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G = M4(2).15D4  order 128 = 27

15th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).15D4, (C22×C4).Q8, C23.56(C2×Q8), C4.158(C4⋊D4), C4.6(C42⋊2C2), (C22×C4).44C23, C22.8(C22⋊Q8), C4.10C42.4C2, M4(2).C4.2C2, (C22×Q8).77C22, (C2×M4(2)).30C22, C2.11(C23.Q8), (C2×C4).266(C2×D4), (C2×C4).353(C4○D4), (C2×C4.10D4).4C2, SmallGroup(128,802)

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

Derived series
C1 — C22×C4 — M4(2).15D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×C4.10D4 — M4(2).15D4
Lower central
C1 — C2 — C22×C4 — M4(2).15D4
Upper central
C1 — C2 — C22×C4 — M4(2).15D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).15D4

Generators and relations for M4(2).15D4
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a6b, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=a4b, bd=db, dcd-1=a4c3 >

Subgroups: 176 in 98 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C4.10D4, C8.C4, C2×M4(2), C22×Q8, C4.10C42, C2×C4.10D4, M4(2).C4, M4(2).15D4
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C4⋊D4, C22⋊Q8, C42⋊2C2, C23.Q8, M4(2).15D4

Character table of M4(2).15D4

 class 12A2B2C2D4A4B4C4D4E4F8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222222288888888888888
ρ111111111111111111111111    trivial
ρ211111111111-1-11-1-111-1-1-1-11    linear of order 2
ρ3111111111-1-1-11-1-111-1-11-111    linear of order 2
ρ4111111111-1-11-1-11-11-11-11-11    linear of order 2
ρ51111111111111-1-1-1-1-1-1-111-1    linear of order 2
ρ611111111111-1-1-111-1-111-1-1-1    linear of order 2
ρ7111111111-1-1-1111-1-111-1-11-1    linear of order 2
ρ8111111111-1-11-11-11-11-111-1-1    linear of order 2
ρ922-2-22-22-2200000-200020000    orthogonal lifted from D4
ρ1022-22-2-222-20000000200000-2    orthogonal lifted from D4
ρ11222-2-2-2-222000200000000-20    orthogonal lifted from D4
ρ1222-22-2-222-20000000-2000002    orthogonal lifted from D4
ρ1322-2-22-22-22000002000-20000    orthogonal lifted from D4
ρ14222-2-2-2-222000-20000000020    orthogonal lifted from D4
ρ1522222-2-2-2-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ1622222-2-2-2-2-22000000000000    symplectic lifted from Q8, Schur index 2
ρ1722-22-22-2-2200002i000-2i00000    complex lifted from C4○D4
ρ18222-2-222-2-2002i00000000-2i00    complex lifted from C4○D4
ρ1922-2-222-22-2000000-2i0002i000    complex lifted from C4○D4
ρ2022-2-222-22-20000002i000-2i000    complex lifted from C4○D4
ρ2122-22-22-2-220000-2i0002i00000    complex lifted from C4○D4
ρ22222-2-222-2-200-2i000000002i00    complex lifted from C4○D4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).15D4
►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)(17 21)(19 23)(25 29)(27 31)

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

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

Matrix representation of M4(2).15D4 ►in GL8(𝔽17)

1212121255010
5555121277
000051200
0000121200
0012120000
001250000
512000000
0550012120
,
10000000
01000000
001600000
000160000
00001000
00000100
000000160
11001616016
,
00010000
001600000
160000000
016000000
000000160
161616161112
000001600
0001160016
,
000001600
00001000
161616161112
00000010
160000000
016000000
001600000
00001011

G:=sub<GL(8,GF(17))| [12,5,0,0,0,0,5,0,12,5,0,0,0,0,12,5,12,5,0,0,12,12,0,5,12,5,0,0,12,5,0,0,5,12,5,12,0,0,0,0,5,12,12,12,0,0,0,12,0,7,0,0,0,0,0,12,10,7,0,0,0,0,0,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,1,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,16,0,0,16,0,0,0,0,0,16,0,16,0,0,0,16,0,0,0,16,0,0,1,0,0,0,0,16,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,1,16,0,0,0,0,0,16,1,0,0,0,0,0,0,0,2,0,16],[0,0,16,0,16,0,0,0,0,0,16,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,1,16,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,2,0,0,0,0,1] >;
 

M4(2).15D4 in GAP, Magma, Sage, TeX 

M_4(2)._{15}D_4
 
% in TeX
 
G:=Group("M4(2).15D4");
 
// GroupNames label
 
G:=SmallGroup(128,802);
 
// by ID
 
G=gap.SmallGroup(128,802);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,352,2019,521,248,2804,4037,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^3,d*a*d^-1=a^3*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^3>;
 
// generators/relations
 

Export

Character table of M4(2).15D4 in TeX

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