direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×M4(2), C4.4C42, C42.9C4, C22.4C42, C42.59C22, C8⋊6(C2×C4), (C4×C8)⋊13C2, C4○2(C8⋊C4), C8⋊C4⋊12C2, C2.6(C2×C42), C42○(C8⋊C4), (C2×C8).98C22, C23.27(C2×C4), C4.33(C22×C4), (C22×C4).10C4, (C2×C42).12C2, C2.2(C2×M4(2)), C42○(C2×M4(2)), (C2×C4).142C23, (C2×M4(2)).16C2, C22.18(C22×C4), (C22×C4).105C22, (C2×C4).84(C2×C4), SmallGroup(64,85)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4×M4(2)
G = < a,b,c | a4=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Subgroups: 81 in 71 conjugacy classes, 61 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C4×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C2×C42, C2×M4(2), C4×M4(2)
►On 32 points
(1 19 26 16)(2 20 27 9)(3 21 28 10)(4 22 29 11)(5 23 30 12)(6 24 31 13)(7 17 32 14)(8 18 25 15)
(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 26)(2 31)(3 28)(4 25)(5 30)(6 27)(7 32)(8 29)(9 24)(10 21)(11 18)(12 23)(13 20)(14 17)(15 22)(16 19)
G:=sub<Sym(32)| (1,19,26,16)(2,20,27,9)(3,21,28,10)(4,22,29,11)(5,23,30,12)(6,24,31,13)(7,17,32,14)(8,18,25,15), (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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19)>;
G:=Group( (1,19,26,16)(2,20,27,9)(3,21,28,10)(4,22,29,11)(5,23,30,12)(6,24,31,13)(7,17,32,14)(8,18,25,15), (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), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,24)(10,21)(11,18)(12,23)(13,20)(14,17)(15,22)(16,19) );
G=PermutationGroup([[(1,19,26,16),(2,20,27,9),(3,21,28,10),(4,22,29,11),(5,23,30,12),(6,24,31,13),(7,17,32,14),(8,18,25,15)], [(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)], [(1,26),(2,31),(3,28),(4,25),(5,30),(6,27),(7,32),(8,29),(9,24),(10,21),(11,18),(12,23),(13,20),(14,17),(15,22),(16,19)]])
C4×M4(2) is a maximal subgroup of
M4(2)⋊C8 C42.3Q8 C42.6Q8 C42.26D4 C42.388D4 C42.9Q8 C8⋊9M4(2) C82⋊15C2 C8⋊6M4(2) C42.47D4 C42.400D4 C42.401D4 D4⋊4M4(2) D4⋊5M4(2) Q8⋊5M4(2) C42.66D4 C42.405D4 C42.406D4 C42.407D4 C42.408D4 C42.376D4 M4(2)⋊1C8 C8⋊1M4(2) C8.6C42 C8⋊C4⋊17C4 C42.427D4 C42.430D4 M4(2)⋊12D4 C42.114D4 C42.115D4 M4(2)⋊13D4 M4(2)⋊7Q8 M4(2)⋊8Q8 C42.128D4 C8.5M4(2) M4(2)○2M4(2) C42.677C23 C42.259C23 C42.260C23 C42.261C23 M4(2)⋊23D4 M4(2).51D4 M4(2)⋊9Q8 C42.290C23 C42.292C23 C42.294C23 D4⋊6M4(2) Q8⋊6M4(2) C42.240D4 C42.241D4 C42.242D4 C42.243D4 C42.244D4 M4(2)⋊7D4 M4(2)⋊8D4 M4(2)⋊9D4 M4(2)⋊5Q8 M4(2)⋊6Q8 C42.255D4 C42.256D4 C42.259D4 C42.260D4 C42.261D4 C42.262D4
D2p.C42: D4.C42 D4.5C42 Dic3⋊5M4(2) D10.6C42 Dic7.C42 ...
C4×M4(2) is a maximal quotient of
C8⋊9M4(2) C82⋊15C2 C8⋊6M4(2) C23.28C42 C43.C2 C23.17C42 C4⋊C8⋊14C4
C42.D2p: C42.378D4 C43.7C2 Dic3⋊5M4(2) D10.6C42 Dic7.C42 ...
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| type | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) |
| kernel | C4×M4(2) | C4×C8 | C8⋊C4 | C2×C42 | C2×M4(2) | C42 | M4(2) | C22×C4 | C4 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 16 | 4 | 8 |
Matrix representation of C4×M4(2) ►in GL3(𝔽17) generated by
| 13 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 4 | 15 |
| 0 | 6 | 13 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 4 | 16 |
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[4,0,0,0,4,6,0,15,13],[1,0,0,0,1,4,0,0,16] >;
C4×M4(2) in GAP, Magma, Sage, TeX
C_4\times M_4(2)
% in TeX
G:=Group("C4xM4(2)"); // GroupNames label
G:=SmallGroup(64,85);
// by ID
G=gap.SmallGroup(64,85);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,103,650,117]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations