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## G = Q8×M4(2)  order 128 = 27

### Direct product of Q8 and M4(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for Q8×M4(2)
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 244 in 194 conjugacy classes, 150 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C42, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C4⋊C8, C2×C42, C2×C4⋊C4, C4×Q8, C4×Q8, C2×M4(2), C2×M4(2), C22×Q8, C4×M4(2), C4⋊M4(2), C8×Q8, C84Q8, C2×C4×Q8, Q8×M4(2)
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, M4(2), C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C2×M4(2), C23×C4, C22×Q8, C2×C4○D4, C2×C4×Q8, C22×M4(2), Q8○M4(2), Q8×M4(2)

Smallest permutation representation of Q8×M4(2)
On 64 points
Generators in S64
(1 26 19 38)(2 27 20 39)(3 28 21 40)(4 29 22 33)(5 30 23 34)(6 31 24 35)(7 32 17 36)(8 25 18 37)(9 45 57 53)(10 46 58 54)(11 47 59 55)(12 48 60 56)(13 41 61 49)(14 42 62 50)(15 43 63 51)(16 44 64 52)
(1 55 19 47)(2 56 20 48)(3 49 21 41)(4 50 22 42)(5 51 23 43)(6 52 24 44)(7 53 17 45)(8 54 18 46)(9 32 57 36)(10 25 58 37)(11 26 59 38)(12 27 60 39)(13 28 61 40)(14 29 62 33)(15 30 63 34)(16 31 64 35)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4R 4S ··· 4X 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 Q8 C4○D4 M4(2) Q8○M4(2) kernel Q8×M4(2) C4×M4(2) C4⋊M4(2) C8×Q8 C8⋊4Q8 C2×C4×Q8 C2×C4⋊C4 C4×Q8 C22×Q8 M4(2) C2×C4 Q8 C2 # reps 1 3 3 2 6 1 6 8 2 4 4 8 2

Matrix representation of Q8×M4(2) in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
,
 1 0 0 0 0 1 0 0 0 0 1 10 0 0 10 16
,
 3 15 0 0 11 14 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 3 16 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,10,0,0,10,16],[3,11,0,0,15,14,0,0,0,0,16,0,0,0,0,16],[1,3,0,0,0,16,0,0,0,0,1,0,0,0,0,1] >;

Q8×M4(2) in GAP, Magma, Sage, TeX

Q_8\times M_4(2)
% in TeX

G:=Group("Q8xM4(2)");
// GroupNames label

G:=SmallGroup(128,1695);
// by ID

G=gap.SmallGroup(128,1695);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,268,2019,521,124]);
// Polycyclic

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

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