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## G = M4(2)⋊4Dic3order 192 = 26·3

### 4th semidirect product of M4(2) and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — M4(2)⋊4Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C2×C4.Dic3 — M4(2)⋊4Dic3
 Lower central C3 — C6 — C2×C6 — M4(2)⋊4Dic3
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2)⋊4Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=cac-1=a5, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 200 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, M4(2)⋊4C4, C2×C4.Dic3, C23.26D6, C6×M4(2), M4(2)⋊4Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, M4(2)⋊4C4, C6.C42, M4(2)⋊4Dic3

Smallest permutation representation of M4(2)⋊4Dic3
On 48 points
Generators in S48
(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)
(1 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 37)(18 34)(19 39)(20 36)(21 33)(22 38)(23 35)(24 40)(25 46)(26 43)(27 48)(28 45)(29 42)(30 47)(31 44)(32 41)
(1 46 35)(2 43 36 6 47 40)(3 48 37)(4 45 38 8 41 34)(5 42 39)(7 44 33)(9 30 24 13 26 20)(10 27 17)(11 32 18 15 28 22)(12 29 19)(14 31 21)(16 25 23)
(1 3)(2 15 6 11)(4 13 8 9)(5 7)(10 12)(14 16)(17 29)(18 47 22 43)(19 27)(20 45 24 41)(21 25)(23 31)(26 38 30 34)(28 36 32 40)(33 42)(35 48)(37 46)(39 44)

G:=sub<Sym(48)| (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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,37)(18,34)(19,39)(20,36)(21,33)(22,38)(23,35)(24,40)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41), (1,46,35)(2,43,36,6,47,40)(3,48,37)(4,45,38,8,41,34)(5,42,39)(7,44,33)(9,30,24,13,26,20)(10,27,17)(11,32,18,15,28,22)(12,29,19)(14,31,21)(16,25,23), (1,3)(2,15,6,11)(4,13,8,9)(5,7)(10,12)(14,16)(17,29)(18,47,22,43)(19,27)(20,45,24,41)(21,25)(23,31)(26,38,30,34)(28,36,32,40)(33,42)(35,48)(37,46)(39,44)>;

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), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11)(17,37)(18,34)(19,39)(20,36)(21,33)(22,38)(23,35)(24,40)(25,46)(26,43)(27,48)(28,45)(29,42)(30,47)(31,44)(32,41), (1,46,35)(2,43,36,6,47,40)(3,48,37)(4,45,38,8,41,34)(5,42,39)(7,44,33)(9,30,24,13,26,20)(10,27,17)(11,32,18,15,28,22)(12,29,19)(14,31,21)(16,25,23), (1,3)(2,15,6,11)(4,13,8,9)(5,7)(10,12)(14,16)(17,29)(18,47,22,43)(19,27)(20,45,24,41)(21,25)(23,31)(26,38,30,34)(28,36,32,40)(33,42)(35,48)(37,46)(39,44) );

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)], [(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11),(17,37),(18,34),(19,39),(20,36),(21,33),(22,38),(23,35),(24,40),(25,46),(26,43),(27,48),(28,45),(29,42),(30,47),(31,44),(32,41)], [(1,46,35),(2,43,36,6,47,40),(3,48,37),(4,45,38,8,41,34),(5,42,39),(7,44,33),(9,30,24,13,26,20),(10,27,17),(11,32,18,15,28,22),(12,29,19),(14,31,21),(16,25,23)], [(1,3),(2,15,6,11),(4,13,8,9),(5,7),(10,12),(14,16),(17,29),(18,47,22,43),(19,27),(20,45,24,41),(21,25),(23,31),(26,38,30,34),(28,36,32,40),(33,42),(35,48),(37,46),(39,44)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 2 2 2 2 1 1 2 2 2 12 12 12 12 2 2 2 4 4 4 4 4 4 12 12 12 12 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - - + - + image C1 C2 C2 C2 C4 C4 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 C4×S3 M4(2)⋊4C4 M4(2)⋊4Dic3 kernel M4(2)⋊4Dic3 C2×C4.Dic3 C23.26D6 C6×M4(2) C2×C3⋊C8 C6.D4 C3×M4(2) C2×M4(2) C2×C12 C2×C12 M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 1 1 1 4 4 4 1 3 1 2 1 2 2 2 4 2 2 4

Matrix representation of M4(2)⋊4Dic3 in GL4(𝔽73) generated by

 0 0 30 60 0 0 13 43 46 0 0 0 0 46 0 0
,
 30 60 0 0 13 43 0 0 0 0 43 13 0 0 60 30
,
 0 1 0 0 72 72 0 0 0 0 0 72 0 0 1 1
,
 14 7 0 0 66 59 0 0 0 0 0 46 0 0 46 0
G:=sub<GL(4,GF(73))| [0,0,46,0,0,0,0,46,30,13,0,0,60,43,0,0],[30,13,0,0,60,43,0,0,0,0,43,60,0,0,13,30],[0,72,0,0,1,72,0,0,0,0,0,1,0,0,72,1],[14,66,0,0,7,59,0,0,0,0,0,46,0,0,46,0] >;

M4(2)⋊4Dic3 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("M4(2):4Dic3");
// GroupNames label

G:=SmallGroup(192,118);
// by ID

G=gap.SmallGroup(192,118);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,184,1123,136,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=c^3,b*a*b=c*a*c^-1=a^5,d*a*d^-1=a^5*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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