direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×M4(2)⋊4C4, M4(2)⋊4C12, (C2×C24)⋊4C4, (C2×C8)⋊2C12, C12.54(C4⋊C4), (C2×C12).38Q8, (C2×C12).510D4, C22⋊C4.1C12, C23.7(C2×C12), (C2×C6).10C42, C22.3(C4×C12), (C3×M4(2))⋊10C4, C42⋊C2.3C6, (C2×M4(2)).9C6, (C6×M4(2)).21C2, C12.106(C22⋊C4), C6.28(C2.C42), (C22×C12).389C22, C4.5(C3×C4⋊C4), (C2×C4).3(C3×Q8), C22.6(C3×C4⋊C4), (C2×C6).23(C4⋊C4), (C2×C4).15(C2×C12), (C2×C4).115(C3×D4), (C3×C22⋊C4).2C4, C4.27(C3×C22⋊C4), (C2×C12).326(C2×C4), (C22×C6).18(C2×C4), (C22×C4).24(C2×C6), C22.9(C3×C22⋊C4), (C2×C6).72(C22⋊C4), C2.9(C3×C2.C42), (C3×C42⋊C2).17C2, SmallGroup(192,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M4(2)⋊4C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=bc, dcd-1=b4c >
Subgroups: 138 in 90 conjugacy classes, 54 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C22×C12, M4(2)⋊4C4, C3×C42⋊C2, C6×M4(2), C3×M4(2)⋊4C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C12, C3×C22⋊C4, C3×C4⋊C4, M4(2)⋊4C4, C3×C2.C42, C3×M4(2)⋊4C4
►On 48 points
(1 38 11)(2 39 12)(3 40 13)(4 33 14)(5 34 15)(6 35 16)(7 36 9)(8 37 10)(17 26 44)(18 27 45)(19 28 46)(20 29 47)(21 30 48)(22 31 41)(23 32 42)(24 25 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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 47)(10 44)(11 41)(12 46)(13 43)(14 48)(15 45)(16 42)(25 40)(26 37)(27 34)(28 39)(29 36)(30 33)(31 38)(32 35)
(2 23 6 19)(3 7)(4 21 8 17)(9 13)(10 44 14 48)(12 42 16 46)(18 22)(26 33 30 37)(27 31)(28 39 32 35)(36 40)(41 45)
G:=sub<Sym(48)| (1,38,11)(2,39,12)(3,40,13)(4,33,14)(5,34,15)(6,35,16)(7,36,9)(8,37,10)(17,26,44)(18,27,45)(19,28,46)(20,29,47)(21,30,48)(22,31,41)(23,32,42)(24,25,43), (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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,47)(10,44)(11,41)(12,46)(13,43)(14,48)(15,45)(16,42)(25,40)(26,37)(27,34)(28,39)(29,36)(30,33)(31,38)(32,35), (2,23,6,19)(3,7)(4,21,8,17)(9,13)(10,44,14,48)(12,42,16,46)(18,22)(26,33,30,37)(27,31)(28,39,32,35)(36,40)(41,45)>;
G:=Group( (1,38,11)(2,39,12)(3,40,13)(4,33,14)(5,34,15)(6,35,16)(7,36,9)(8,37,10)(17,26,44)(18,27,45)(19,28,46)(20,29,47)(21,30,48)(22,31,41)(23,32,42)(24,25,43), (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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,47)(10,44)(11,41)(12,46)(13,43)(14,48)(15,45)(16,42)(25,40)(26,37)(27,34)(28,39)(29,36)(30,33)(31,38)(32,35), (2,23,6,19)(3,7)(4,21,8,17)(9,13)(10,44,14,48)(12,42,16,46)(18,22)(26,33,30,37)(27,31)(28,39,32,35)(36,40)(41,45) );
G=PermutationGroup([[(1,38,11),(2,39,12),(3,40,13),(4,33,14),(5,34,15),(6,35,16),(7,36,9),(8,37,10),(17,26,44),(18,27,45),(19,28,46),(20,29,47),(21,30,48),(22,31,41),(23,32,42),(24,25,43)], [(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)], [(1,22),(2,19),(3,24),(4,21),(5,18),(6,23),(7,20),(8,17),(9,47),(10,44),(11,41),(12,46),(13,43),(14,48),(15,45),(16,42),(25,40),(26,37),(27,34),(28,39),(29,36),(30,33),(31,38),(32,35)], [(2,23,6,19),(3,7),(4,21,8,17),(9,13),(10,44,14,48),(12,42,16,46),(18,22),(26,33,30,37),(27,31),(28,39,32,35),(36,40),(41,45)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | ··· | 6H | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 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 | C3 | C4 | C4 | C4 | C6 | C6 | C12 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | M4(2)⋊4C4 | C3×M4(2)⋊4C4 |
| kernel | C3×M4(2)⋊4C4 | C3×C42⋊C2 | C6×M4(2) | M4(2)⋊4C4 | C3×C22⋊C4 | C2×C24 | C3×M4(2) | C42⋊C2 | C2×M4(2) | C22⋊C4 | C2×C8 | M4(2) | C2×C12 | C2×C12 | C2×C4 | C2×C4 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 4 | 8 | 8 | 8 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C3×M4(2)⋊4C4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 27 | 46 |
| 0 | 0 | 0 | 46 |
| 1 | 72 | 0 | 0 |
| 2 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 71 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 71 | 1 |
| 1 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 71 | 1 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,0,1,2,0,0,72,72,27,0,0,0,46,46,0,0],[72,71,0,0,0,1,0,0,0,0,72,71,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,72,71,0,0,1,1] >;
C3×M4(2)⋊4C4 in GAP, Magma, Sage, TeX
C_3\times M_4(2)\rtimes_4C_4
% in TeX
G:=Group("C3xM4(2):4C4"); // GroupNames label
G:=SmallGroup(192,150);
// by ID
G=gap.SmallGroup(192,150);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,2111,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b*c,d*c*d^-1=b^4*c>;
// generators/relations