D8.Dic3 - GroupNames

G = D₈.Dic₃ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₁₂.₁₁D₈, C₂₄.₁₄D₄, D₈.₂Dic₃, (C₃×D₈).₂C₄, (C₆×D₈).₁C₂, (C₂×D₈).₅S₃, (C₂×C₈).₄₉D₆, C₂₄.₂₅(C₂×C₄), C₁₂.C₈⋊₁C₂, C₂₄.C₄⋊₂C₂, C₈.₁(C₂×Dic₃), C₄.₁₄(D₄⋊S₃), (C₂×C₁₂).₁₁₅D₄, C₃⋊₃(M₅(2)⋊C₂), C₈.₂₄(C₃⋊D₄), (C₂×C₆).₃₀SD₁₆, (C₂×C₂₄).₂₉C₂², C₆.₂₇(D₄⋊C₄), C₁₂.₁₄(C₂²⋊C₄), C₄.₂(C₆.D₄), C₂.₇(D₄⋊Dic₃), C₂².₆(D₄.S₃), (C₂×C₄).₂₄(C₃⋊D₄), SmallGroup(192,122)

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

Export

Character table of D₈.Dic₃ in TeX

G = D8.Dic3 order 192 = 26·3

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

G = D₈.Dic₃ order 192 = 2⁶·3

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