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G = D8.Dic3order 192 = 26·3

2nd non-split extension by D8 of Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D8.Dic3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C24.C4 — D8.Dic3
 Lower central C3 — C6 — C12 — C24 — D8.Dic3
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×D8

Generators and relations for D8.Dic3
G = < a,b,c,d | a8=b2=1, c6=a4, d2=a4c3, bab=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a5b, dcd-1=c5 >

Subgroups: 184 in 62 conjugacy classes, 27 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C8, C2×C4, D4, C23, C12, C2×C6, C2×C6, C16, C2×C8, M4(2), D8, D8, C2×D4, C3⋊C8, C24, C2×C12, C3×D4, C22×C6, C8.C4, M5(2), C2×D8, C3⋊C16, C4.Dic3, C2×C24, C3×D8, C3×D8, C6×D4, M5(2)⋊C2, C12.C8, C24.C4, C6×D8, D8.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C2×Dic3, C3⋊D4, D4⋊C4, D4⋊S3, D4.S3, C6.D4, M5(2)⋊C2, D4⋊Dic3, D8.Dic3

Character table of D8.Dic3

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 8E 12A 12B 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 8 8 2 2 2 2 2 2 8 8 8 8 2 2 4 24 24 4 4 12 12 12 12 4 4 4 4 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 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 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 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 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 i -i 1 -1 -i i -i i 1 -1 1 -1 linear of order 4 ρ6 1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 -i i 1 -1 i -i i -i 1 -1 1 -1 linear of order 4 ρ7 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 i -i 1 -1 i -i i -i 1 -1 1 -1 linear of order 4 ρ8 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 -i i 1 -1 -i i -i i 1 -1 1 -1 linear of order 4 ρ9 2 2 2 0 0 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 0 0 2 2 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 -1 2 2 -1 -1 -1 1 1 1 1 2 2 2 0 0 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ11 2 2 2 2 2 -1 2 2 -1 -1 -1 -1 -1 -1 -1 2 2 2 0 0 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 -2 0 0 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 -2 2 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ13 2 2 -2 0 0 2 2 -2 -2 -2 2 0 0 0 0 2 2 -2 0 0 2 -2 0 0 0 0 -2 2 -2 2 orthogonal lifted from D4 ρ14 2 2 -2 0 0 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 -2 2 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ15 2 2 -2 -2 2 -1 2 -2 1 1 -1 1 -1 -1 1 -2 -2 2 0 0 -1 1 0 0 0 0 -1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ16 2 2 -2 2 -2 -1 2 -2 1 1 -1 -1 1 1 -1 -2 -2 2 0 0 -1 1 0 0 0 0 -1 1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ17 2 2 2 0 0 -1 2 2 -1 -1 -1 √-3 -√-3 √-3 -√-3 -2 -2 -2 0 0 -1 -1 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ18 2 2 2 0 0 -1 2 2 -1 -1 -1 -√-3 √-3 -√-3 √-3 -2 -2 -2 0 0 -1 -1 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ19 2 2 2 0 0 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 -2 -2 √-2 -√-2 -√-2 √-2 0 0 0 0 complex lifted from SD16 ρ20 2 2 2 0 0 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 -2 -2 -√-2 √-2 √-2 -√-2 0 0 0 0 complex lifted from SD16 ρ21 2 2 -2 0 0 -1 2 -2 1 1 -1 √-3 √-3 -√-3 -√-3 2 2 -2 0 0 -1 1 0 0 0 0 1 -1 1 -1 complex lifted from C3⋊D4 ρ22 2 2 -2 0 0 -1 2 -2 1 1 -1 -√-3 -√-3 √-3 √-3 2 2 -2 0 0 -1 1 0 0 0 0 1 -1 1 -1 complex lifted from C3⋊D4 ρ23 4 4 -4 0 0 -2 -4 4 2 2 -2 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 orthogonal lifted from M5(2)⋊C2 ρ25 4 -4 0 0 0 4 0 0 0 0 -4 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 orthogonal lifted from M5(2)⋊C2 ρ26 4 4 4 0 0 -2 -4 -4 -2 -2 -2 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ27 4 -4 0 0 0 -2 0 0 2√-3 -2√-3 2 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 √-6 -√2 -√-6 √2 complex faithful ρ28 4 -4 0 0 0 -2 0 0 2√-3 -2√-3 2 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 -√-6 √2 √-6 -√2 complex faithful ρ29 4 -4 0 0 0 -2 0 0 -2√-3 2√-3 2 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 -√-6 -√2 √-6 √2 complex faithful ρ30 4 -4 0 0 0 -2 0 0 -2√-3 2√-3 2 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 √-6 √2 -√-6 -√2 complex faithful

Smallest permutation representation of D8.Dic3
On 48 points
Generators in S48
(1 22 10 19 7 16 4 13)(2 23 11 20 8 17 5 14)(3 24 12 21 9 18 6 15)(25 37 28 40 31 43 34 46)(26 38 29 41 32 44 35 47)(27 39 30 42 33 45 36 48)
(1 13)(2 20)(3 15)(4 22)(5 17)(6 24)(7 19)(8 14)(9 21)(10 16)(11 23)(12 18)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)(37 43)(39 45)(41 47)
(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 31 10 28 7 25 4 34)(2 36 11 33 8 30 5 27)(3 29 12 26 9 35 6 32)(13 37 22 46 19 43 16 40)(14 42 23 39 20 48 17 45)(15 47 24 44 21 41 18 38)

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

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

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

Matrix representation of D8.Dic3 in GL4(𝔽7) generated by

 5 1 0 1 2 5 5 1 0 3 0 5 6 2 1 4
,
 4 5 0 3 1 6 5 5 0 4 0 2 5 2 1 4
,
 3 6 6 6 3 0 3 6 4 0 5 3 3 5 3 6
,
 4 0 4 3 0 4 4 2 2 6 6 6 5 3 3 0
G:=sub<GL(4,GF(7))| [5,2,0,6,1,5,3,2,0,5,0,1,1,1,5,4],[4,1,0,5,5,6,4,2,0,5,0,1,3,5,2,4],[3,3,4,3,6,0,0,5,6,3,5,3,6,6,3,6],[4,0,2,5,0,4,6,3,4,4,6,3,3,2,6,0] >;

D8.Dic3 in GAP, Magma, Sage, TeX

D_8.{\rm Dic}_3
% in TeX

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

G:=SmallGroup(192,122);
// by ID

G=gap.SmallGroup(192,122);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,387,184,675,794,80,1684,851,102,6278]);
// Polycyclic

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

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