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## G = M4(2)⋊4Q8order 128 = 27

### 2nd semidirect product of M4(2) and Q8 acting via Q8/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2)⋊4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2)⋊4Q8
 Lower central C1 — C2 — C2×C4 — M4(2)⋊4Q8
 Upper central C1 — C22 — C42⋊C2 — M4(2)⋊4Q8
 Jennings C1 — C2 — C2 — C2×C4 — M4(2)⋊4Q8

Generators and relations for M4(2)⋊4Q8
G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a3, cbc-1=a4b, bd=db, dcd-1=c-1 >

Subgroups: 284 in 170 conjugacy classes, 108 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×4], C23, C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×8], C4⋊C4 [×14], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×6], C2.D8 [×2], C2.D8 [×8], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C82M4(2), C2×C2.D8, C23.25D4, M4(2)⋊C4 [×2], C8.5Q8 [×2], C82Q8 [×2], C8⋊Q8 [×4], C23.41C23 [×2], M4(2)⋊4Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, D4○D8, Q8○D8, M4(2)⋊4Q8

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 ··· 8 2 2 2 2 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 Q8 Q8 D4○D8 Q8○D8 kernel M4(2)⋊4Q8 C8○2M4(2) C2×C2.D8 C23.25D4 M4(2)⋊C4 C8.5Q8 C8⋊2Q8 C8⋊Q8 C23.41C23 C22⋊C4 C4⋊C4 C2×C8 M4(2) C2 C2 # reps 1 1 1 1 2 2 2 4 2 2 2 4 4 2 2

Matrix representation of M4(2)⋊4Q8 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 9 5 16 0 0 2 0 0 0 0 0 6 0 11 8
,
 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 1 0 16
,
 10 13 0 0 0 0 4 7 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 15 0 0 16 0 0 0 0 0 0 1 0 16
,
 0 13 0 0 0 0 13 0 0 0 0 0 0 0 9 16 0 0 0 0 14 8 0 0 0 0 0 1 8 15 0 0 7 8 7 9

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,6,0,0,0,9,0,0,0,0,2,5,0,11,0,0,0,16,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,16,0,0,0,0,0,0,16],[10,4,0,0,0,0,13,7,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,1,0,0,1,0,0,0,0,0,0,15,0,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,9,14,0,7,0,0,16,8,1,8,0,0,0,0,8,7,0,0,0,0,15,9] >;

M4(2)⋊4Q8 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4Q_8
% in TeX

G:=Group("M4(2):4Q8");
// GroupNames label

G:=SmallGroup(128,1896);
// by ID

G=gap.SmallGroup(128,1896);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,520,1018,4037,124]);
// Polycyclic

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

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