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G = M4(2)⋊8Q8  order 128 = 27

6th semidirect product of M4(2) and Q8 acting via Q8/C4=C2

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

Aliases: M4(2)⋊8Q8, C42.127D4, C4.33(C4⋊Q8), C22.14(C4×Q8), C4.8(C4.D4), C4.84(C22⋊Q8), (C22×Q8).12C4, C4.52(C4.4D4), C4.8(C4.10D4), (C4×M4(2)).27C2, C4⋊M4(2).35C2, C23.201(C22×C4), (C2×C42).339C22, (C22×C4).706C23, C22.C42.12C2, (C2×M4(2)).218C22, C2.19(C23.67C23), (C2×C4⋊C4).25C4, (C2×C4⋊Q8).17C2, (C2×C4).13(C2×Q8), (C2×C4).70(C4○D4), (C2×C4).1368(C2×D4), (C2×C4⋊C4).99C22, (C22×C4).27(C2×C4), C2.29(C2×C4.D4), C2.27(C2×C4.10D4), (C2×C4).265(C22⋊C4), C22.306(C2×C22⋊C4), SmallGroup(128,729)

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

Derived series
C1 — C23 — M4(2)⋊8Q8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C4×M4(2) — M4(2)⋊8Q8
Lower central
C1 — C2 — C23 — M4(2)⋊8Q8
Upper central
C1 — C22 — C2×C42 — M4(2)⋊8Q8
Jennings
C1 — C2 — C2 — C22×C4 — M4(2)⋊8Q8

Generators and relations for M4(2)⋊8Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 244 in 130 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C4×C8, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C4⋊Q8, C2×M4(2), C22×Q8, C22.C42, C4×M4(2), C4⋊M4(2), C2×C4⋊Q8, M4(2)⋊8Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C23.67C23, C2×C4.D4, C2×C4.10D4, M4(2)⋊8Q8

Smallest permutation representation of M4(2)⋊8Q8
►On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 48)(18 45)(19 42)(20 47)(21 44)(22 41)(23 46)(24 43)(25 35)(26 40)(27 37)(28 34)(29 39)(30 36)(31 33)(32 38)(49 63)(50 60)(51 57)(52 62)(53 59)(54 64)(55 61)(56 58)

(1 29 15 39)(2 30 16 40)(3 31 9 33)(4 32 10 34)(5 25 11 35)(6 26 12 36)(7 27 13 37)(8 28 14 38)(17 55 44 57)(18 56 45 58)(19 49 46 59)(20 50 47 60)(21 51 48 61)(22 52 41 62)(23 53 42 63)(24 54 43 64)

(1 58 15 56)(2 53 16 63)(3 64 9 54)(4 51 10 61)(5 62 11 52)(6 49 12 59)(7 60 13 50)(8 55 14 57)(17 38 44 28)(18 29 45 39)(19 36 46 26)(20 27 47 37)(21 34 48 32)(22 25 41 35)(23 40 42 30)(24 31 43 33)
 
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,48)(18,45)(19,42)(20,47)(21,44)(22,41)(23,46)(24,43)(25,35)(26,40)(27,37)(28,34)(29,39)(30,36)(31,33)(32,38)(49,63)(50,60)(51,57)(52,62)(53,59)(54,64)(55,61)(56,58), (1,29,15,39)(2,30,16,40)(3,31,9,33)(4,32,10,34)(5,25,11,35)(6,26,12,36)(7,27,13,37)(8,28,14,38)(17,55,44,57)(18,56,45,58)(19,49,46,59)(20,50,47,60)(21,51,48,61)(22,52,41,62)(23,53,42,63)(24,54,43,64), (1,58,15,56)(2,53,16,63)(3,64,9,54)(4,51,10,61)(5,62,11,52)(6,49,12,59)(7,60,13,50)(8,55,14,57)(17,38,44,28)(18,29,45,39)(19,36,46,26)(20,27,47,37)(21,34,48,32)(22,25,41,35)(23,40,42,30)(24,31,43,33)>;
 
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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,48)(18,45)(19,42)(20,47)(21,44)(22,41)(23,46)(24,43)(25,35)(26,40)(27,37)(28,34)(29,39)(30,36)(31,33)(32,38)(49,63)(50,60)(51,57)(52,62)(53,59)(54,64)(55,61)(56,58), (1,29,15,39)(2,30,16,40)(3,31,9,33)(4,32,10,34)(5,25,11,35)(6,26,12,36)(7,27,13,37)(8,28,14,38)(17,55,44,57)(18,56,45,58)(19,49,46,59)(20,50,47,60)(21,51,48,61)(22,52,41,62)(23,53,42,63)(24,54,43,64), (1,58,15,56)(2,53,16,63)(3,64,9,54)(4,51,10,61)(5,62,11,52)(6,49,12,59)(7,60,13,50)(8,55,14,57)(17,38,44,28)(18,29,45,39)(19,36,46,26)(20,27,47,37)(21,34,48,32)(22,25,41,35)(23,40,42,30)(24,31,43,33) );
 
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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10),(17,48),(18,45),(19,42),(20,47),(21,44),(22,41),(23,46),(24,43),(25,35),(26,40),(27,37),(28,34),(29,39),(30,36),(31,33),(32,38),(49,63),(50,60),(51,57),(52,62),(53,59),(54,64),(55,61),(56,58)], [(1,29,15,39),(2,30,16,40),(3,31,9,33),(4,32,10,34),(5,25,11,35),(6,26,12,36),(7,27,13,37),(8,28,14,38),(17,55,44,57),(18,56,45,58),(19,49,46,59),(20,50,47,60),(21,51,48,61),(22,52,41,62),(23,53,42,63),(24,54,43,64)], [(1,58,15,56),(2,53,16,63),(3,64,9,54),(4,51,10,61),(5,62,11,52),(6,49,12,59),(7,60,13,50),(8,55,14,57),(17,38,44,28),(18,29,45,39),(19,36,46,26),(20,27,47,37),(21,34,48,32),(22,25,41,35),(23,40,42,30),(24,31,43,33)]])
 

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K4L4M4N8A···8H8I8J8K8L
order1222224···44444448···88888
size1111222···24488884···48888

32 irreducible representations

dim111111122244
type++++++-+-
imageC1C2C2C2C2C4C4D4Q8C4○D4C4.D4C4.10D4
kernelM4(2)⋊8Q8C22.C42C4×M4(2)C4⋊M4(2)C2×C4⋊Q8C2×C4⋊C4C22×Q8C42M4(2)C2×C4C4C4
# reps141114444422

Matrix representation of M4(2)⋊8Q8 ►in GL6(𝔽17)

400000
040000
000010
000001
000100
0016000
,
100000
010000
001000
000100
0000160
0000016
,
4130000
0130000
0016000
0001600
0000160
0000016
,
400000
8130000
0041100
00111300
0000411
00001113

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[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],[4,0,0,0,0,0,13,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[4,8,0,0,0,0,0,13,0,0,0,0,0,0,4,11,0,0,0,0,11,13,0,0,0,0,0,0,4,11,0,0,0,0,11,13] >;
 

M4(2)⋊8Q8 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_8Q_8
 
% in TeX
 
G:=Group("M4(2):8Q8");
 
// GroupNames label
 
G:=SmallGroup(128,729);
 
// by ID
 
G=gap.SmallGroup(128,729);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,1018,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=c^2,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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