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G = M4(2)⋊11D4  order 128 = 27

5th semidirect product of M4(2) and D4 acting via D4/C4=C2

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

Aliases: M4(2)⋊11D4, C42.377C23, (C2×C8)⋊18D4, C8.4(C2×D4), C8⋊3D4⋊6C2, C8⋊4D4⋊18C2, C4⋊C4.241D4, (C4×C8)⋊28C22, C2.22(D4○D8), (C22×D8)⋊19C2, (C2×D8)⋊21C22, C4⋊1D4⋊7C22, C22⋊C4.81D4, C4.13(C22×D4), C8⋊C4⋊49C22, C8.12D4⋊16C2, C4.42(C4⋊1D4), (C2×C4).353C24, (C2×C8).560C23, (C2×Q16)⋊45C22, C4.4D4⋊9C22, C23.455(C2×D4), C8○2M4(2)⋊13C2, (C2×SD16)⋊17C22, (C2×D4).119C23, (C2×Q8).107C23, C22.29C24⋊13C2, C22.19(C4⋊1D4), (C22×C8).272C22, C22.613(C22×D4), (C22×C4).1043C23, (C22×D4).379C22, C42⋊C2.326C22, (C2×M4(2)).273C22, (C2×C4○D8)⋊22C2, (C2×C8⋊C22)⋊25C2, (C2×C4).139(C2×D4), C2.32(C2×C4⋊1D4), (C2×C4○D4).159C22, SmallGroup(128,1887)

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

Derived series
C1 — C2×C4 — M4(2)⋊11D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C8○2M4(2) — M4(2)⋊11D4
Lower central
C1 — C2 — C2×C4 — M4(2)⋊11D4
Upper central
C1 — C22 — C42⋊C2 — M4(2)⋊11D4
Jennings
C1 — C2 — C2 — C2×C4 — M4(2)⋊11D4

Generators and relations for M4(2)⋊11D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a-1, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 716 in 298 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×C8, C8⋊C4, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×C8, C2×M4(2), C2×D8, C2×D8, C2×D8, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C22×D4, C2×C4○D4, C8○2M4(2), C8⋊4D4, C8.12D4, C8⋊3D4, C22.29C24, C22×D8, C2×C4○D8, C2×C8⋊C22, M4(2)⋊11D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C22×D4, C2×C4⋊1D4, D4○D8, M4(2)⋊11D4

Smallest permutation representation of M4(2)⋊11D4
►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)(26 30)(28 32)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E···8J
order1222222···2444444444488888···8
size1111228···8222244448822224···4

32 irreducible representations

dim11111111122224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4D4○D8
kernelM4(2)⋊11D4C8○2M4(2)C8⋊4D4C8.12D4C8⋊3D4C22.29C24C22×D8C2×C4○D8C2×C8⋊C22C22⋊C4C4⋊C4C2×C8M4(2)C2
# reps11224211222444

Matrix representation of M4(2)⋊11D4 ►in GL6(𝔽17)

1150000
1160000
0000143
00001414
0031400
003300
,
1600000
0160000
001000
000100
0000160
0000016
,
1620000
1610000
000010
000001
001000
000100
,
1150000
0160000
00001414
0000143
00141400
0014300

G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,14,14,0,0,0,0,3,14,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,14,14,0,0,0,0,14,3,0,0] >;
 

M4(2)⋊11D4 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_{11}D_4
 
% in TeX
 
G:=Group("M4(2):11D4");
 
// GroupNames label
 
G:=SmallGroup(128,1887);
 
// by ID
 
G=gap.SmallGroup(128,1887);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,184,521,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^-1,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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