direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23⋊2D6, C24⋊10D6, C6⋊2C22≀C2, D6⋊12(C2×D4), (C2×D4)⋊35D6, (S3×C24)⋊3C2, (C2×C12)⋊9C23, (C22×D4)⋊8S3, (C22×C6)⋊11D4, (C22×C4)⋊29D6, D6⋊C4⋊71C22, (C6×D4)⋊55C22, (C22×S3)⋊15D4, C23⋊7(C3⋊D4), (C22×C6)⋊5C23, C23⋊4(C22×S3), (C2×C6).294C24, (C23×C6)⋊12C22, (C2×Dic3)⋊3C23, C22.146(S3×D4), C6.141(C22×D4), (S3×C23)⋊21C22, (C22×C12)⋊43C22, C6.D4⋊60C22, C22.307(S3×C23), (C22×S3).238C23, (C22×Dic3)⋊32C22, (D4×C2×C6)⋊15C2, (C2×C6)⋊7(C2×D4), C3⋊3(C2×C22≀C2), C2.101(C2×S3×D4), (C2×D6⋊C4)⋊41C2, (C2×C4)⋊4(C22×S3), C22⋊4(C2×C3⋊D4), (C2×C3⋊D4)⋊43C22, (C22×C3⋊D4)⋊12C2, C2.14(C22×C3⋊D4), (C2×C6.D4)⋊27C2, SmallGroup(192,1358)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1960 in 662 conjugacy classes, 143 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×14], C3, C4 [×6], C22, C22 [×10], C22 [×86], S3 [×8], C6, C6 [×6], C6 [×6], C2×C4 [×2], C2×C4 [×10], D4 [×24], C23, C23 [×8], C23 [×86], Dic3 [×4], C12 [×2], D6 [×8], D6 [×56], C2×C6, C2×C6 [×10], C2×C6 [×22], C22⋊C4 [×12], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24 [×18], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×S3 [×12], C22×S3 [×64], C22×C6, C22×C6 [×8], C22×C6 [×10], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4, C22×D4 [×2], C25, D6⋊C4 [×8], C6.D4 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23 [×6], S3×C23 [×12], C23×C6 [×2], C2×C22≀C2, C2×D6⋊C4 [×2], C23⋊2D6 [×8], C2×C6.D4, C22×C3⋊D4 [×2], D4×C2×C6, S3×C24, C2×C23⋊2D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C3⋊D4 [×4], C22×S3 [×7], C22≀C2 [×4], C22×D4 [×3], S3×D4 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C22≀C2, C23⋊2D6 [×4], C2×S3×D4 [×2], C22×C3⋊D4, C2×C23⋊2D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 48 points
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(25 44)(26 45)(27 46)(28 47)(29 48)(30 43)
(1 44)(2 12)(3 46)(4 8)(5 48)(6 10)(7 14)(9 16)(11 18)(13 45)(15 47)(17 43)(19 31)(20 40)(21 33)(22 42)(23 35)(24 38)(25 32)(26 41)(27 34)(28 37)(29 36)(30 39)
(1 4)(2 5)(3 6)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 16)(14 17)(15 18)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 46)(8 47)(9 48)(10 43)(11 44)(12 45)(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 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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)
G:=sub<Sym(48)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(25,44)(26,45)(27,46)(28,47)(29,48)(30,43), (1,44)(2,12)(3,46)(4,8)(5,48)(6,10)(7,14)(9,16)(11,18)(13,45)(15,47)(17,43)(19,31)(20,40)(21,33)(22,42)(23,35)(24,38)(25,32)(26,41)(27,34)(28,37)(29,36)(30,39), (1,4)(2,5)(3,6)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,16)(14,17)(15,18)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,46)(8,47)(9,48)(10,43)(11,44)(12,45)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38), (1,2,3,4,5,6)(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), (1,14)(2,13)(3,18)(4,17)(5,16)(6,15)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(19,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(25,44)(26,45)(27,46)(28,47)(29,48)(30,43), (1,44)(2,12)(3,46)(4,8)(5,48)(6,10)(7,14)(9,16)(11,18)(13,45)(15,47)(17,43)(19,31)(20,40)(21,33)(22,42)(23,35)(24,38)(25,32)(26,41)(27,34)(28,37)(29,36)(30,39), (1,4)(2,5)(3,6)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,16)(14,17)(15,18)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,46)(8,47)(9,48)(10,43)(11,44)(12,45)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,39)(32,40)(33,41)(34,42)(35,37)(36,38), (1,2,3,4,5,6)(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), (1,14)(2,13)(3,18)(4,17)(5,16)(6,15)(7,47)(8,46)(9,45)(10,44)(11,43)(12,48)(19,25)(20,30)(21,29)(22,28)(23,27)(24,26)(31,37)(32,42)(33,41)(34,40)(35,39)(36,38) );
G=PermutationGroup([(1,32),(2,33),(3,34),(4,35),(5,36),(6,31),(7,22),(8,23),(9,24),(10,19),(11,20),(12,21),(13,41),(14,42),(15,37),(16,38),(17,39),(18,40),(25,44),(26,45),(27,46),(28,47),(29,48),(30,43)], [(1,44),(2,12),(3,46),(4,8),(5,48),(6,10),(7,14),(9,16),(11,18),(13,45),(15,47),(17,43),(19,31),(20,40),(21,33),(22,42),(23,35),(24,38),(25,32),(26,41),(27,34),(28,37),(29,36),(30,39)], [(1,4),(2,5),(3,6),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,16),(14,17),(15,18),(19,27),(20,28),(21,29),(22,30),(23,25),(24,26),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42)], [(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,46),(8,47),(9,48),(10,43),(11,44),(12,45),(19,30),(20,25),(21,26),(22,27),(23,28),(24,29),(31,39),(32,40),(33,41),(34,42),(35,37),(36,38)], [(1,2,3,4,5,6),(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)], [(1,14),(2,13),(3,18),(4,17),(5,16),(6,15),(7,47),(8,46),(9,45),(10,44),(11,43),(12,48),(19,25),(20,30),(21,29),(22,28),(23,27),(24,26),(31,37),(32,42),(33,41),(34,40),(35,39),(36,38)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 4 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,9,2],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | ··· | 2U | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C3⋊D4 | S3×D4 |
| kernel | C2×C23⋊2D6 | C2×D6⋊C4 | C23⋊2D6 | C2×C6.D4 | C22×C3⋊D4 | D4×C2×C6 | S3×C24 | C22×D4 | C22×S3 | C22×C6 | C22×C4 | C2×D4 | C24 | C23 | C22 |
| # reps | 1 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 8 | 4 | 1 | 4 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3\rtimes_2D_6
% in TeX
G:=Group("C2xC2^3:2D6"); // GroupNames label
G:=SmallGroup(192,1358);
// by ID
G=gap.SmallGroup(192,1358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=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=e^-1>;
// generators/relations