direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D8⋊C22, C24.47C23, C12.84C24, C4○D8⋊3C6, D8⋊4(C2×C6), C8⋊C22⋊6C6, Q16⋊4(C2×C6), C4.68(C6×D4), SD16⋊3(C2×C6), C8.C22⋊6C6, C8.2(C22×C6), C4.7(C23×C6), (C2×C24)⋊22C22, C12.473(C2×D4), (C2×C12).527D4, (C6×D4)⋊67C22, (C3×D8)⋊20C22, M4(2)⋊5(C2×C6), (C2×M4(2))⋊5C6, (C6×Q8)⋊56C22, D4.4(C22×C6), (C22×C6).38D4, C22.25(C6×D4), C23.20(C3×D4), Q8.8(C22×C6), (C6×M4(2))⋊10C2, (C3×Q16)⋊18C22, (C3×D4).37C23, C6.205(C22×D4), (C3×Q8).38C23, (C2×C12).686C23, (C3×SD16)⋊19C22, (C3×M4(2))⋊26C22, (C22×C12).467C22, (C2×C8)⋊3(C2×C6), C2.29(D4×C2×C6), C4○D4⋊7(C2×C6), (C2×C4○D4)⋊16C6, (C3×C4○D8)⋊10C2, (C6×C4○D4)⋊28C2, (C2×D4)⋊16(C2×C6), (C2×Q8)⋊18(C2×C6), (C3×C8⋊C22)⋊13C2, (C2×C4).138(C3×D4), (C2×C6).421(C2×D4), (C3×C4○D4)⋊25C22, (C3×C8.C22)⋊13C2, (C22×C4).83(C2×C6), (C2×C4).47(C22×C6), SmallGroup(192,1464)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 402 in 262 conjugacy classes, 158 normal (22 characteristic)
C1, C2, C2 [×7], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×9], C6, C6 [×7], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C12 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×9], C2×C8 [×2], M4(2) [×4], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C24 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×10], C3×D4 [×4], C3×D4 [×10], C3×Q8 [×4], C3×Q8 [×2], C22×C6, C22×C6 [×2], C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C2×C24 [×2], C3×M4(2) [×4], C3×D8 [×4], C3×SD16 [×8], C3×Q16 [×4], C22×C12, C22×C12 [×2], C6×D4 [×2], C6×D4 [×2], C6×Q8 [×2], C3×C4○D4 [×8], C3×C4○D4 [×4], D8⋊C22, C6×M4(2), C3×C4○D8 [×4], C3×C8⋊C22 [×4], C3×C8.C22 [×4], C6×C4○D4 [×2], C3×D8⋊C22
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C22×D4, C6×D4 [×6], C23×C6, D8⋊C22, D4×C2×C6, C3×D8⋊C22
Generators and relations
G = < a,b,c,d,e | a3=b8=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd=b5, be=eb, dcd=ece=b4c, de=ed >
►On 48 points
(1 39 18)(2 40 19)(3 33 20)(4 34 21)(5 35 22)(6 36 23)(7 37 24)(8 38 17)(9 47 26)(10 48 27)(11 41 28)(12 42 29)(13 43 30)(14 44 31)(15 45 32)(16 46 25)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 44)(42 43)(45 48)(46 47)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
G:=sub<Sym(48)| (1,39,18)(2,40,19)(3,33,20)(4,34,21)(5,35,22)(6,36,23)(7,37,24)(8,38,17)(9,47,26)(10,48,27)(11,41,28)(12,42,29)(13,43,30)(14,44,31)(15,45,32)(16,46,25), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)>;
G:=Group( (1,39,18)(2,40,19)(3,33,20)(4,34,21)(5,35,22)(6,36,23)(7,37,24)(8,38,17)(9,47,26)(10,48,27)(11,41,28)(12,42,29)(13,43,30)(14,44,31)(15,45,32)(16,46,25), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,18)(19,24)(20,23)(21,22)(25,26)(27,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)(45,48)(46,47), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35) );
G=PermutationGroup([(1,39,18),(2,40,19),(3,33,20),(4,34,21),(5,35,22),(6,36,23),(7,37,24),(8,38,17),(9,47,26),(10,48,27),(11,41,28),(12,42,29),(13,43,30),(14,44,31),(15,45,32),(16,46,25)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,18),(19,24),(20,23),(21,22),(25,26),(27,32),(28,31),(29,30),(33,36),(34,35),(37,40),(38,39),(41,44),(42,43),(45,48),(46,47)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 |
| 71 | 0 | 0 | 72 |
| 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 71 | 0 | 72 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 46 | 46 | 0 |
| 27 | 0 | 46 | 27 |
| 0 | 0 | 27 | 46 |
| 0 | 0 | 54 | 46 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,1,72,71,0,0,1,0,1,0,0,0,0,1,0,72],[0,0,1,0,1,1,72,71,1,0,0,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,72,0,1,1,0,72],[0,27,0,0,46,0,0,0,46,46,27,54,0,27,46,46] >;
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | D8⋊C22 | C3×D8⋊C22 |
| kernel | C3×D8⋊C22 | C6×M4(2) | C3×C4○D8 | C3×C8⋊C22 | C3×C8.C22 | C6×C4○D4 | D8⋊C22 | C2×M4(2) | C4○D8 | C8⋊C22 | C8.C22 | C2×C4○D4 | C2×C12 | C22×C6 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 8 | 8 | 8 | 4 | 3 | 1 | 6 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_3\times D_8\rtimes C_2^2
% in TeX
G:=Group("C3xD8:C2^2"); // GroupNames label
G:=SmallGroup(192,1464);
// by ID
G=gap.SmallGroup(192,1464);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,2102,360,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d=b^5,b*e=e*b,d*c*d=e*c*e=b^4*c,d*e=e*d>;
// generators/relations