direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2xQ8oM4(2), C4.21C25, C8.22C24, M4(2):14C23, D4o(C2xM4(2)), M4(2)o(C2xD4), Q8o(C2xM4(2)), M4(2)o(C2xQ8), (C2xC8):11C23, C8oD4:20C22, C4o(Q8oM4(2)), C24.90(C2xC4), C4.43(C23xC4), C2.15(C24xC4), M4(2)o2(C4oD4), (C2xC4).605C24, (C22xC8):58C22, C4oD4.36C23, D4.27(C22xC4), (C22xD4).45C4, C22.8(C23xC4), (C22xQ8).35C4, Q8.28(C22xC4), M4(2)o2(C2xM4(2)), (C22xM4(2)):28C2, (C2xM4(2)):80C22, (C23xC4).621C22, C23.112(C22xC4), (C22xC4).1220C23, (C2xC8oD4):28C2, C4oD4o(C2xM4(2)), M4(2)o(C2xC4oD4), (C2xQ8)o(C2xM4(2)), (C2xC4oD4).33C4, C4oD4.35(C2xC4), (C2xD4).240(C2xC4), (C2xQ8).215(C2xC4), (C2xM4(2))o(C2xM4(2)), (C2xC4).284(C22xC4), (C22xC4).373(C2xC4), (C22xC4oD4).28C2, (C2xC4oD4).334C22, (C2xM4(2))o(C2xC4oD4), SmallGroup(128,2304)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2xQ8oM4(2)
G = < a,b,c,d,e | a2=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
Subgroups: 812 in 730 conjugacy classes, 684 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C22xC8, C2xM4(2), C8oD4, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C22xM4(2), C2xC8oD4, Q8oM4(2), C22xC4oD4, C2xQ8oM4(2)
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C24, C23xC4, C25, Q8oM4(2), C24xC4, C2xQ8oM4(2)
►On 32 points
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)
(1 32 5 28)(2 25 6 29)(3 26 7 30)(4 27 8 31)(9 24 13 20)(10 17 14 21)(11 18 15 22)(12 19 16 23)
(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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
G:=sub<Sym(32)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30), (1,32,5,28)(2,25,6,29)(3,26,7,30)(4,27,8,31)(9,24,13,20)(10,17,14,21)(11,18,15,22)(12,19,16,23), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)>;
G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30), (1,32,5,28)(2,25,6,29)(3,26,7,30)(4,27,8,31)(9,24,13,20)(10,17,14,21)(11,18,15,22)(12,19,16,23), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31) );
G=PermutationGroup([[(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,21,5,17),(2,22,6,18),(3,23,7,19),(4,24,8,20),(9,31,13,27),(10,32,14,28),(11,25,15,29),(12,26,16,30)], [(1,32,5,28),(2,25,6,29),(3,26,7,30),(4,27,8,31),(9,24,13,20),(10,17,14,21),(11,18,15,22),(12,19,16,23)], [(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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2Q | 4A | 4B | 4C | 4D | 4E | ··· | 4R | 8A | ··· | 8AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | Q8oM4(2) |
| kernel | C2xQ8oM4(2) | C22xM4(2) | C2xC8oD4 | Q8oM4(2) | C22xC4oD4 | C22xD4 | C22xQ8 | C2xC4oD4 | C2 |
| # reps | 1 | 6 | 8 | 16 | 1 | 6 | 2 | 24 | 4 |
Matrix representation of C2xQ8oM4(2) ►in GL5(F17)
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 4 | 12 | 4 |
| 0 | 8 | 1 | 0 | 5 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 8 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 13 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 10 | 13 | 9 |
| 0 | 0 | 0 | 15 | 1 |
| 0 | 4 | 11 | 0 | 11 |
| 0 | 8 | 1 | 0 | 5 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 14 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,8,0,0,0,4,1,0,0,0,12,0,4,8,0,4,5,0,13],[1,0,0,0,0,0,13,0,0,0,0,16,4,0,0,0,0,0,4,0,0,0,0,13,13],[1,0,0,0,0,0,12,0,4,8,0,10,0,11,1,0,13,15,0,0,0,9,1,11,5],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,14,0,0,16] >;
C2xQ8oM4(2) in GAP, Magma, Sage, TeX
C_2\times Q_8\circ M_4(2)
% in TeX
G:=Group("C2xQ8oM4(2)"); // GroupNames label
G:=SmallGroup(128,2304);
// by ID
G=gap.SmallGroup(128,2304);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,723,2019,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations