D8:5Dic3 - GroupNames

G = D₈:₅Dic₃ order 192 = 2⁶·3

The semidirect product of D₈ and Dic₃ acting through Inn(D₈)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₈:₅Dic₃, Q₁₆:₅Dic₃, SD₁₆:₃Dic₃, (C₃xD₈):₃C₄, C₃:₅(C₈oD₈), C₃:C₈.₂₃D₄, (C₃xQ₁₆):₃C₄, C₄oD₈.₅S₃, C₆.₉₉(C₄xD₄), C₂₄.₂₀(C₂xC₄), C₄oD₄.₃₈D₆, (C₈xDic₃):₂C₂, (C₃xSD₁₆):₄C₄, (C₂xC₈).₂₅₄D₆, C₄.₂₁₇(S₃xD₄), C₂₄.C₄:₈C₂, C₁₂.₃₇₆(C₂xD₄), D₄.Dic₃:₃C₂, Q₈.₈(C₂xDic₃), D₄.₃(C₂xDic₃), C₈.₁₁(C₂xDic₃), C₂.₁₆(D₄xDic₃), Q₈:₃Dic₃:₄C₂, C₁₂.₇₇(C₂²xC₄), (C₂xC₂₄).₄₄C₂², C₄.₇(C₂²xDic₃), (C₂xC₁₂).₄₆₇C₂³, C₂².₃(D₄:₂S₃), C₄.Dic₃.₂₂C₂², (C₄xDic₃).₂₄₆C₂², (C₃xC₄oD₈).₂C₂, (C₃xD₄).₁₀(C₂xC₄), (C₃xQ₈).₁₀(C₂xC₄), (C₂xC₆).₁₁(C₄oD₄), (C₂xC₃:C₈).₂₈₀C₂², (C₃xC₄oD₄).₉C₂², (C₂xC₄).₅₅₄(C₂²xS₃), SmallGroup(192,755)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — D₈:₅Dic₃

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₂xC₃:C₈ — D₄.Dic₃ — D₈:₅Dic₃

Lower central

C₃ — C₆ — C₁₂ — D₈:₅Dic₃

Upper central

C₁ — C₄ — C₂xC₄ — C₄oD₈

Generators and relations for D₈:₅Dic₃
G = < a,b,c,d | a⁸=b²=c⁶=1, d²=c³, bab=a^-1, ac=ca, ad=da, cbc^-1=a⁴b, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 216 in 106 conjugacy classes, 53 normal (31 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₄², C₂xC₈, C₂xC₈, M₄(2), D₈, SD₁₆, Q₁₆, C₄oD₄, C₃:C₈, C₃:C₈, C₂₄, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₄xC₈, C₄wrC₂, C₈.C₄, C₈oD₄, C₄oD₈, C₂xC₃:C₈, C₂xC₃:C₈, C₄.Dic₃, C₄.Dic₃, C₄xDic₃, C₂xC₂₄, C₃xD₈, C₃xSD₁₆, C₃xQ₁₆, C₃xC₄oD₄, C₈oD₈, C₈xDic₃, C₂₄.C₄, Q₈:₃Dic₃, D₄.Dic₃, C₃xC₄oD₈, D₈:₅Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, Dic₃, D₆, C₂²xC₄, C₂xD₄, C₄oD₄, C₂xDic₃, C₂²xS₃, C₄xD₄, S₃xD₄, D₄:₂S₃, C₂²xDic₃, C₈oD₈, D₄xDic₃, D₈:₅Dic₃

Smallest permutation representation of D₈:₅Dic₃
►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)

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+				+	+	+	-	-	-	+			+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₄	D₆	Dic₃	Dic₃	Dic₃	D₆	C₄oD₄	C₈oD₈	S₃xD₄	D₄:₂S₃	D₈:₅Dic₃
kernel	D₈:₅Dic₃	C₈xDic₃	C₂₄.C₄	Q₈:₃Dic₃	D₄.Dic₃	C₃xC₄oD₈	C₃xD₈	C₃xSD₁₆	C₃xQ₁₆	C₄oD₈	C₃:C₈	C₂xC₈	D₈	SD₁₆	Q₁₆	C₄oD₄	C₂xC₆	C₃	C₄	C₂²	C₁
# reps	1	1	1	2	2	1	2	4	2	1	2	1	1	2	1	2	2	8	1	1	4

Matrix representation of D₈:₅Dic₃ ►in GL₄(F₇₃) generated by

1	0	0	0
0	1	0	0
0	0	51	0
0	0	0	63

72	0	0	0
0	72	0	0
0	0	0	63
0	0	51	0

0	1	0	0
72	72	0	0
0	0	1	0
0	0	0	72

1	0	0	0
72	72	0	0
0	0	72	0
0	0	0	46

G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,51,0,0,0,0,63],[72,0,0,0,0,72,0,0,0,0,0,51,0,0,63,0],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,72],[1,72,0,0,0,72,0,0,0,0,72,0,0,0,0,46] >;

D₈:₅Dic₃ in GAP, Magma, Sage, TeX

D_8\rtimes_5{\rm Dic}_3

% in TeX

G:=Group("D8:5Dic3");

// GroupNames label

G:=SmallGroup(192,755);

// by ID

G=gap.SmallGroup(192,755);

# by ID

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

// 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,a*d=d*a,c*b*c^-1=a^4*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;

// generators/relations

G = D8:5Dic3 order 192 = 26·3

The semidirect product of D8 and Dic3 acting through Inn(D8)

G = D₈:₅Dic₃ order 192 = 2⁶·3

The semidirect product of D₈ and Dic₃ acting through Inn(D₈)