direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C24.4C4, C24.6C12, C4.70(C6×D4), C22⋊C8⋊11C6, (C2×C6)⋊7M4(2), (C23×C6).6C4, (C2×C24)⋊37C22, C12.475(C2×D4), (C2×C12).513D4, (C2×M4(2))⋊6C6, (C23×C4).13C6, C2.6(C6×M4(2)), (C6×M4(2))⋊24C2, (C23×C12).23C2, (C22×C12).20C4, (C22×C4).15C12, C23.34(C2×C12), C6.48(C2×M4(2)), C22⋊3(C3×M4(2)), (C2×C12).982C23, C12.107(C22⋊C4), C22.41(C22×C12), (C22×C12).496C22, (C2×C8)⋊7(C2×C6), (C3×C22⋊C8)⋊28C2, (C2×C4).71(C2×C12), (C2×C4).118(C3×D4), C6.98(C2×C22⋊C4), C4.21(C3×C22⋊C4), C2.10(C6×C22⋊C4), (C2×C12).333(C2×C4), (C22×C4).99(C2×C6), (C2×C6).80(C22⋊C4), (C2×C4).150(C22×C6), (C22×C6).115(C2×C4), (C2×C6).232(C22×C4), C22.17(C3×C22⋊C4), SmallGroup(192,840)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24.4C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 290 in 190 conjugacy classes, 90 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C24, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C22⋊C8, C2×M4(2), C23×C4, C2×C24, C3×M4(2), C22×C12, C22×C12, C22×C12, C23×C6, C24.4C4, C3×C22⋊C8, C6×M4(2), C23×C12, C3×C24.4C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, M4(2), C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C2×M4(2), C3×C22⋊C4, C3×M4(2), C22×C12, C6×D4, C24.4C4, C6×C22⋊C4, C6×M4(2), C3×C24.4C4
►On 48 points
(1 14 39)(2 15 40)(3 16 33)(4 9 34)(5 10 35)(6 11 36)(7 12 37)(8 13 38)(17 29 41)(18 30 42)(19 31 43)(20 32 44)(21 25 45)(22 26 46)(23 27 47)(24 28 48)
(2 24)(4 18)(6 20)(8 22)(9 30)(11 32)(13 26)(15 28)(34 42)(36 44)(38 46)(40 48)
(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 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)
G:=sub<Sym(48)| (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (2,24)(4,18)(6,20)(8,22)(9,30)(11,32)(13,26)(15,28)(34,42)(36,44)(38,46)(40,48), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (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)>;
G:=Group( (1,14,39)(2,15,40)(3,16,33)(4,9,34)(5,10,35)(6,11,36)(7,12,37)(8,13,38)(17,29,41)(18,30,42)(19,31,43)(20,32,44)(21,25,45)(22,26,46)(23,27,47)(24,28,48), (2,24)(4,18)(6,20)(8,22)(9,30)(11,32)(13,26)(15,28)(34,42)(36,44)(38,46)(40,48), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47), (1,23)(2,24)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (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) );
G=PermutationGroup([[(1,14,39),(2,15,40),(3,16,33),(4,9,34),(5,10,35),(6,11,36),(7,12,37),(8,13,38),(17,29,41),(18,30,42),(19,31,43),(20,32,44),(21,25,45),(22,26,46),(23,27,47),(24,28,48)], [(2,24),(4,18),(6,20),(8,22),(9,30),(11,32),(13,26),(15,28),(34,42),(36,44),(38,46),(40,48)], [(1,5),(3,7),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47)], [(1,23),(2,24),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48)], [(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)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | ··· | 6R | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12T | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | M4(2) | C3×D4 | C3×M4(2) |
| kernel | C3×C24.4C4 | C3×C22⋊C8 | C6×M4(2) | C23×C12 | C24.4C4 | C22×C12 | C23×C6 | C22⋊C8 | C2×M4(2) | C23×C4 | C22×C4 | C24 | C2×C12 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 6 | 2 | 8 | 4 | 2 | 12 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C24.4C4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,1,0,0,0,0,0,0,46,0,0,1,0] >;
C3×C24.4C4 in GAP, Magma, Sage, TeX
C_3\times C_2^4._4C_4
% in TeX
G:=Group("C3xC2^4.4C4"); // GroupNames label
G:=SmallGroup(192,840);
// by ID
G=gap.SmallGroup(192,840);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=1,f^4=e,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations