direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×M4(2).8C22, C4.50(C6×D4), (C6×D4).19C4, (C2×D4).7C12, C4.D4⋊6C6, (C2×C12).517D4, C12.457(C2×D4), C4.10D4⋊6C6, (C2×M4(2))⋊9C6, C23.5(C2×C12), (C22×C4).7C12, (C6×M4(2))⋊27C2, (C22×C12).12C4, M4(2).8(C2×C6), (C2×C12).608C23, (C6×D4).285C22, (C6×Q8).249C22, C12.115(C22⋊C4), C22.10(C22×C12), (C22×C12).409C22, (C3×M4(2)).42C22, (C2×C4).6(C2×C12), (C2×C4○D4).9C6, (C2×C12).19(C2×C4), (C6×C4○D4).17C2, (C2×D4).43(C2×C6), (C2×C4).121(C3×D4), C2.16(C6×C22⋊C4), C4.22(C3×C22⋊C4), (C2×C4).3(C22×C6), (C2×Q8).46(C2×C6), (C3×C4.D4)⋊13C2, C6.104(C2×C22⋊C4), (C22×C4).33(C2×C6), (C22×C6).12(C2×C4), C22.3(C3×C22⋊C4), (C3×C4.10D4)⋊13C2, (C2×C6).30(C22⋊C4), (C2×C6).163(C22×C4), SmallGroup(192,846)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M4(2).8C22
G = < a,b,c,d,e | a3=b8=c2=d2=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b5, dbd=bc, cd=dc, ece-1=b4c, ede-1=b4cd >
Subgroups: 242 in 150 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, M4(2).8C22, C3×C4.D4, C3×C4.10D4, C6×M4(2), C6×C4○D4, C3×M4(2).8C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C3×C22⋊C4, C22×C12, C6×D4, M4(2).8C22, C6×C22⋊C4, C3×M4(2).8C22
►On 48 points
(1 33 22)(2 34 23)(3 35 24)(4 36 17)(5 37 18)(6 38 19)(7 39 20)(8 40 21)(9 44 30)(10 45 31)(11 46 32)(12 47 25)(13 48 26)(14 41 27)(15 42 28)(16 43 29)
(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)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)
(1 41)(2 46)(3 47)(4 44)(5 45)(6 42)(7 43)(8 48)(9 17)(10 18)(11 23)(12 24)(13 21)(14 22)(15 19)(16 20)(25 35)(26 40)(27 33)(28 38)(29 39)(30 36)(31 37)(32 34)
(1 8 3 2 5 4 7 6)(9 16 11 10 13 12 15 14)(17 20 19 22 21 24 23 18)(25 28 27 30 29 32 31 26)(33 40 35 34 37 36 39 38)(41 44 43 46 45 48 47 42)
G:=sub<Sym(48)| (1,33,22)(2,34,23)(3,35,24)(4,36,17)(5,37,18)(6,38,19)(7,39,20)(8,40,21)(9,44,30)(10,45,31)(11,46,32)(12,47,25)(13,48,26)(14,41,27)(15,42,28)(16,43,29), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (1,41)(2,46)(3,47)(4,44)(5,45)(6,42)(7,43)(8,48)(9,17)(10,18)(11,23)(12,24)(13,21)(14,22)(15,19)(16,20)(25,35)(26,40)(27,33)(28,38)(29,39)(30,36)(31,37)(32,34), (1,8,3,2,5,4,7,6)(9,16,11,10,13,12,15,14)(17,20,19,22,21,24,23,18)(25,28,27,30,29,32,31,26)(33,40,35,34,37,36,39,38)(41,44,43,46,45,48,47,42)>;
G:=Group( (1,33,22)(2,34,23)(3,35,24)(4,36,17)(5,37,18)(6,38,19)(7,39,20)(8,40,21)(9,44,30)(10,45,31)(11,46,32)(12,47,25)(13,48,26)(14,41,27)(15,42,28)(16,43,29), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48), (1,41)(2,46)(3,47)(4,44)(5,45)(6,42)(7,43)(8,48)(9,17)(10,18)(11,23)(12,24)(13,21)(14,22)(15,19)(16,20)(25,35)(26,40)(27,33)(28,38)(29,39)(30,36)(31,37)(32,34), (1,8,3,2,5,4,7,6)(9,16,11,10,13,12,15,14)(17,20,19,22,21,24,23,18)(25,28,27,30,29,32,31,26)(33,40,35,34,37,36,39,38)(41,44,43,46,45,48,47,42) );
G=PermutationGroup([[(1,33,22),(2,34,23),(3,35,24),(4,36,17),(5,37,18),(6,38,19),(7,39,20),(8,40,21),(9,44,30),(10,45,31),(11,46,32),(12,47,25),(13,48,26),(14,41,27),(15,42,28),(16,43,29)], [(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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48)], [(1,41),(2,46),(3,47),(4,44),(5,45),(6,42),(7,43),(8,48),(9,17),(10,18),(11,23),(12,24),(13,21),(14,22),(15,19),(16,20),(25,35),(26,40),(27,33),(28,38),(29,39),(30,36),(31,37),(32,34)], [(1,8,3,2,5,4,7,6),(9,16,11,10,13,12,15,14),(17,20,19,22,21,24,23,18),(25,28,27,30,29,32,31,26),(33,40,35,34,37,36,39,38),(41,44,43,46,45,48,47,42)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | D4 | C3×D4 | M4(2).8C22 | C3×M4(2).8C22 |
| kernel | C3×M4(2).8C22 | C3×C4.D4 | C3×C4.10D4 | C6×M4(2) | C6×C4○D4 | M4(2).8C22 | C22×C12 | C6×D4 | C4.D4 | C4.10D4 | C2×M4(2) | C2×C4○D4 | C22×C4 | C2×D4 | C2×C12 | C2×C4 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 8 | 8 | 4 | 8 | 2 | 4 |
Matrix representation of C3×M4(2).8C22 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 46 | 14 |
| 0 | 1 | 0 | 14 |
| 72 | 72 | 0 | 72 |
| 0 | 71 | 0 | 72 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 71 | 1 |
| 0 | 46 | 1 | 59 |
| 0 | 46 | 0 | 59 |
| 27 | 27 | 0 | 27 |
| 0 | 54 | 0 | 27 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,0,72,0,1,1,72,71,46,0,0,0,14,14,72,72],[1,0,0,0,0,1,0,0,0,0,72,0,1,1,0,72],[0,1,0,0,1,0,0,0,1,1,72,71,0,0,0,1],[0,0,27,0,46,46,27,54,1,0,0,0,59,59,27,27] >;
C3×M4(2).8C22 in GAP, Magma, Sage, TeX
C_3\times M_4(2)._8C_2^2
% in TeX
G:=Group("C3xM4(2).8C2^2"); // GroupNames label
G:=SmallGroup(192,846);
// by ID
G=gap.SmallGroup(192,846);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,520,4204,3036,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^5,d*b*d=b*c,c*d=d*c,e*c*e^-1=b^4*c,e*d*e^-1=b^4*c*d>;
// generators/relations