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G = M4(2).9D4order 128 = 27

9th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).9D4, C4.71C22≀C2, (C2×D4).110D4, (C2×Q8).101D4, C4.29(C4⋊D4), M4(2).C45C2, (C22×C4).42C23, M4(2)⋊4C411C2, C23.132(C4○D4), C23.38D430C2, C22.70(C4⋊D4), C42⋊C22.1C2, (C22×Q8).70C22, C42⋊C2.62C22, C4.24(C22.D4), C22.11(C4.4D4), C2.22(C23.10D4), (C2×M4(2)).231C22, C22.58(C22.D4), M4(2).8C22.8C2, (C2×C4).262(C2×D4), (C2×C4.10D4)⋊6C2, (C2×C8.C22).7C2, (C2×C4).345(C4○D4), (C2×C4○D4).62C22, SmallGroup(128,781)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — M4(2).9D4
C1C2C22C2×C4C22×C4C2×M4(2)M4(2).8C22 — M4(2).9D4
C1C2C22×C4 — M4(2).9D4
C1C2C22×C4 — M4(2).9D4
C1C2C2C22×C4 — M4(2).9D4

Generators and relations for M4(2).9D4
 G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=a-1, dad-1=a5b, cbc-1=a4b, bd=db, dcd-1=c3 >

Subgroups: 264 in 127 conjugacy classes, 40 normal (34 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×8], D4 [×4], Q8 [×8], C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8 [×4], M4(2) [×4], M4(2) [×6], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4.D4, C4.10D4 [×3], Q8⋊C4 [×2], C4≀C2 [×2], C8.C4 [×2], C42⋊C2, C2×M4(2) [×4], C2×SD16, C2×Q16, C8.C22 [×4], C22×Q8, C2×C4○D4, M4(2)⋊4C4, C2×C4.10D4, M4(2).8C22, C23.38D4, C42⋊C22, M4(2).C4, C2×C8.C22, M4(2).9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, M4(2).9D4

Character table of M4(2).9D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 11222822228888888888888
ρ111111111111111111111111    trivial
ρ211111-1111111-1111-1-1-1-1-1-11    linear of order 2
ρ31111111111-1111-1-1-1-1-111-1-1    linear of order 2
ρ411111-11111-11-11-1-1111-1-11-1    linear of order 2
ρ51111111111-1-11-1-11-111-1-1-11    linear of order 2
ρ611111-11111-1-1-1-1-111-1-11111    linear of order 2
ρ711111111111-11-11-11-1-1-1-11-1    linear of order 2
ρ811111-111111-1-1-11-1-11111-1-1    linear of order 2
ρ922-2-2202-22-200000020000-20    orthogonal lifted from D4
ρ1022-2-220-22-22020-2000000000    orthogonal lifted from D4
ρ1122-22-222-2-2200-20000000000    orthogonal lifted from D4
ρ1222-2-220-22-220-202000000000    orthogonal lifted from D4
ρ13222-2-20-2-2220000000002-200    orthogonal lifted from D4
ρ1422-2-2202-22-2000000-2000020    orthogonal lifted from D4
ρ15222-2-20-2-222000000000-2200    orthogonal lifted from D4
ρ1622-22-2-22-2-220020000000000    orthogonal lifted from D4
ρ1722-22-20-222-2000002i000000-2i    complex lifted from C4○D4
ρ18222-2-2022-2-2-2i0002i00000000    complex lifted from C4○D4
ρ1922-22-20-222-200000-2i0000002i    complex lifted from C4○D4
ρ20222220-2-2-2-20000000-2i2i0000    complex lifted from C4○D4
ρ21222220-2-2-2-200000002i-2i0000    complex lifted from C4○D4
ρ22222-2-2022-2-22i000-2i00000000    complex lifted from C4○D4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).9D4
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)
(1 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)
(1 18 32 9 5 22 28 13)(2 17 25 16 6 21 29 12)(3 24 26 15 7 20 30 11)(4 23 27 14 8 19 31 10)
(1 9 5 13)(2 21 6 17)(3 15 7 11)(4 19 8 23)(10 27 14 31)(12 25 16 29)(18 28 22 32)(20 26 24 30)

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

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

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

Matrix representation of M4(2).9D4 in GL8(𝔽17)

70706868
3030167167
1106011968
12050110167
93101412161413
54141312161413
0393511413
13454511413
,
001620000
001610000
115000000
116000000
61101000016
0117100010
071160100
1070616000
,
791080000
3101470000
79790000
3103100000
74116133414
84016341413
13451133133
04913434
,
791080000
3101470000
1081080000
1471470000
1611013133414
01913341413
1216413414414
81601314131413

G:=sub<GL(8,GF(17))| [7,3,11,12,9,5,0,13,0,0,0,0,3,4,3,4,7,3,6,5,10,14,9,5,0,0,0,0,14,13,3,4,6,16,11,1,12,12,5,5,8,7,9,10,16,16,1,1,6,16,6,16,14,14,14,14,8,7,8,7,13,13,13,13],[0,0,1,1,6,0,0,10,0,0,15,16,11,11,7,7,16,16,0,0,0,7,11,0,2,1,0,0,10,10,6,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0],[7,3,7,3,7,8,13,0,9,10,9,10,4,4,4,4,10,14,7,3,1,0,5,9,8,7,9,10,16,16,1,1,0,0,0,0,13,3,13,3,0,0,0,0,3,4,3,4,0,0,0,0,4,14,13,3,0,0,0,0,14,13,3,4],[7,3,10,14,16,0,12,8,9,10,8,7,1,1,16,16,10,14,10,14,10,9,4,0,8,7,8,7,13,13,13,13,0,0,0,0,13,3,4,14,0,0,0,0,3,4,14,13,0,0,0,0,4,14,4,14,0,0,0,0,14,13,14,13] >;

M4(2).9D4 in GAP, Magma, Sage, TeX

M_4(2)._9D_4
% in TeX

G:=Group("M4(2).9D4");
// GroupNames label

G:=SmallGroup(128,781);
// by ID

G=gap.SmallGroup(128,781);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2019,1018,248,2804,172,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^5*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of M4(2).9D4 in TeX

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