p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2).4D4, (C2×C8).43D4, (C2×D4).96D4, (C22×D8)⋊3C2, (C2×Q8).87D4, C4.13C22≀C2, C4.33(C4⋊1D4), C4.C42⋊8C2, C2.18(D4.4D4), C23.272(C4○D4), C22.61(C4⋊D4), C2.23(C23⋊2D4), (C22×C8).110C22, (C22×C4).716C23, (C22×D4).68C22, (C2×M4(2)).21C22, (C2×C8⋊C22)⋊4C2, (C2×C4).254(C2×D4), (C2×C4.D4)⋊3C2, (C22×C8)⋊C2⋊18C2, (C2×C4○D4).53C22, SmallGroup(128,750)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).4D4
G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=a3b, dad=ab, cbc-1=a4b, bd=db, dcd=a4c3 >
Subgroups: 504 in 191 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×2], C4 [×2], C4, C22, C22 [×2], C22 [×21], C8 [×7], C2×C4 [×2], C2×C4 [×4], C2×C4 [×3], D4 [×16], Q8 [×2], C23, C23 [×13], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], D8 [×16], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×11], C2×Q8, C4○D4 [×4], C24 [×2], C22⋊C8 [×2], C4.D4 [×4], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×D8 [×8], C2×SD16 [×2], C8⋊C22 [×8], C22×D4 [×2], C2×C4○D4, C4.C42, (C22×C8)⋊C2, C2×C4.D4 [×2], C22×D8, C2×C8⋊C22 [×2], M4(2).4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C4⋊1D4, C23⋊2D4, D4.4D4 [×2], M4(2).4D4
Character table of M4(2).4D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
ρ9 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ15 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ16 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | orthogonal lifted from D4 |
ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | complex lifted from C4○D4 |
ρ22 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | complex lifted from C4○D4 |
ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
(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 22)(2 19)(3 24)(4 21)(5 18)(6 23)(7 20)(8 17)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)
(1 25 24 9 5 29 20 13)(2 10 17 30 6 14 21 26)(3 27 18 11 7 31 22 15)(4 12 19 32 8 16 23 28)
(1 32)(2 11)(3 30)(4 9)(5 28)(6 15)(7 26)(8 13)(10 18)(12 24)(14 22)(16 20)(17 27)(19 25)(21 31)(23 29)
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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26), (1,25,24,9,5,29,20,13)(2,10,17,30,6,14,21,26)(3,27,18,11,7,31,22,15)(4,12,19,32,8,16,23,28), (1,32)(2,11)(3,30)(4,9)(5,28)(6,15)(7,26)(8,13)(10,18)(12,24)(14,22)(16,20)(17,27)(19,25)(21,31)(23,29)>;
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), (1,22)(2,19)(3,24)(4,21)(5,18)(6,23)(7,20)(8,17)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26), (1,25,24,9,5,29,20,13)(2,10,17,30,6,14,21,26)(3,27,18,11,7,31,22,15)(4,12,19,32,8,16,23,28), (1,32)(2,11)(3,30)(4,9)(5,28)(6,15)(7,26)(8,13)(10,18)(12,24)(14,22)(16,20)(17,27)(19,25)(21,31)(23,29) );
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)], [(1,22),(2,19),(3,24),(4,21),(5,18),(6,23),(7,20),(8,17),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26)], [(1,25,24,9,5,29,20,13),(2,10,17,30,6,14,21,26),(3,27,18,11,7,31,22,15),(4,12,19,32,8,16,23,28)], [(1,32),(2,11),(3,30),(4,9),(5,28),(6,15),(7,26),(8,13),(10,18),(12,24),(14,22),(16,20),(17,27),(19,25),(21,31),(23,29)])
Matrix representation of M4(2).4D4 ►in GL6(𝔽17)
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 2 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 16 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 |
0 | 0 | 0 | 0 | 0 | 16 |
13 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 6 | 11 |
0 | 0 | 0 | 0 | 3 | 11 |
0 | 0 | 0 | 11 | 0 | 0 |
0 | 0 | 14 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
13 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 6 | 11 | 0 | 0 |
0 | 0 | 3 | 11 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 11 |
0 | 0 | 0 | 0 | 14 | 0 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,16,0,0,0,0,0,2,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,6,3,0,0,0,0,11,11,0,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,6,3,0,0,0,0,11,11,0,0,0,0,0,0,0,14,0,0,0,0,11,0] >;
M4(2).4D4 in GAP, Magma, Sage, TeX
M_4(2)._4D_4
% in TeX
G:=Group("M4(2).4D4");
// GroupNames label
G:=SmallGroup(128,750);
// by ID
G=gap.SmallGroup(128,750);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,718,172,4037,2028,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=a^4*c^3>;
// generators/relations
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