direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×M4(2).C4, M4(2).1C12, C8.4(C2×C12), C4.78(C6×D4), C8.C4⋊3C6, C24.39(C2×C4), C12.65(C4⋊C4), (C2×C12).44Q8, (C2×C12).523D4, C12.483(C2×D4), (C22×C6).5Q8, C23.5(C3×Q8), C22.2(C6×Q8), C4.29(C22×C12), (C3×M4(2)).2C4, (C2×M4(2)).2C6, (C2×C24).198C22, C12.187(C22×C4), (C2×C12).903C23, (C6×M4(2)).34C2, M4(2).10(C2×C6), (C22×C12).415C22, (C3×M4(2)).44C22, C2.16(C6×C4⋊C4), C4.16(C3×C4⋊C4), C6.72(C2×C4⋊C4), (C2×C4).7(C3×Q8), (C2×C8).17(C2×C6), (C2×C6).15(C2×Q8), (C2×C6).28(C4⋊C4), (C2×C4).26(C2×C12), (C2×C4).126(C3×D4), C22.12(C3×C4⋊C4), (C3×C8.C4)⋊12C2, (C2×C12).199(C2×C4), (C22×C4).39(C2×C6), (C2×C4).78(C22×C6), SmallGroup(192,863)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M4(2).C4
G = < a,b,c,d | a3=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >
Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C24, C2×C12, C2×C12, C22×C6, C8.C4, C2×M4(2), C2×M4(2), C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, M4(2).C4, C3×C8.C4, C6×M4(2), C6×M4(2), C3×M4(2).C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, M4(2).C4, C6×C4⋊C4, C3×M4(2).C4
►On 48 points
(1 29 21)(2 30 22)(3 31 23)(4 32 24)(5 25 17)(6 26 18)(7 27 19)(8 28 20)(9 39 44)(10 40 45)(11 33 46)(12 34 47)(13 35 48)(14 36 41)(15 37 42)(16 38 43)
(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 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)
(1 41 7 43 5 45 3 47)(2 48 8 42 6 44 4 46)(9 32 11 30 13 28 15 26)(10 31 12 29 14 27 16 25)(17 40 23 34 21 36 19 38)(18 39 24 33 22 35 20 37)
G:=sub<Sym(48)| (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,25,17)(6,26,18)(7,27,19)(8,28,20)(9,39,44)(10,40,45)(11,33,46)(12,34,47)(13,35,48)(14,36,41)(15,37,42)(16,38,43), (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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48), (1,41,7,43,5,45,3,47)(2,48,8,42,6,44,4,46)(9,32,11,30,13,28,15,26)(10,31,12,29,14,27,16,25)(17,40,23,34,21,36,19,38)(18,39,24,33,22,35,20,37)>;
G:=Group( (1,29,21)(2,30,22)(3,31,23)(4,32,24)(5,25,17)(6,26,18)(7,27,19)(8,28,20)(9,39,44)(10,40,45)(11,33,46)(12,34,47)(13,35,48)(14,36,41)(15,37,42)(16,38,43), (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), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48), (1,41,7,43,5,45,3,47)(2,48,8,42,6,44,4,46)(9,32,11,30,13,28,15,26)(10,31,12,29,14,27,16,25)(17,40,23,34,21,36,19,38)(18,39,24,33,22,35,20,37) );
G=PermutationGroup([[(1,29,21),(2,30,22),(3,31,23),(4,32,24),(5,25,17),(6,26,18),(7,27,19),(8,28,20),(9,39,44),(10,40,45),(11,33,46),(12,34,47),(13,35,48),(14,36,41),(15,37,42),(16,38,43)], [(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)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48)], [(1,41,7,43,5,45,3,47),(2,48,8,42,6,44,4,46),(9,32,11,30,13,28,15,26),(10,31,12,29,14,27,16,25),(17,40,23,34,21,36,19,38),(18,39,24,33,22,35,20,37)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 8A | ··· | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | Q8 | C3×D4 | C3×Q8 | C3×Q8 | M4(2).C4 | C3×M4(2).C4 |
| kernel | C3×M4(2).C4 | C3×C8.C4 | C6×M4(2) | M4(2).C4 | C3×M4(2) | C8.C4 | C2×M4(2) | M4(2) | C2×C12 | C2×C12 | C22×C6 | C2×C4 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 4 | 3 | 2 | 8 | 8 | 6 | 16 | 2 | 1 | 1 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C3×M4(2).C4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 46 | 54 | 0 | 0 |
| 59 | 27 | 0 | 0 |
| 59 | 27 | 0 | 1 |
| 46 | 27 | 27 | 0 |
| 72 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 72 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 71 | 0 |
| 0 | 0 | 72 | 1 |
| 60 | 0 | 72 | 0 |
| 60 | 27 | 72 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,59,59,46,54,27,27,27,0,0,0,27,0,0,1,0],[72,72,72,0,0,1,0,0,0,0,1,0,0,0,0,72],[1,0,60,60,0,0,0,27,71,72,72,72,0,1,0,0] >;
C3×M4(2).C4 in GAP, Magma, Sage, TeX
C_3\times M_4(2).C_4
% in TeX
G:=Group("C3xM4(2).C4"); // GroupNames label
G:=SmallGroup(192,863);
// by ID
G=gap.SmallGroup(192,863);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1059,4204,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations