C2^3.8Dic6 - GroupNames

G = C₂³.₈Dic₆ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₈Dic₆, M₄(2).₃₀D₆, C₁₂.₃₇(C₄⋊C₄), (C₂×C₁₂).₂₈Q₈, C₁₂.₄₄₁(C₂×D₄), (C₂×C₁₂).₄₈₃D₄, (C₂×C₄).₁₆Dic₆, C₄.Dic₃.₆C₄, (C₂²×C₆).₁₆Q₈, C₁₂.₆₈(C₂²×C₄), (C₂²×C₄).₁₅₀D₆, C₁₂.₅₃D₄⋊₁₃C₂, C₂².₄(C₂×Dic₆), C₃⋊₂(M₄(2).C₄), C₄.₂₁(Dic₃⋊C₄), (C₂×C₁₂).₄₁₅C₂³, (C₆×M₄(2)).₂₇C₂, (C₂×M₄(2)).₁₆S₃, C₄.Dic₃.₄₁C₂², C₂².₁₇(Dic₃⋊C₄), (C₂²×C₁₂).₁₈₃C₂², (C₃×M₄(2)).₃₃C₂², C₃⋊C₈.₅(C₂×C₄), C₄.₉₀(S₃×C₂×C₄), C₆.₅₂(C₂×C₄⋊C₄), (C₂×C₄).₄₉(C₄×S₃), (C₂×C₆).₁₁(C₂×Q₈), (C₂×C₆).₅₄(C₄⋊C₄), C₄.₁₃₁(C₂×C₃⋊D₄), (C₂×C₁₂).₁₀₃(C₂×C₄), (C₂×C₃⋊C₈).₁₄₃C₂², C₂.₁₉(C₂×Dic₃⋊C₄), (C₂×C₄).₁₉₄(C₃⋊D₄), (C₂×C₄).₅₁₁(C₂²×S₃), (C₂×C₄.Dic₃).₂₄C₂, SmallGroup(192,683)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₂³.₈Dic₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₂×C₃⋊C₈ — C₂×C₄.Dic₃ — C₂³.₈Dic₆

Lower central

C₃ — C₆ — C₁₂ — C₂³.₈Dic₆

Upper central

C₁ — C₄ — C₂²×C₄ — C₂×M₄(2)

Generators and relations for C₂³.₈Dic₆
G = < a,b,c,d,e | a²=b²=c²=1, d¹²=c, e²=bd⁶, ab=ba, eae^-1=ac=ca, ad=da, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=bcd¹¹ >

Subgroups: 184 in 102 conjugacy classes, 59 normal (39 characteristic)
C₁, C₂, C₂^[×3], C₃, C₄^[×4], C₂²^[×3], C₂², C₆, C₆^[×3], C₈^[×8], C₂×C₄^[×6], C₂³, C₁₂^[×4], C₂×C₆^[×3], C₂×C₆, C₂×C₈^[×4], M₄(2)^[×2], M₄(2)^[×8], C₂²×C₄, C₃⋊C₈^[×4], C₃⋊C₈^[×2], C₂₄^[×2], C₂×C₁₂^[×6], C₂²×C₆, C₈.C₄^[×4], C₂×M₄(2), C₂×M₄(2)^[×2], C₂×C₃⋊C₈^[×2], C₂×C₃⋊C₈, C₄.Dic₃^[×6], C₄.Dic₃, C₂×C₂₄, C₃×M₄(2)^[×2], C₃×M₄(2), C₂²×C₁₂, M₄(2).C₄, C₁₂.₅₃D₄^[×4], C₂×C₄.Dic₃^[×2], C₆×M₄(2), C₂³.₈Dic₆
Quotients: C₁, C₂^[×7], C₄^[×4], C₂²^[×7], S₃, C₂×C₄^[×6], D₄^[×2], Q₈^[×2], C₂³, D₆^[×3], C₄⋊C₄^[×4], C₂²×C₄, C₂×D₄, C₂×Q₈, Dic₆^[×2], C₄×S₃^[×2], C₃⋊D₄^[×2], C₂²×S₃, C₂×C₄⋊C₄, Dic₃⋊C₄^[×4], C₂×Dic₆, S₃×C₂×C₄, C₂×C₃⋊D₄, M₄(2).C₄, C₂×Dic₃⋊C₄, C₂³.₈Dic₆

Smallest permutation representation of C₂³.₈Dic₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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	D₆	Dic₆	C₄×S₃	C₃⋊D₄	Dic₆	M₄(2).C₄	C₂³.₈Dic₆
kernel	C₂³.₈Dic₆	C₁₂.₅₃D₄	C₂×C₄.Dic₃	C₆×M₄(2)	C₄.Dic₃	C₂×M₄(2)	C₂×C₁₂	C₂×C₁₂	C₂²×C₆	M₄(2)	C₂²×C₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂³	C₃	C₁
# reps	1	4	2	1	8	1	2	1	1	2	1	2	4	4	2	2	4

Matrix representation of C₂³.₈Dic₆ ►in GL₆(𝔽₇₃)

1	0	0	0	0	0
0	1	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

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

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

64	0	0	0	0	0
13	65	0	0	0	0
0	0	72	71	0	0
0	0	60	1	0	0
0	0	59	72	0	46
0	0	1	1	72	0

33	25	0	0	0	0
44	40	0	0	0	0
0	0	1	0	71	0
0	0	0	0	1	1
0	0	14	0	72	0
0	0	13	27	1	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[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],[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],[64,13,0,0,0,0,0,65,0,0,0,0,0,0,72,60,59,1,0,0,71,1,72,1,0,0,0,0,0,72,0,0,0,0,46,0],[33,44,0,0,0,0,25,40,0,0,0,0,0,0,1,0,14,13,0,0,0,0,0,27,0,0,71,1,72,1,0,0,0,1,0,0] >;

C₂³.₈Dic₆ in GAP, Magma, Sage, TeX

C_2^3._8{\rm Dic}_6

% in TeX

G:=Group("C2^3.8Dic6");

// GroupNames label

G:=SmallGroup(192,683);

// by ID

G=gap.SmallGroup(192,683);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,58,136,438,102,6278]);

// 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,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^11>;

// generators/relations

G = C23.8Dic6 order 192 = 26·3

6th non-split extension by C23 of Dic6 acting via Dic6/C6=C22

G = C₂³.₈Dic₆ order 192 = 2⁶·3

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