)(18 44 24 38)(25 26 31 32)(27 36 33 30)(28 29 34 35)
G:=sub<Sym(48)| (2,30)(4,32)(6,34)(8,36)(10,26)(12,28)(14,47)(16,37)(18,39)(20,41)(22,43)(24,45), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45), (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)(13,43,19,37)(14,48,20,42)(15,41,21,47)(16,46,22,40)(17,39,23,45)(18,44,24,38)(25,26,31,32)(27,36,33,30)(28,29,34,35)>;
G:=Group( (2,30)(4,32)(6,34)(8,36)(10,26)(12,28)(14,47)(16,37)(18,39)(20,41)(22,43)(24,45), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,25)(24,26), (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), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45), (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)(13,43,19,37)(14,48,20,42)(15,41,21,47)(16,46,22,40)(17,39,23,45)(18,44,24,38)(25,26,31,32)(27,36,33,30)(28,29,34,35) );
G=PermutationGroup([(2,30),(4,32),(6,34),(8,36),(10,26),(12,28),(14,47),(16,37),(18,39),(20,41),(22,43),(24,45)], [(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,25),(24,26)], [(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)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,25),(10,26),(11,27),(12,28),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45)], [(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),(13,43,19,37),(14,48,20,42),(15,41,21,47),(16,46,22,40),(17,39,23,45),(18,44,24,38),(25,26,31,32),(27,36,33,30),(28,29,34,35)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 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 | 0 | 12 | 0 |
| 0 | 0 | 0 | 8 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 2 | 2 |
| 0 | 0 | 8 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 2 | 2 |
| 0 | 0 | 0 | 8 | 0 | 11 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,8,0,0,0,0,1,0,8,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,5,0,0,0,0,8,0,0,0,0,0,0,0,5,8,0,0,0,0,5,0,0,0,0,0,2,11,8,5,0,0,2,0,8,0],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,5,8,0,0,0,0,2,0,8,0,0,0,2,11,8,5] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 3 | 4A | ··· | 4H | 4I | ··· | 4T | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | C4×S3 | 2+ (1+4) | D4⋊6D6 |
| kernel | C24.35D6 | C23.16D6 | S3×C22⋊C4 | Dic3⋊4D4 | C4×C3⋊D4 | C2×C6.D4 | C6×C22⋊C4 | C22×C3⋊D4 | C2×C3⋊D4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 16 | 1 | 4 | 2 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._{35}D_6 % in TeX
G:=Group("C2^4.35D6"); // GroupNames label
G:=SmallGroup(192,1045);
// by ID
G=gap.SmallGroup(192,1045);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,570,80,6278]);
// 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,b*c=c*b,f*b*f^-1=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