p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8o2M4(2), C4.5C42, M4(2):5C4, C22.5C42, C42.60C22, C8o(C4:C4), (C2xC8):9C4, (C4xC8):14C2, C8o(C8:C4), C4:C4.10C4, C8o(C22:C4), C8.12(C2xC4), C8:C4:13C2, (C2xC8)oM4(2), C8o(C2xM4(2)), C2.1(C8oD4), C22:C4.6C4, C2.7(C2xC42), C8o(C42:C2), C4.34(C22xC4), (C22xC8).15C2, (C2xC8).99C22, C23.15(C2xC4), (C2xC4).143C23, C42:C2.14C2, (C2xM4(2)).17C2, C22.19(C22xC4), (C22xC4).106C22, (C2xC8)o(C8:C4), (C2xC4).34(C2xC4), (C2xC8)o(C42:C2), SmallGroup(64,86)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8o2M4(2)
G = < a,b,c | a8=c2=1, b4=a4, ab=ba, ac=ca, cbc=a4b >
Subgroups: 73 in 65 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C23, C42, C22:C4, C4:C4, C2xC8, C2xC8, M4(2), C22xC4, C4xC8, C8:C4, C42:C2, C22xC8, C2xM4(2), C8o2M4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, C42, C22xC4, C2xC42, C8oD4, C8o2M4(2)
►On 32 points
(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)
(1 23 25 16 5 19 29 12)(2 24 26 9 6 20 30 13)(3 17 27 10 7 21 31 14)(4 18 28 11 8 22 32 15)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
G:=sub<Sym(32)| (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), (1,23,25,16,5,19,29,12)(2,24,26,9,6,20,30,13)(3,17,27,10,7,21,31,14)(4,18,28,11,8,22,32,15), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)>;
G:=Group( (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), (1,23,25,16,5,19,29,12)(2,24,26,9,6,20,30,13)(3,17,27,10,7,21,31,14)(4,18,28,11,8,22,32,15), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24) );
G=PermutationGroup([[(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)], [(1,23,25,16,5,19,29,12),(2,24,26,9,6,20,30,13),(3,17,27,10,7,21,31,14),(4,18,28,11,8,22,32,15)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24)]])
C8o2M4(2) is a maximal subgroup of
M4(2).C8 M5(2):7C4 C23.5C42 C8.(C4:C4) M4(2).3Q8 M4(2).24D4 C4.10D4:3C4 C4.D4:3C4 M4(2).5Q8 M4(2).6Q8 M4(2).27D4 M4(2).30D4 M4(2).31D4 M4(2).32D4 M4(2).33D4 C4:C4.7C8 M4(2).1C8 C8.12M4(2) C8.19M4(2) M4(2)o2M4(2) C42.262C23 C42.678C23 C42.264C23 C42.265C23 M4(2):22D4 C42.275C23 C42.276C23 C42.277C23 C42.278C23 C42.279C23 C42.280C23 C42.281C23 C42.283C23 M4(2)oD8 C42.286C23 C42.287C23 M4(2):9Q8 C42.291C23 C42.292C23 C42.293C23 C42.294C23 C42.366C23 C42.367C23 M4(2).20D4 M4(2):3Q8 M4(2):4Q8 C42.385C23 C42.387C23 C42.389C23 C42.390C23 Dic5.C42
D2p.C42: Q8.C42 D4.3C42 C4xC8oD4 D4.5C42 D6.C42 D6.4C42 D10.5C42 D10.7C42 ...
C4p.C42: C8.16C42 C8.14C42 C8.5C42 C16o2M5(2) C8.23C42 C12.5C42 C12.12C42 C12.7C42 ...
C8:pD4:C2: M4(2):10D4 M4(2):11D4 C42.386C23 C42.388C23 C42.391C23 ...
C8o2M4(2) is a maximal quotient of
C8xM4(2) C82:C2 C8:9M4(2) C23.27C42 C82:15C2 C82:2C2 C23.29C42 (C4xC8):12C4 C8xC22:C4 C23.36C42 C23.17C42 C8xC4:C4 C4:C8:13C4 C4:C8:14C4 Dic5.C42
C4p.C42: C2.C43 C12.5C42 C12.12C42 C12.7C42 C20.35C42 C20.42C42 C20.37C42 D10.C42 ...
C42.D2p: C42.379D4 C42.45Q8 D6.C42 D6.4C42 D10.5C42 D10.7C42 D14.C42 D14.4C42 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 8A | ··· | 8H | 8I | ··· | 8T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8oD4 |
| kernel | C8o2M4(2) | C4xC8 | C8:C4 | C42:C2 | C22xC8 | C2xM4(2) | C22:C4 | C4:C4 | C2xC8 | M4(2) | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C8o2M4(2) ►in GL3(F17) generated by
| 13 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 15 |
| 1 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 8 | 0 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [13,0,0,0,15,0,0,0,15],[1,0,0,0,0,8,0,8,0],[16,0,0,0,1,0,0,0,16] >;
C8o2M4(2) in GAP, Magma, Sage, TeX
C_8\circ_2M_4(2)
% in TeX
G:=Group("C8o2M4(2)"); // GroupNames label
G:=SmallGroup(64,86);
// by ID
G=gap.SmallGroup(64,86);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,332,117]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^4=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b>;
// generators/relations