C2^3.9Dic6 - GroupNames

G = C₂³.₉Dic₆ order 192 = 2⁶·3

7^th non-split extension by C₂³ of Dic₆ acting via Dic₆/C₆=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₉Dic₆, M₄(2).₁Dic₃, C₂₄.₆(C₂×C₄), (C₂×C₈).₇₆D₆, C₄.₈₇(C₂×D₁₂), C₁₂.₂₂(C₄⋊C₄), (C₂×C₁₂).₂₉Q₈, C₈.₄(C₂×Dic₃), C₁₂.₃₀₅(C₂×D₄), (C₂×C₄).₁₅₀D₁₂, (C₂×C₁₂).₁₆₉D₄, C₄.₇(C₄⋊Dic₃), C₂₄.C₄⋊₁₃C₂, (C₂×C₄).₁₇Dic₆, (C₂²×C₆).₁₇Q₈, (C₂×C₂₄).₆₂C₂², (C₂²×C₄).₁₅₁D₆, (C₆×M₄(2)).₂C₂, (C₃×M₄(2)).₁C₄, (C₂×M₄(2)).₂S₃, C₂².₉(C₂×Dic₆), C₃⋊₃(M₄(2).C₄), C₁₂.₁₇₄(C₂²×C₄), (C₂×C₁₂).₇₉₆C₂³, C₂².₇(C₄⋊Dic₃), C₄.₂₈(C₂²×Dic₃), C₄.Dic₃.₃₆C₂², (C₂²×C₁₂).₁₈₄C₂², C₆.₅₃(C₂×C₄⋊C₄), (C₂×C₆).₄₁(C₂×Q₈), (C₂×C₆).₁₇(C₄⋊C₄), C₂.₁₅(C₂×C₄⋊Dic₃), (C₂×C₁₂).₁₀₄(C₂×C₄), (C₂×C₄).₂₂(C₂×Dic₃), (C₂×C₄).₇₂₁(C₂²×S₃), (C₂×C₄.Dic₃).₂₅C₂, SmallGroup(192,684)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₂³.₉Dic₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₄.Dic₃ — C₂×C₄.Dic₃ — C₂³.₉Dic₆

Lower central

C₃ — C₆ — C₁₂ — C₂³.₉Dic₆

Upper central

C₁ — C₄ — C₂²×C₄ — C₂×M₄(2)

Generators and relations for C₂³.₉Dic₆
G = < a,b,c,d,e | a²=b²=c²=1, d¹²=c, e²=bd⁶, ab=ba, dad^-1=ac=ca, ae=ea, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d¹¹ >

Subgroups: 184 in 102 conjugacy classes, 67 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₃⋊C₈, C₂₄, C₂×C₁₂, C₂×C₁₂, C₂²×C₆, C₈.C₄, C₂×M₄(2), C₂×M₄(2), C₂×C₃⋊C₈, C₄.Dic₃, C₄.Dic₃, C₂×C₂₄, C₃×M₄(2), C₂²×C₁₂, M₄(2).C₄, C₂₄.C₄, C₂×C₄.Dic₃, C₆×M₄(2), C₂³.₉Dic₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, Q₈, C₂³, Dic₃, D₆, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₆, D₁₂, C₂×Dic₃, C₂²×S₃, C₂×C₄⋊C₄, C₄⋊Dic₃, C₂×Dic₆, C₂×D₁₂, C₂²×Dic₃, M₄(2).C₄, C₂×C₄⋊Dic₃, C₂³.₉Dic₆

Smallest permutation representation of C₂³.₉Dic₆
►On 48 points

Generators in S₄₈

(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)

(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	8A	8B	8C	8D	8E	···	8L	12A	12B	12C	12D	12E	12F	24A	···	24H
order	1	2	2	2	2	3	4	4	4	4	4	6	6	6	6	6	8	8	8	8	8	···	8	12	12	12	12	12	12	24	···	24
size	1	1	2	2	2	2	1	1	2	2	2	2	2	2	4	4	4	4	4	4	12	···	12	2	2	2	2	4	4	4	···	4

42 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+		+	+	-	-	+	-	+	-	+	-
image	C₁	C₂	C₂	C₂	C₄	S₃	D₄	Q₈	Q₈	D₆	Dic₃	D₆	Dic₆	D₁₂	Dic₆	M₄(2).C₄	C₂³.₉Dic₆
kernel	C₂³.₉Dic₆	C₂₄.C₄	C₂×C₄.Dic₃	C₆×M₄(2)	C₃×M₄(2)	C₂×M₄(2)	C₂×C₁₂	C₂×C₁₂	C₂²×C₆	C₂×C₈	M₄(2)	C₂²×C₄	C₂×C₄	C₂×C₄	C₂³	C₃	C₁
# reps	1	4	2	1	8	1	2	1	1	2	4	1	2	4	2	2	4

Matrix representation of C₂³.₉Dic₆ ►in GL₆(𝔽₇₃)

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

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

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

72	1	0	0	0	0
72	0	0	0	0	0
0	0	72	71	0	0
0	0	60	1	0	0
0	0	13	72	0	27
0	0	1	1	72	0

27	46	0	0	0	0
0	46	0	0	0	0
0	0	1	0	71	0
0	0	0	0	1	1
0	0	14	0	72	0
0	0	59	46	1	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,72,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,1,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,60,13,1,0,0,71,1,72,1,0,0,0,0,0,72,0,0,0,0,27,0],[27,0,0,0,0,0,46,46,0,0,0,0,0,0,1,0,14,59,0,0,0,0,0,46,0,0,71,1,72,1,0,0,0,1,0,0] >;

C₂³.₉Dic₆ in GAP, Magma, Sage, TeX

C_2^3._9{\rm Dic}_6

% in TeX

G:=Group("C2^3.9Dic6");

// GroupNames label

G:=SmallGroup(192,684);

// by ID

G=gap.SmallGroup(192,684);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,100,136,1684,102,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=b*d^6,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^11>;

// generators/relations

G = C23.9Dic6 order 192 = 26·3

7th non-split extension by C23 of Dic6 acting via Dic6/C6=C22

G = C₂³.₉Dic₆ order 192 = 2⁶·3

7^th non-split extension by C₂³ of Dic₆ acting via Dic₆/C₆=C₂²