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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, (C2xC8).43D4, (C2xD4).96D4, (C22xD8):3C2, (C2xQ8).87D4, C4.13C22wrC2, C4.33(C4:1D4), C4.C42:8C2, C2.18(D4.4D4), C23.272(C4oD4), C22.61(C4:D4), C2.23(C23:2D4), (C22xC8).110C22, (C22xC4).716C23, (C22xD4).68C22, (C2xM4(2)).21C22, (C2xC8:C22):4C2, (C2xC4).254(C2xD4), (C2xC4.D4):3C2, (C22xC8):C2:18C2, (C2xC4oD4).53C22, SmallGroup(128,750)

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

Derived series
C1 — C22xC4 — M4(2).4D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C22xD4 — C2xC8:C22 — M4(2).4D4
Lower central
C1 — C2 — C22xC4 — M4(2).4D4
Upper central
C1 — C22 — C22xC4 — M4(2).4D4
Jennings
C1 — C2 — C2 — C22xC4 — 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, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C22:C8, C4.D4, C22xC8, C2xM4(2), C2xM4(2), C2xD8, C2xSD16, C8:C22, C22xD4, C2xC4oD4, C4.C42, (C22xC8):C2, C2xC4.D4, C22xD8, C2xC8:C22, M4(2).4D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C22wrC2, C4:D4, C4:1D4, C23:2D4, D4.4D4, 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 C4oD4
ρ2222222200000-2-2-2-2000000-2i002i0    complex lifted from C4oD4
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 24)(2 21)(3 18)(4 23)(5 20)(6 17)(7 22)(8 19)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)

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

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

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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