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G = M4(2).50D4order 128 = 27

14th non-split extension by M4(2) of D4 acting via D4/C22=C2

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

Aliases: M4(2).50D4, (C2×D4).84D4, (C2×Q8).76D4, C22.13(C4×D4), (C22×C4).72D4, C22.D4.C4, C4.102C22≀C2, C4.12(C4⋊D4), Q8○M4(2).6C2, (C22×C4).38C23, C23.72(C22×C4), M4(2)⋊4C417C2, C23.18(C22⋊C4), (C22×Q8).26C22, C42⋊C2.36C22, C4.16(C22.D4), C2.54(C23.23D4), (C2×M4(2)).197C22, C23.38C23.3C2, M4(2).8C22.7C2, (C2×D4).93(C2×C4), (C2×C4).249(C2×D4), C22⋊C4.7(C2×C4), (C22×C4).22(C2×C4), (C2×C4).333(C4○D4), (C2×C4.10D4)⋊21C2, (C2×C4).22(C22⋊C4), (C2×C4○D4).32C22, C22.53(C2×C22⋊C4), SmallGroup(128,647)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — M4(2).50D4
C1C2C4C2×C4C22×C4C2×C4○D4Q8○M4(2) — M4(2).50D4
C1C2C23 — M4(2).50D4
C1C2C22×C4 — M4(2).50D4
C1C2C2C22×C4 — M4(2).50D4

Generators and relations for M4(2).50D4
 G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, cac-1=a5b, dad-1=ab, bc=cb, dbd-1=a4b, dcd-1=a6bc3 >

Subgroups: 284 in 154 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×6], Q8 [×6], C23, C23 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×5], C2×C8 [×8], M4(2) [×4], M4(2) [×10], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×6], C4○D4 [×4], C4.D4, C4.10D4 [×5], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×4], C4.4D4, C4⋊Q8, C2×M4(2) [×4], C2×M4(2) [×2], C8○D4 [×4], C22×Q8, C2×C4○D4, M4(2)⋊4C4 [×2], C2×C4.10D4 [×2], M4(2).8C22, C23.38C23, Q8○M4(2), M4(2).50D4
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).50D4

Character table of M4(2).50D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H8I8J8K8L
 size 11222442222448888444444448888
ρ111111111111111111111111111111    trivial
ρ211111-1-11111-1-1-111-111-1-1-1-111-11-11    linear of order 2
ρ311111-1-11111-1-11-1-1111-1-1-1-1111-11-1    linear of order 2
ρ41111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ511111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ611111-1-11111-1-1-111-1-1-11111-1-11-11-1    linear of order 2
ρ711111-1-11111-1-11-1-11-1-11111-1-1-11-11    linear of order 2
ρ81111111111111-1-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ9111111-1-1-1-1-1-11-1-111-iii-ii-ii-ii-i-ii    linear of order 4
ρ10111111-1-1-1-1-1-11-1-111i-i-ii-ii-ii-iii-i    linear of order 4
ρ11111111-1-1-1-1-1-1111-1-1i-i-ii-ii-iii-i-ii    linear of order 4
ρ12111111-1-1-1-1-1-1111-1-1-iii-ii-ii-i-iii-i    linear of order 4
ρ1311111-11-1-1-1-11-11-11-1-ii-ii-iii-i-i-iii    linear of order 4
ρ1411111-11-1-1-1-11-11-11-1i-ii-ii-i-iiii-i-i    linear of order 4
ρ1511111-11-1-1-1-11-1-11-11i-ii-ii-i-ii-i-iii    linear of order 4
ρ1611111-11-1-1-1-11-1-11-11-ii-ii-iii-iii-i-i    linear of order 4
ρ1722-22-2-2-2-222-2220000000000000000    orthogonal lifted from D4
ρ1822-2-2200-22-220000002-200002-20000    orthogonal lifted from D4
ρ1922-22-2-222-2-22-220000000000000000    orthogonal lifted from D4
ρ2022-22-22-22-2-222-20000000000000000    orthogonal lifted from D4
ρ21222-2-200-2-22200000000-2-222000000    orthogonal lifted from D4
ρ2222-2-2200-22-22000000-220000-220000    orthogonal lifted from D4
ρ2322-22-222-222-2-2-20000000000000000    orthogonal lifted from D4
ρ24222-2-200-2-2220000000022-2-2000000    orthogonal lifted from D4
ρ25222-2-20022-2-2000000002i-2i-2i2i000000    complex lifted from C4○D4
ρ2622-2-22002-22-2000000-2i-2i00002i2i0000    complex lifted from C4○D4
ρ27222-2-20022-2-200000000-2i2i2i-2i000000    complex lifted from C4○D4
ρ2822-2-22002-22-20000002i2i0000-2i-2i0000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of M4(2).50D4
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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)
(1 30 25 2 5 26 29 6)(3 28 27 8 7 32 31 4)(9 22 21 14 13 18 17 10)(11 20 23 12 15 24 19 16)
(1 18 25 14 5 22 29 10)(2 13 30 17 6 9 26 21)(3 24 27 12 7 20 31 16)(4 11 32 23 8 15 28 19)

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

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

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

Matrix representation of M4(2).50D4 in GL8(𝔽17)

00000010
00000001
00001000
00000100
01000000
160000000
00010000
001600000
,
00100000
00010000
10000000
01000000
000000160
000000016
000016000
000001600
,
00000100
000016000
000000016
00000010
10000000
01000000
001600000
000160000
,
00000004
00000040
00000400
00004000
00400000
000130000
40000000
013000000

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

M4(2).50D4 in GAP, Magma, Sage, TeX

M_4(2)._{50}D_4
% in TeX

G:=Group("M4(2).50D4");
// GroupNames label

G:=SmallGroup(128,647);
// by ID

G=gap.SmallGroup(128,647);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,1018,521,2804,1411,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2*b,b*a*b=a^5,c*a*c^-1=a^5*b,d*a*d^-1=a*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^6*b*c^3>;
// generators/relations

Export

Character table of M4(2).50D4 in TeX

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