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## G = M4(2).6D4order 128 = 27

### 6th non-split extension by M4(2) of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — M4(2).6D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C2×C8.C22 — M4(2).6D4
 Lower central C1 — C2 — C22×C4 — M4(2).6D4
 Upper central C1 — C22 — C22×C4 — M4(2).6D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).6D4

Generators and relations for M4(2).6D4
G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=a5, cac-1=a-1b, dad-1=ab, cbc-1=a4b, bd=db, dcd-1=c3 >

Subgroups: 344 in 171 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×5], C8 [×7], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×4], Q8 [×14], C23, C23, C2×C8 [×2], C2×C8 [×5], M4(2) [×4], M4(2) [×4], SD16 [×8], Q16 [×16], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×4], C22⋊C8 [×2], C4.10D4 [×4], C22×C8, C2×M4(2), C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×8], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4, C4.C42, (C22×C8)⋊C2, C2×C4.10D4 [×2], C22×Q16, C2×C8.C22 [×2], M4(2).6D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, D4.5D4 [×2], M4(2).6D4

Character table of M4(2).6D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 1 1 2 2 8 2 2 2 2 8 8 8 8 8 4 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 1 linear of order 2 ρ6 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 linear of order 2 ρ9 2 -2 -2 2 2 -2 0 -2 2 2 -2 2 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 -2 2 0 2 2 -2 -2 0 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 2 -2 2 -2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 -2 2 -2 2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 2 0 0 0 0 -2 orthogonal lifted from D4 ρ13 2 -2 -2 2 -2 2 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 orthogonal lifted from D4 ρ14 2 -2 -2 2 2 -2 0 -2 2 2 -2 -2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 -2 -2 2 2 -2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 -2 0 -2 2 -2 2 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 2 -2 -2 2 2 -2 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 2 2 -2 2 -2 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 2 0 2 2 -2 -2 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 0 -2 2 -2 2 0 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 2i 0 0 -2i 0 complex lifted from C4○D4 ρ22 2 2 2 2 2 2 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2i 0 0 2i 0 complex lifted from C4○D4 ρ23 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2

Smallest permutation representation of M4(2).6D4
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 21)(2 18)(3 23)(4 20)(5 17)(6 22)(7 19)(8 24)(9 38)(10 35)(11 40)(12 37)(13 34)(14 39)(15 36)(16 33)(25 53)(26 50)(27 55)(28 52)(29 49)(30 54)(31 51)(32 56)(41 63)(42 60)(43 57)(44 62)(45 59)(46 64)(47 61)(48 58)
(1 57 10 26 5 61 14 30)(2 46 11 53 6 42 15 49)(3 59 12 28 7 63 16 32)(4 48 13 55 8 44 9 51)(17 43 39 50 21 47 35 54)(18 60 40 29 22 64 36 25)(19 45 33 52 23 41 37 56)(20 62 34 31 24 58 38 27)
(1 57 5 61)(2 48 6 44)(3 63 7 59)(4 46 8 42)(9 53 13 49)(10 30 14 26)(11 51 15 55)(12 28 16 32)(17 47 21 43)(18 58 22 62)(19 45 23 41)(20 64 24 60)(25 34 29 38)(27 40 31 36)(33 56 37 52)(35 54 39 50)

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

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

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

Matrix representation of M4(2).6D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 4 4 0 0 0 13 0 13 0 0 0 13 0 0 0 0 13 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 15 0 0 16 0 16 1 0 0 16 16 0 1 0 0 0 0 0 16
,
 16 2 0 0 0 0 16 1 0 0 0 0 0 0 0 0 7 7 0 0 12 0 0 10 0 0 0 5 5 5 0 0 5 5 12 12
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 10 0 10 7 0 0 12 0 0 10 0 0 0 5 12 5 0 0 5 5 5 12

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,13,0,0,4,13,13,0,0,0,4,0,0,0,0,0,4,13,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,16,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,15,1,1,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,12,0,5,0,0,0,0,5,5,0,0,7,0,5,12,0,0,7,10,5,12],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,10,12,0,5,0,0,0,0,5,5,0,0,10,0,12,5,0,0,7,10,5,12] >;

M4(2).6D4 in GAP, Magma, Sage, TeX

M_4(2)._6D_4
% in TeX

G:=Group("M4(2).6D4");
// GroupNames label

G:=SmallGroup(128,752);
// by ID

G=gap.SmallGroup(128,752);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,352,2019,1018,521,248,2804,718,172,4037,2028,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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