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G = M4(2).46D4  order 128 = 27

10th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).46D4, C4.1(C4×D4), (C2×Q16)⋊3C4, C4○D4.7D4, (C2×C8).28D4, (C2×SD16)⋊1C4, C4.95C22≀C2, C22.53(C4×D4), D4.5(C22⋊C4), Q8○M4(2).4C2, Q8.5(C22⋊C4), C4.10C42⋊2C2, C4.135(C4⋊D4), M4(2)⋊4C4⋊3C2, (C22×C4).29C23, C23.122(C4○D4), C23.38D4⋊23C2, C22.49(C4⋊D4), C42⋊C22.4C2, (C22×Q8).21C22, C42⋊C2.27C22, C2.41(C23.23D4), (C2×M4(2)).189C22, C22.5(C22.D4), (C2×C8).5(C2×C4), (C2×D4).84(C2×C4), (C2×C4).238(C2×D4), C4.18(C2×C22⋊C4), (C2×Q8).72(C2×C4), (C2×C8.C22).2C2, (C2×C4).324(C4○D4), (C2×C4.10D4)⋊18C2, (C2×C4).189(C22×C4), (C2×C4○D4).23C22, SmallGroup(128,634)

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

Derived series
C1 — C2×C4 — M4(2).46D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — Q8○M4(2) — M4(2).46D4
Lower central
C1 — C2 — C2×C4 — M4(2).46D4
Upper central
C1 — C2 — C22×C4 — M4(2).46D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).46D4

Generators and relations for M4(2).46D4
 G = < a,b,c,d | a8=b2=c4=1, d2=a4, bab=a5, cac-1=a5b, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=a6c-1 >

Subgroups: 284 in 152 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.10D4, Q8⋊C4, C4≀C2, C42⋊C2, C2×M4(2), C2×M4(2), C8○D4, C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C4.10C42, M4(2)⋊4C4, C2×C4.10D4, C23.38D4, C42⋊C22, Q8○M4(2), C2×C8.C22, M4(2).46D4
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).46D4

Character table of M4(2).46D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222442222448888444444448888
ρ111111111111111111111111111111    trivial
ρ211111-1-11111-1-11-11-11-1-1111-1-111-1-1    linear of order 2
ρ311111-1-11111-1-1-11-111-1-1111-1-1-1-111    linear of order 2
ρ41111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ511111111111111-11-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ611111-1-11111-1-11111-111-1-1-111-1-1-1-1    linear of order 2
ρ711111-1-11111-1-1-1-1-1-1-111-1-1-1111111    linear of order 2
ρ81111111111111-11-11-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ911-1-1111-11-11-1-11i-1-iii-i-ii-ii-i-ii1-1    linear of order 4
ρ1011-1-11-1-1-11-11111-i-1ii-ii-ii-i-ii-ii-11    linear of order 4
ρ1111-1-1111-11-11-1-11-i-1i-i-iii-ii-iii-i1-1    linear of order 4
ρ1211-1-11-1-1-11-11111i-1-i-ii-ii-iii-ii-i-11    linear of order 4
ρ1311-1-11-1-1-11-1111-1i1-ii-ii-ii-i-iii-i1-1    linear of order 4
ρ1411-1-1111-11-11-1-1-1-i1iii-i-ii-ii-ii-i-11    linear of order 4
ρ1511-1-11-1-1-11-1111-1-i1i-ii-ii-iii-i-ii1-1    linear of order 4
ρ1611-1-1111-11-11-1-1-1i1-i-i-iii-ii-ii-ii-11    linear of order 4
ρ1722-22-2002-2-22000000022000-2-20000    orthogonal lifted from D4
ρ18222-2-2-2222-2-22-20000000000000000    orthogonal lifted from D4
ρ1922-22-22-2-222-22-20000000000000000    orthogonal lifted from D4
ρ2022-22-2-22-222-2-220000000000000000    orthogonal lifted from D4
ρ21222-2-22-222-2-2-220000000000000000    orthogonal lifted from D4
ρ2222-2-22002-22-2000000200-2-22000000    orthogonal lifted from D4
ρ2322-2-22002-22-2000000-20022-2000000    orthogonal lifted from D4
ρ2422-22-2002-2-220000000-2-2000220000    orthogonal lifted from D4
ρ25222-2-200-2-2220000000-2i2i0002i-2i0000    complex lifted from C4○D4
ρ262222200-2-2-2-20000002i002i-2i-2i000000    complex lifted from C4○D4
ρ272222200-2-2-2-2000000-2i00-2i2i2i000000    complex lifted from C4○D4
ρ28222-2-200-2-22200000002i-2i000-2i2i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).46D4
►On 32 points
Generators in S32
(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)

(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)

(1 26 21 11)(2 27 18 16)(3 32 23 9)(4 25 20 14)(5 30 17 15)(6 31 22 12)(7 28 19 13)(8 29 24 10)

(1 14 5 10)(2 11 6 15)(3 12 7 16)(4 9 8 13)(17 29 21 25)(18 26 22 30)(19 27 23 31)(20 32 24 28)
 
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32), (1,26,21,11)(2,27,18,16)(3,32,23,9)(4,25,20,14)(5,30,17,15)(6,31,22,12)(7,28,19,13)(8,29,24,10), (1,14,5,10)(2,11,6,15)(3,12,7,16)(4,9,8,13)(17,29,21,25)(18,26,22,30)(19,27,23,31)(20,32,24,28)>;
 
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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32), (1,26,21,11)(2,27,18,16)(3,32,23,9)(4,25,20,14)(5,30,17,15)(6,31,22,12)(7,28,19,13)(8,29,24,10), (1,14,5,10)(2,11,6,15)(3,12,7,16)(4,9,8,13)(17,29,21,25)(18,26,22,30)(19,27,23,31)(20,32,24,28) );
 
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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32)], [(1,26,21,11),(2,27,18,16),(3,32,23,9),(4,25,20,14),(5,30,17,15),(6,31,22,12),(7,28,19,13),(8,29,24,10)], [(1,14,5,10),(2,11,6,15),(3,12,7,16),(4,9,8,13),(17,29,21,25),(18,26,22,30),(19,27,23,31),(20,32,24,28)]])
 

Matrix representation of M4(2).46D4 ►in GL8(𝔽17)

00100000
000160000
40000000
013000000
00000010
000000016
000013000
00000400
,
10000000
01000000
001600000
000160000
000016000
000001600
00000010
00000001
,
00001000
00000100
00000010
00000001
01000000
10000000
00010000
00100000
,
00000010
00000001
000013000
000001300
001300000
000130000
160000000
016000000

G:=sub<GL(8,GF(17))| [0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0] >;
 

M4(2).46D4 in GAP, Magma, Sage, TeX 

M_4(2)._{46}D_4
 
% in TeX
 
G:=Group("M4(2).46D4");
 
// GroupNames label
 
G:=SmallGroup(128,634);
 
// by ID
 
G=gap.SmallGroup(128,634);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,1411,718,172,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4,b*a*b=a^5,c*a*c^-1=a^5*b,d*a*d^-1=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*c^-1>;
 
// generators/relations
 

Export

Character table of M4(2).46D4 in TeX

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