p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.313C23, M4(2).3C23, 2- (1+4).7C22, 2+ (1+4).9C22, C4○D4⋊15D4, (C2×D4)⋊22D4, (C2×Q8)⋊18D4, C4○(D4⋊4D4), D4⋊4D4⋊8C2, D4.52(C2×D4), C4⋊Q8⋊51C22, C4≀C2⋊11C22, Q8.52(C2×D4), C4○(D4.8D4), C4○(D4.9D4), D4.8D4⋊8C2, D4.9D4⋊8C2, C8⋊C22⋊6C22, (C2×C4).10C24, C4○D4.5C23, C23.21(C2×D4), C4.55(C22×D4), C4.130C22≀C2, C4○(D4.10D4), D4.10D4⋊8C2, D8⋊C22⋊5C2, C4⋊1D4⋊31C22, C2.C25⋊1C2, (C2×C42)⋊38C22, C22.5C22≀C2, (C2×D4).35C23, (C22×C4).110D4, C4.D4⋊8C22, C8.C22⋊7C22, (C2×Q8).27C23, C4.10D4⋊8C22, C4.4D4⋊51C22, (C2×M4(2))⋊9C22, C22.34(C22×D4), C22.26C24⋊1C2, (C22×C4).969C23, M4(2).8C22⋊1C2, (C2×C4≀C2)⋊8C2, (C2×C4○D4)⋊6C22, C2.55(C2×C22≀C2), (C2×C4).1097(C2×D4), SmallGroup(128,1750)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 708 in 362 conjugacy classes, 106 normal (26 characteristic)
C1, C2, C2 [×10], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×33], D4 [×4], D4 [×36], Q8 [×4], Q8 [×10], C23, C23 [×2], C23 [×7], C42 [×2], C42, C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×D4 [×23], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×8], C4○D4 [×40], C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×8], C2×C42, C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C2×M4(2) [×2], C4○D8 [×4], C8⋊C22 [×4], C8⋊C22 [×2], C8.C22 [×4], C8.C22 [×2], C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4 [×6], 2+ (1+4) [×2], 2+ (1+4) [×4], 2- (1+4) [×2], 2- (1+4) [×2], M4(2).8C22, C2×C4≀C2 [×2], D4⋊4D4 [×2], D4.8D4 [×2], D4.9D4 [×2], D4.10D4 [×2], C22.26C24, D8⋊C22 [×2], C2.C25, C42.313C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C42.313C23
Generators and relations
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, cac=ab=ba, dad=a-1, ae=ea, cbc=dbd=b-1, be=eb, dcd=b-1c, ce=ec, de=ed >
►On 16 points - transitive group 16T264
(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 8 6 7)(9 12 11 10)(13 16 15 14)
(1 9)(2 11)(3 12)(4 10)(5 15)(6 13)(7 14)(8 16)
(3 4)(7 8)(9 12)(10 11)(13 16)(14 15)
(1 6 2 5)(3 8 4 7)(9 13 11 15)(10 14 12 16)
G:=sub<Sym(16)| (9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,16,15,14), (1,9)(2,11)(3,12)(4,10)(5,15)(6,13)(7,14)(8,16), (3,4)(7,8)(9,12)(10,11)(13,16)(14,15), (1,6,2,5)(3,8,4,7)(9,13,11,15)(10,14,12,16)>;
G:=Group( (9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,16,15,14), (1,9)(2,11)(3,12)(4,10)(5,15)(6,13)(7,14)(8,16), (3,4)(7,8)(9,12)(10,11)(13,16)(14,15), (1,6,2,5)(3,8,4,7)(9,13,11,15)(10,14,12,16) );
G=PermutationGroup([(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(13,16,15,14)], [(1,9),(2,11),(3,12),(4,10),(5,15),(6,13),(7,14),(8,16)], [(3,4),(7,8),(9,12),(10,11),(13,16),(14,15)], [(1,6,2,5),(3,8,4,7),(9,13,11,15),(10,14,12,16)])
G:=TransitiveGroup(16,264);
►On 16 points - transitive group 16T303
(9 10 11 12)(13 14 15 16)
(1 4 2 3)(5 7 6 8)(9 12 11 10)(13 16 15 14)
(1 9)(2 11)(3 12)(4 10)(5 13)(6 15)(7 14)(8 16)
(1 5)(2 6)(3 7)(4 8)(9 16)(10 15)(11 14)(12 13)
(1 3 2 4)(5 7 6 8)(9 12 11 10)(13 14 15 16)
G:=sub<Sym(16)| (9,10,11,12)(13,14,15,16), (1,4,2,3)(5,7,6,8)(9,12,11,10)(13,16,15,14), (1,9)(2,11)(3,12)(4,10)(5,13)(6,15)(7,14)(8,16), (1,5)(2,6)(3,7)(4,8)(9,16)(10,15)(11,14)(12,13), (1,3,2,4)(5,7,6,8)(9,12,11,10)(13,14,15,16)>;
G:=Group( (9,10,11,12)(13,14,15,16), (1,4,2,3)(5,7,6,8)(9,12,11,10)(13,16,15,14), (1,9)(2,11)(3,12)(4,10)(5,13)(6,15)(7,14)(8,16), (1,5)(2,6)(3,7)(4,8)(9,16)(10,15)(11,14)(12,13), (1,3,2,4)(5,7,6,8)(9,12,11,10)(13,14,15,16) );
G=PermutationGroup([(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,7,6,8),(9,12,11,10),(13,16,15,14)], [(1,9),(2,11),(3,12),(4,10),(5,13),(6,15),(7,14),(8,16)], [(1,5),(2,6),(3,7),(4,8),(9,16),(10,15),(11,14),(12,13)], [(1,3,2,4),(5,7,6,8),(9,12,11,10),(13,14,15,16)])
G:=TransitiveGroup(16,303);
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 1 | 0 | 4 | 3 |
| 2 | 0 | 2 | 0 |
| 0 | 0 | 1 | 0 |
| 3 | 1 | 0 | 0 |
| 3 | 2 | 3 | 4 |
| 0 | 1 | 2 | 1 |
| 2 | 1 | 4 | 1 |
| 1 | 1 | 0 | 2 |
| 0 | 3 | 2 | 2 |
| 0 | 0 | 0 | 4 |
| 3 | 1 | 0 | 4 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 2 | 2 |
| 0 | 4 | 3 | 4 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [1,2,0,3,0,0,0,1,4,2,1,0,3,0,0,0],[3,0,2,1,2,1,1,1,3,2,4,0,4,1,1,2],[0,0,3,0,3,0,1,4,2,0,0,0,2,4,4,0],[4,0,0,0,0,4,0,0,2,3,1,0,2,4,0,1],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4O | 4P | 8A | 8B | 8C | 8D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 8 | 8 | 8 | 8 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | C42.313C23 |
| kernel | C42.313C23 | M4(2).8C22 | C2×C4≀C2 | D4⋊4D4 | D4.8D4 | D4.9D4 | D4.10D4 | C22.26C24 | D8⋊C22 | C2.C25 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{313}C_2^3 % in TeX
G:=Group("C4^2.313C2^3"); // GroupNames label
G:=SmallGroup(128,1750);
// by ID
G=gap.SmallGroup(128,1750);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,248,2804,1411,718,172,2028]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b^2,c*a*c=a*b=b*a,d*a*d=a^-1,a*e=e*a,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b^-1*c,c*e=e*c,d*e=e*d>;
// generators/relations