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G = M4(2)⋊21D4  order 128 = 27

8th semidirect product of M4(2) and D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: M4(2)⋊21D4, C22≀C2.C4, (C2×D4).83D4, C24.9(C2×C4), (C2×Q8).75D4, C22.12(C4×D4), (C22×C4).71D4, C4.101C22≀C2, C4.11(C4⋊D4), Q8○M4(2)⋊10C2, (C22×C4).37C23, C23.71(C22×C4), M4(2)⋊4C4⋊16C2, C23.17(C22⋊C4), C22.29C24.3C2, (C22×D4).35C22, C42⋊C2.35C22, C4.15(C22.D4), C2.53(C23.23D4), (C2×M4(2)).196C22, M4(2).8C22⋊11C2, (C2×D4).92(C2×C4), (C2×C4).248(C2×D4), C22⋊C4.6(C2×C4), (C2×C4.D4)⋊21C2, (C2×C4).332(C4○D4), (C2×C4).21(C22⋊C4), (C2×C4○D4).31C22, C22.52(C2×C22⋊C4), SmallGroup(128,646)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — M4(2)⋊21D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2)⋊21D4
Lower central
C1 — C2 — C23 — M4(2)⋊21D4
Upper central
C1 — C2 — C22×C4 — M4(2)⋊21D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2)⋊21D4

Generators and relations for M4(2)⋊21D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a5b, dad=ab, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 396 in 170 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4.D4, C4.10D4, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C2×M4(2), C2×M4(2), C8○D4, C22×D4, C2×C4○D4, M4(2)⋊4C4, C2×C4.D4, M4(2).8C22, C22.29C24, Q8○M4(2), M4(2)⋊21D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, M4(2)⋊21D4

Character table of M4(2)⋊21D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222448822224488444444448888
ρ111111111111111111111111111111    trivial
ρ211111-1-1-1-11111-1-11111-1-1-1-1111-11-1    linear of order 2
ρ31111111-1-1111111-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ411111-1-1111111-1-1-1-1-1-11111-1-11-11-1    linear of order 2
ρ511111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ611111-1-1-1-11111-1-111-1-11111-1-1-11-11    linear of order 2
ρ71111111-1-1111111-1-111111111-1-1-1-1    linear of order 2
ρ811111-1-1111111-1-1-1-111-1-1-1-111-11-11    linear of order 2
ρ9111111-1-11-1-1-1-11-1-11-iii-ii-ii-ii-i-ii    linear of order 4
ρ10111111-1-11-1-1-1-11-1-11i-i-ii-ii-ii-iii-i    linear of order 4
ρ1111111-11-11-1-1-1-1-111-1i-ii-ii-i-iiii-i-i    linear of order 4
ρ1211111-11-11-1-1-1-1-111-1-ii-ii-iii-i-i-iii    linear of order 4
ρ1311111-111-1-1-1-1-1-11-11i-ii-ii-i-ii-i-iii    linear of order 4
ρ1411111-111-1-1-1-1-1-11-11-ii-ii-iii-iii-i-i    linear of order 4
ρ15111111-11-1-1-1-1-11-11-1-iii-ii-ii-i-iii-i    linear of order 4
ρ16111111-11-1-1-1-1-11-11-1i-i-ii-ii-iii-i-ii    linear of order 4
ρ17222-2-20000-22-22000000-2-222000000    orthogonal lifted from D4
ρ1822-2-220000-222-200002-200002-20000    orthogonal lifted from D4
ρ19222-2-20000-22-2200000022-2-2000000    orthogonal lifted from D4
ρ2022-22-2-220022-2-22-200000000000000    orthogonal lifted from D4
ρ2122-22-22-20022-2-2-2200000000000000    orthogonal lifted from D4
ρ2222-22-22200-2-222-2-200000000000000    orthogonal lifted from D4
ρ2322-2-220000-222-20000-220000-220000    orthogonal lifted from D4
ρ2422-22-2-2-200-2-2222200000000000000    orthogonal lifted from D4
ρ25222-2-200002-22-2000000-2i2i2i-2i000000    complex lifted from C4○D4
ρ2622-2-2200002-2-220000-2i-2i00002i2i0000    complex lifted from C4○D4
ρ27222-2-200002-22-20000002i-2i-2i2i000000    complex lifted from C4○D4
ρ2822-2-2200002-2-2200002i2i0000-2i-2i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of M4(2)⋊21D4
►On 16 points - transitive group 16T216
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)

(2 12 6 16)(3 7)(4 10 8 14)(11 15)

(1 13)(2 4)(3 11)(5 9)(6 8)(7 15)(10 16)(12 14)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (2,12,6,16)(3,7)(4,10,8,14)(11,15), (1,13)(2,4)(3,11)(5,9)(6,8)(7,15)(10,16)(12,14)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (2,12,6,16)(3,7)(4,10,8,14)(11,15), (1,13)(2,4)(3,11)(5,9)(6,8)(7,15)(10,16)(12,14) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10)], [(2,12,6,16),(3,7),(4,10,8,14),(11,15)], [(1,13),(2,4),(3,11),(5,9),(6,8),(7,15),(10,16),(12,14)]])
 
G:=TransitiveGroup(16,216);
 

►On 16 points - transitive group 16T278
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 5)(3 7)(10 14)(12 16)

(1 9 5 13)(2 14)(3 15 7 11)(4 12)(6 10)(8 16)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 16)(12 14)(13 15)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(10,14)(12,16), (1,9,5,13)(2,14)(3,15,7,11)(4,12)(6,10)(8,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,14)(13,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(10,14)(12,16), (1,9,5,13)(2,14)(3,15,7,11)(4,12)(6,10)(8,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,14)(13,15) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(3,7),(10,14),(12,16)], [(1,9,5,13),(2,14),(3,15,7,11),(4,12),(6,10),(8,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,16),(12,14),(13,15)]])
 
G:=TransitiveGroup(16,278);
 

Matrix representation of M4(2)⋊21D4 ►in GL8(ℤ)

000000-10
0000000-1
0000-1000
00000-100
01000000
-10000000
00010000
00-100000
,
00-100000
000-10000
-10000000
0-1000000
00000010
00000001
00001000
00000100
,
0-1000000
10000000
00010000
00-100000
00000001
000000-10
00000-100
00001000
,
10000000
0-1000000
00100000
000-10000
000000-10
00000001
0000-1000
00000100

G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0] >;
 

M4(2)⋊21D4 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_{21}D_4
 
% in TeX
 
G:=Group("M4(2):21D4");
 
// GroupNames label
 
G:=SmallGroup(128,646);
 
// by ID
 
G=gap.SmallGroup(128,646);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,521,2804,1411,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a^5*b,d*a*d=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of M4(2)⋊21D4 in TeX

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