C2xQ8:3Dic3 - GroupNames

G = C₂×Q₈⋊₃Dic₃ order 192 = 2⁶·3

Direct product of C₂ and Q₈⋊₃Dic₃

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×Q₈⋊₃Dic₃, C₆⋊₃C₄≀C₂, (C₆×D₄)⋊₈C₄, (C₆×Q₈)⋊₈C₄, C₄○D₄⋊₅Dic₃, C₄○D₄.₅₂D₆, (C₂×D₄)⋊₈Dic₃, (C₂×Q₈)⋊₈Dic₃, Q₈⋊₆(C₂×Dic₃), D₄⋊₅(C₂×Dic₃), C₁₂.₄₅₂(C₂×D₄), (C₂×C₁₂).₁₉₇D₄, C₁₂.₈₅(C₂²×C₄), (C₂²×C₄).₃₇₃D₆, (C₂²×C₆).₁₁₃D₄, C₁₂.₉₉(C₂²⋊C₄), (C₂×C₁₂).₄₈₁C₂³, (C₄×Dic₃)⋊₆₃C₂², C₂³.₇₃(C₃⋊D₄), C₄.Dic₃⋊₂₃C₂², C₄.₁₅(C₂²×Dic₃), C₄.₃₃(C₆.D₄), (C₂²×C₁₂).₂₀₇C₂², C₂².₅(C₆.D₄), C₃⋊₄(C₂×C₄≀C₂), (C₃×C₄○D₄)⋊₃C₄, (C₂×C₄×Dic₃)⋊₄C₂, (C₃×D₄)⋊₁₈(C₂×C₄), (C₃×Q₈)⋊₁₇(C₂×C₄), (C₆×C₄○D₄).₄C₂, (C₂×C₆).₃₉(C₂×D₄), (C₂×C₄○D₄).₁₀S₃, C₄.₁₄₃(C₂×C₃⋊D₄), C₆.₈₄(C₂×C₂²⋊C₄), (C₂×C₁₂).₁₂₅(C₂×C₄), (C₂×C₄.Dic₃)⋊₂₁C₂, (C₂×C₄).₅₄(C₂×Dic₃), C₂².₁₁(C₂×C₃⋊D₄), (C₂×C₄).₂₈₂(C₃⋊D₄), (C₂×C₄).₅₆₆(C₂²×S₃), C₂.₂₀(C₂×C₆.D₄), (C₃×C₄○D₄).₄₁C₂², (C₂×C₆).₁₁₇(C₂²⋊C₄), SmallGroup(192,794)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₂×Q₈⋊₃Dic₃

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₄.Dic₃ — C₂×C₄.Dic₃ — C₂×Q₈⋊₃Dic₃

Lower central

C₃ — C₆ — C₁₂ — C₂×Q₈⋊₃Dic₃

Upper central

C₁ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄

Generators and relations for C₂×Q₈⋊₃Dic₃
G = < a,b,c,d,e | a²=b⁴=d⁶=1, c²=b², e²=d³, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=b^-1, bd=db, be=eb, dcd^-1=b²c, ece^-1=b^-1c, ede^-1=d^-1 >

Subgroups: 360 in 170 conjugacy classes, 71 normal (43 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₃⋊C₈, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₄≀C₂, C₂×C₄², C₂×M₄(2), C₂×C₄○D₄, C₂×C₃⋊C₈, C₄.Dic₃, C₄.Dic₃, C₄×Dic₃, C₄×Dic₃, C₂²×Dic₃, C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₆×Q₈, C₃×C₄○D₄, C₃×C₄○D₄, C₂×C₄≀C₂, Q₈⋊₃Dic₃, C₂×C₄.Dic₃, C₂×C₄×Dic₃, C₆×C₄○D₄, C₂×Q₈⋊₃Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, Dic₃, D₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×Dic₃, C₃⋊D₄, C₂²×S₃, C₄≀C₂, C₂×C₂²⋊C₄, C₆.D₄, C₂²×Dic₃, C₂×C₃⋊D₄, C₂×C₄≀C₂, Q₈⋊₃Dic₃, C₂×C₆.D₄, C₂×Q₈⋊₃Dic₃

Smallest permutation representation of C₂×Q₈⋊₃Dic₃
►On 48 points

Generators in S₄₈

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

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

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

(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)

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	4
type	+	+	+	+	+				+	+	+	+	-	-	-	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₄	D₄	D₆	Dic₃	Dic₃	Dic₃	D₆	C₃⋊D₄	C₃⋊D₄	C₄≀C₂	Q₈⋊₃Dic₃
kernel	C₂×Q₈⋊₃Dic₃	Q₈⋊₃Dic₃	C₂×C₄.Dic₃	C₂×C₄×Dic₃	C₆×C₄○D₄	C₆×D₄	C₆×Q₈	C₃×C₄○D₄	C₂×C₄○D₄	C₂×C₁₂	C₂²×C₆	C₂²×C₄	C₂×D₄	C₂×Q₈	C₄○D₄	C₄○D₄	C₂×C₄	C₂³	C₆	C₂
# reps	1	4	1	1	1	2	2	4	1	3	1	1	1	1	2	2	6	2	8	4

Matrix representation of C₂×Q₈⋊₃Dic₃ ►in GL₅(𝔽₇₃)

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

1	0	0	0	0
0	46	27	0	0
0	0	27	0	0
0	0	0	72	0
0	0	0	0	72

1	0	0	0	0
0	27	0	0	0
0	54	46	0	0
0	0	0	43	13
0	0	0	60	30

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

27	0	0	0	0
0	27	60	0	0
0	0	1	0	0
0	0	0	72	0
0	0	0	1	1

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,46,0,0,0,0,27,27,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,27,54,0,0,0,0,46,0,0,0,0,0,43,60,0,0,0,13,30],[72,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,72,1,0,0,0,72,0],[27,0,0,0,0,0,27,0,0,0,0,60,1,0,0,0,0,0,72,1,0,0,0,0,1] >;

C₂×Q₈⋊₃Dic₃ in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3{\rm Dic}_3

% in TeX

G:=Group("C2xQ8:3Dic3");

// GroupNames label

G:=SmallGroup(192,794);

// by ID

G=gap.SmallGroup(192,794);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,136,1684,438,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;

// generators/relations

G = C2×Q8⋊3Dic3 order 192 = 26·3

Direct product of C2 and Q8⋊3Dic3

G = C₂×Q₈⋊₃Dic₃ order 192 = 2⁶·3

Direct product of C₂ and Q₈⋊₃Dic₃