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G = M4(2).47D4order 128 = 27

11st non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).47D4, (C2×D8)⋊3C4, C4.2(C4×D4), C4○D4.8D4, (C2×C8).29D4, (C2×SD16)⋊2C4, C4.96C22≀C2, Q8○M4(2)⋊8C2, C22.54(C4×D4), D4.6(C22⋊C4), Q8.6(C22⋊C4), C4.10C423C2, C4.136(C4⋊D4), M4(2)⋊4C44C2, (C22×C4).30C23, C42⋊C2213C2, C23.123(C4○D4), C23.37D423C2, C22.50(C4⋊D4), (C22×D4).29C22, C42⋊C2.28C22, C2.42(C23.23D4), (C2×M4(2)).190C22, C22.6(C22.D4), (C2×C8).6(C2×C4), (C2×D4).85(C2×C4), (C2×C4).239(C2×D4), (C2×C8⋊C22).2C2, C4.19(C2×C22⋊C4), (C2×Q8).73(C2×C4), (C2×C4.D4)⋊18C2, (C2×C4).325(C4○D4), (C2×C4).190(C22×C4), (C2×C4○D4).24C22, SmallGroup(128,635)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2).47D4
C1C2C22C2×C4C22×C4C2×C4○D4Q8○M4(2) — M4(2).47D4
C1C2C2×C4 — M4(2).47D4
C1C2C22×C4 — M4(2).47D4
C1C2C2C22×C4 — M4(2).47D4

Generators and relations for M4(2).47D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=ab, dad=a5b, cbc-1=a4b, bd=db, dcd=a2c-1 >

Subgroups: 364 in 162 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×3], C22 [×3], C22 [×12], C8 [×7], C2×C4 [×6], C2×C4 [×6], D4 [×2], D4 [×11], Q8 [×2], Q8, C23, C23 [×7], C42, C22⋊C4, C4⋊C4, C2×C8 [×4], C2×C8 [×6], M4(2) [×2], M4(2) [×11], D8 [×4], SD16 [×4], C22×C4, C22×C4, C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C4.D4 [×2], D4⋊C4 [×2], C4≀C2 [×2], C42⋊C2, C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×4], C22×D4, C2×C4○D4, C4.10C42, M4(2)⋊4C4, C2×C4.D4, C23.37D4, C42⋊C22, Q8○M4(2), C2×C8⋊C22, M4(2).47D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, M4(2).47D4

Character table of M4(2).47D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222448822224488444444448888
ρ111111111111111111111111111111    trivial
ρ21111111-1-111111111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ311111-1-1111111-1-1-1-1-1-1-1111-1111-1-1    linear of order 2
ρ411111-1-1-1-11111-1-1-1-1111-1-1-11-11111    linear of order 2
ρ51111111-1-1111111-1-111111111-1-1-1-1    linear of order 2
ρ6111111111111111-1-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ711111-1-1-1-11111-1-111-1-1-1111-11-1-111    linear of order 2
ρ811111-1-1111111-1-111111-1-1-11-1-1-1-1-1    linear of order 2
ρ911-11-111-111-11-1-1-1-ii-i-iii-iii-i-ii-11    linear of order 4
ρ1011-11-1-1-1-111-11-111i-iii-ii-ii-i-i-ii1-1    linear of order 4
ρ1111-11-111-111-11-1-1-1i-iii-i-ii-i-iii-i-11    linear of order 4
ρ1211-11-1-1-1-111-11-111-ii-i-ii-ii-iiii-i1-1    linear of order 4
ρ1311-11-1111-11-11-1-1-1-iiii-i-ii-i-ii-ii1-1    linear of order 4
ρ1411-11-1-1-11-11-11-111i-i-i-ii-ii-iii-ii-11    linear of order 4
ρ1511-11-1111-11-11-1-1-1i-i-i-iii-iii-ii-i1-1    linear of order 4
ρ1611-11-1-1-11-11-11-111-iiii-ii-ii-i-ii-i-11    linear of order 4
ρ1722-22-20000-22-220000000-2-22020000    orthogonal lifted from D4
ρ18222-2-2-220022-2-2-2200000000000000    orthogonal lifted from D4
ρ19222-2-20000-2-2220000-222000-200000    orthogonal lifted from D4
ρ2022-2-222-2002-2-22-2200000000000000    orthogonal lifted from D4
ρ2122-2-22-22002-2-222-200000000000000    orthogonal lifted from D4
ρ22222-2-22-20022-2-22-200000000000000    orthogonal lifted from D4
ρ23222-2-20000-2-22200002-2-2000200000    orthogonal lifted from D4
ρ2422-22-20000-22-22000000022-20-20000    orthogonal lifted from D4
ρ2522-2-220000-222-200002i-2i2i000-2i00000    complex lifted from C4○D4
ρ2622-2-220000-222-20000-2i2i-2i0002i00000    complex lifted from C4○D4
ρ27222220000-2-2-2-20000000-2i2i2i0-2i0000    complex lifted from C4○D4
ρ28222220000-2-2-2-200000002i-2i-2i02i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    orthogonal faithful

Permutation representations of M4(2).47D4
On 16 points - transitive group 16T267
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 13 5 9)(2 14)(3 11 7 15)(4 12)(6 10)(8 16)
(2 6)(3 7)(9 11)(10 12)(13 15)(14 16)

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

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

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

G:=TransitiveGroup(16,267);

Matrix representation of M4(2).47D4 in GL8(ℤ)

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

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

M4(2).47D4 in GAP, Magma, Sage, TeX

M_4(2)._{47}D_4
% in TeX

G:=Group("M4(2).47D4");
// GroupNames label

G:=SmallGroup(128,635);
// by ID

G=gap.SmallGroup(128,635);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,521,248,1411,718,172,1027]);
// 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*b,d*a*d=a^5*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=a^2*c^-1>;
// generators/relations

Export

Character table of M4(2).47D4 in TeX

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