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G = M4(2)⋊8Q8order 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

C1C23 — M4(2)⋊8Q8
C1C2C4C2×C4C22×C4C2×C42C4×M4(2) — M4(2)⋊8Q8
C1C2C23 — M4(2)⋊8Q8
C1C22C2×C42 — M4(2)⋊8Q8
C1C2C2C22×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 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×12], Q8 [×8], C23, C42 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], M4(2) [×6], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×8], C4×C8, C8⋊C4, C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×4], C2×M4(2) [×4], C22×Q8 [×2], C22.C42 [×4], C4×M4(2), C4⋊M4(2), C2×C4⋊Q8, M4(2)⋊8Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C4.D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], 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 44)(18 41)(19 46)(20 43)(21 48)(22 45)(23 42)(24 47)(25 33)(26 38)(27 35)(28 40)(29 37)(30 34)(31 39)(32 36)(49 57)(50 62)(51 59)(52 64)(53 61)(54 58)(55 63)(56 60)
(1 31 15 39)(2 32 16 40)(3 25 9 33)(4 26 10 34)(5 27 11 35)(6 28 12 36)(7 29 13 37)(8 30 14 38)(17 53 48 57)(18 54 41 58)(19 55 42 59)(20 56 43 60)(21 49 44 61)(22 50 45 62)(23 51 46 63)(24 52 47 64)
(1 58 15 54)(2 51 16 63)(3 64 9 52)(4 49 10 61)(5 62 11 50)(6 55 12 59)(7 60 13 56)(8 53 14 57)(17 38 48 30)(18 31 41 39)(19 36 42 28)(20 29 43 37)(21 34 44 26)(22 27 45 35)(23 40 46 32)(24 25 47 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)(17,44)(18,41)(19,46)(20,43)(21,48)(22,45)(23,42)(24,47)(25,33)(26,38)(27,35)(28,40)(29,37)(30,34)(31,39)(32,36)(49,57)(50,62)(51,59)(52,64)(53,61)(54,58)(55,63)(56,60), (1,31,15,39)(2,32,16,40)(3,25,9,33)(4,26,10,34)(5,27,11,35)(6,28,12,36)(7,29,13,37)(8,30,14,38)(17,53,48,57)(18,54,41,58)(19,55,42,59)(20,56,43,60)(21,49,44,61)(22,50,45,62)(23,51,46,63)(24,52,47,64), (1,58,15,54)(2,51,16,63)(3,64,9,52)(4,49,10,61)(5,62,11,50)(6,55,12,59)(7,60,13,56)(8,53,14,57)(17,38,48,30)(18,31,41,39)(19,36,42,28)(20,29,43,37)(21,34,44,26)(22,27,45,35)(23,40,46,32)(24,25,47,33)>;

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

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

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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