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G = M4(2).15D4order 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, (C22xC4).Q8, C23.56(C2xQ8), C4.158(C4:D4), C4.6(C42:2C2), (C22xC4).44C23, C22.8(C22:Q8), C4.10C42.4C2, M4(2).C4.2C2, (C22xQ8).77C22, (C2xM4(2)).30C22, C2.11(C23.Q8), (C2xC4).266(C2xD4), (C2xC4).353(C4oD4), (C2xC4.10D4).4C2, SmallGroup(128,802)

Series: Derived Chief Lower central Upper central Jennings

C1C22xC4 — M4(2).15D4
C1C2C22C23C22xC4C22xQ8C2xC4.10D4 — M4(2).15D4
C1C2C22xC4 — M4(2).15D4
C1C2C22xC4 — M4(2).15D4
C1C2C2C22xC4 — 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, C2xC4, C2xC4, Q8, C23, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C2xQ8, C4.10D4, C8.C4, C2xM4(2), C22xQ8, C4.10C42, C2xC4.10D4, M4(2).C4, M4(2).15D4
Quotients: C1, C2, C22, D4, Q8, C23, C2xD4, C2xQ8, C4oD4, 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 C4oD4
ρ18222-2-222-2-2002i00000000-2i00    complex lifted from C4oD4
ρ1922-2-222-22-2000000-2i0002i000    complex lifted from C4oD4
ρ2022-2-222-22-20000002i000-2i000    complex lifted from C4oD4
ρ2122-22-22-2-220000-2i0002i00000    complex lifted from C4oD4
ρ22222-2-222-2-200-2i000000002i00    complex lifted from C4oD4
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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