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## G = D8⋊4Dic3order 192 = 26·3

### 4th semidirect product of D8 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D8⋊4Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — D4.Dic3 — D8⋊4Dic3
 Lower central C3 — C6 — C12 — D8⋊4Dic3
 Upper central C1 — C4 — C2×C4 — C4○D8

Generators and relations for D84Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a5, cbc-1=a4b, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 216 in 104 conjugacy classes, 53 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Dic3, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, C3⋊C8, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C8.26D4, C24⋊C4, C24.C4, Q83Dic3, D4.Dic3, C3×C4○D8, D84Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, S3×D4, D42S3, C22×Dic3, C8.26D4, D4×Dic3, D84Dic3

Smallest permutation representation of D84Dic3
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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 35)(10 34)(11 33)(12 40)(13 39)(14 38)(15 37)(16 36)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)
(1 46 40)(2 47 33)(3 48 34)(4 41 35)(5 42 36)(6 43 37)(7 44 38)(8 45 39)(9 24 26 13 20 30)(10 17 27 14 21 31)(11 18 28 15 22 32)(12 19 29 16 23 25)
(2 6)(4 8)(9 32 13 28)(10 29 14 25)(11 26 15 30)(12 31 16 27)(17 19 21 23)(18 24 22 20)(33 43)(34 48)(35 45)(36 42)(37 47)(38 44)(39 41)(40 46)

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - - + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D4 D6 Dic3 Dic3 Dic3 D6 C4○D4 S3×D4 D4⋊2S3 C8.26D4 D8⋊4Dic3 kernel D8⋊4Dic3 C24⋊C4 C24.C4 Q8⋊3Dic3 D4.Dic3 C3×C4○D8 C3×D8 C3×SD16 C3×Q16 C4○D8 C3⋊C8 C2×C8 D8 SD16 Q16 C4○D4 C2×C6 C4 C22 C3 C1 # reps 1 1 1 2 2 1 2 4 2 1 2 1 1 2 1 2 2 1 1 2 4

Matrix representation of D84Dic3 in GL4(𝔽5) generated by

 0 4 0 0 3 0 0 0 0 0 2 2 0 0 2 3
,
 0 0 4 0 0 0 4 4 4 0 0 0 1 4 0 0
,
 3 3 0 0 1 3 0 0 0 0 4 2 0 0 2 0
,
 3 0 0 0 0 2 0 0 0 0 4 0 0 0 2 1
G:=sub<GL(4,GF(5))| [0,3,0,0,4,0,0,0,0,0,2,2,0,0,2,3],[0,0,4,1,0,0,0,4,4,4,0,0,0,4,0,0],[3,1,0,0,3,3,0,0,0,0,4,2,0,0,2,0],[3,0,0,0,0,2,0,0,0,0,4,2,0,0,0,1] >;

D84Dic3 in GAP, Magma, Sage, TeX

D_8\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("D8:4Dic3");
// GroupNames label

G:=SmallGroup(192,756);
// by ID

G=gap.SmallGroup(192,756);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,758,219,136,851,438,102,6278]);
// Polycyclic

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

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