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G = M4(2).4D4  order 128 = 27

4th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Aliases: M4(2).4D4, (C2×C8).43D4, (C2×D4).96D4, (C22×D8)⋊3C2, (C2×Q8).87D4, C4.13C22≀C2, C4.33(C4⋊1D4), C4.C42⋊8C2, C2.18(D4.4D4), C23.272(C4○D4), C22.61(C4⋊D4), C2.23(C23⋊2D4), (C22×C8).110C22, (C22×C4).716C23, (C22×D4).68C22, (C2×M4(2)).21C22, (C2×C8⋊C22)⋊4C2, (C2×C4).254(C2×D4), (C2×C4.D4)⋊3C2, (C22×C8)⋊C2⋊18C2, (C2×C4○D4).53C22, SmallGroup(128,750)

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

Derived series
C1 — C22×C4 — M4(2).4D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C8⋊C22 — M4(2).4D4
Lower central
C1 — C2 — C22×C4 — M4(2).4D4
Upper central
C1 — C22 — C22×C4 — M4(2).4D4
Jennings
C1 — C2 — C2 — C22×C4 — M4(2).4D4

Generators and relations for M4(2).4D4
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, cac-1=a3b, dad=ab, cbc-1=a4b, bd=db, dcd=a4c3 >

Subgroups: 504 in 191 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×2], C4 [×2], C4, C22, C22 [×2], C22 [×21], C8 [×7], C2×C4 [×2], C2×C4 [×4], C2×C4 [×3], D4 [×16], Q8 [×2], C23, C23 [×13], C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], D8 [×16], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×11], C2×Q8, C4○D4 [×4], C24 [×2], C22⋊C8 [×2], C4.D4 [×4], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×D8 [×8], C2×SD16 [×2], C8⋊C22 [×8], C22×D4 [×2], C2×C4○D4, C4.C42, (C22×C8)⋊C2, C2×C4.D4 [×2], C22×D8, C2×C8⋊C22 [×2], M4(2).4D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C4⋊1D4, C23⋊2D4, D4.4D4 [×2], M4(2).4D4

Character table of M4(2).4D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J
 size 11112288888222284444888888
ρ111111111111111111111111111    trivial
ρ211111111-1111111-11111-1-1-1-1-1-1    linear of order 2
ρ3111111-1-1-1-1-11111-111111-111-11    linear of order 2
ρ4111111-1-11-1-1111111111-11-1-11-1    linear of order 2
ρ5111111-11-11-11111-1-1-1-1-111-1-111    linear of order 2
ρ6111111-1111-111111-1-1-1-1-1-111-1-1    linear of order 2
ρ71111111-1-1-111111-1-1-1-1-1-11111-1    linear of order 2
ρ81111111-11-1111111-1-1-1-11-1-1-1-11    linear of order 2
ρ92-2-222-22000-22-2-2200000000000    orthogonal lifted from D4
ρ102-2-222-200000-222-200000002-200    orthogonal lifted from D4
ρ112222-2-20000022-2-2022-2-2000000    orthogonal lifted from D4
ρ122-2-222-2-200022-2-2200000000000    orthogonal lifted from D4
ρ132-2-222-200000-222-20000000-2200    orthogonal lifted from D4
ρ142222-2-200-200-2-22220000000000    orthogonal lifted from D4
ρ152222-2-20000022-2-20-2-222000000    orthogonal lifted from D4
ρ162-2-22-22020-202-22-200000000000    orthogonal lifted from D4
ρ172-2-22-2200000-22-2200000-200002    orthogonal lifted from D4
ρ182-2-22-2200000-22-220000020000-2    orthogonal lifted from D4
ρ192-2-22-220-20202-22-200000000000    orthogonal lifted from D4
ρ202222-2-200200-2-222-20000000000    orthogonal lifted from D4
ρ2122222200000-2-2-2-20000002i00-2i0    complex lifted from C4○D4
ρ2222222200000-2-2-2-2000000-2i002i0    complex lifted from C4○D4
ρ234-44-40000000000002√2-2√200000000    orthogonal lifted from D4.4D4
ρ2444-4-4000000000000002√2-2√2000000    orthogonal lifted from D4.4D4
ρ254-44-4000000000000-2√22√200000000    orthogonal lifted from D4.4D4
ρ2644-4-400000000000000-2√22√2000000    orthogonal lifted from D4.4D4

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

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

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

Matrix representation of M4(2).4D4 ►in GL6(𝔽17)

1600000
0160000
0000162
000001
001000
0011600
,
100000
010000
001000
000100
0000160
0000016
,
1300000
040000
0000611
0000311
0001100
0014000
,
040000
1300000
0061100
0031100
0000011
0000140

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,16,0,0,0,0,0,2,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,14,0,0,0,0,11,0,0,0,6,3,0,0,0,0,11,11,0,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,6,3,0,0,0,0,11,11,0,0,0,0,0,0,0,14,0,0,0,0,11,0] >;
 

M4(2).4D4 in GAP, Magma, Sage, TeX 

M_4(2)._4D_4
 
% in TeX
 
G:=Group("M4(2).4D4");
 
// GroupNames label
 
G:=SmallGroup(128,750);
 
// by ID
 
G=gap.SmallGroup(128,750);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,521,248,2804,718,172,4037,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=d^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=a^3*b,d*a*d=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d=a^4*c^3>;
 
// generators/relations
 

Export

Character table of M4(2).4D4 in TeX

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