C2xC2^3.6D6 - GroupNames

G = C₂×C₂³.₆D₆ order 192 = 2⁶·3

Direct product of C₂ and C₂³.₆D₆

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

Aliases: C₂×C₂³.₆D₆, C₂⁴.₂₂D₆, C₂³.₂₀D₁₂, C₆⋊₁(C₂³⋊C₄), (S₃×C₂³)⋊₄C₄, C₂²⋊C₄⋊₃₆D₆, C₂³.₂₂(C₄×S₃), (C₂²×C₆).₆₆D₄, (C₂²×Dic₃)⋊₅C₄, C₂².₁₄(C₂×D₁₂), C₂².₄₄(D₆⋊C₄), C₂³.₈₁(C₃⋊D₄), (C₂³×C₆).₃₇C₂², C₂³.₈₂(C₂²×S₃), C₆.D₄⋊₄₂C₂², (C₂²×C₆).₁₁₁C₂³, C₃⋊₂(C₂×C₂³⋊C₄), (C₂×C₃⋊D₄)⋊₃C₄, C₂.₈(C₂×D₆⋊C₄), (C₂×C₂²⋊C₄)⋊₁S₃, (C₆×C₂²⋊C₄)⋊₁C₂, C₂².₁₈(S₃×C₂×C₄), (C₂²×S₃)⋊₂(C₂×C₄), (C₂×Dic₃)⋊₂(C₂×C₄), (C₂×C₆).₄₃₃(C₂×D₄), C₆.₃₅(C₂×C₂²⋊C₄), (C₂×C₆.D₄)⋊₁C₂, (C₂×C₆).₁₂(C₂²×C₄), (C₂²×C₆).₅₂(C₂×C₄), (C₂²×C₃⋊D₄).₁C₂, C₂².₂₆(C₂×C₃⋊D₄), (C₂×C₆).₅₆(C₂²⋊C₄), (C₃×C₂²⋊C₄)⋊₄₄C₂², (C₂×C₃⋊D₄).₈₃C₂², SmallGroup(192,513)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂×C₂³.₆D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂×C₃⋊D₄ — C₂²×C₃⋊D₄ — C₂×C₂³.₆D₆

Lower central

C₃ — C₆ — C₂×C₆ — C₂×C₂³.₆D₆

Upper central

C₁ — C₂² — C₂⁴ — C₂×C₂²⋊C₄

Generators and relations for C₂×C₂³.₆D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁶=b, f²=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf^-1=bd=db, be=eb, ece^-1=fcf^-1=cd=dc, de=ed, df=fd, fef^-1=cde⁵ >

Subgroups: 680 in 210 conjugacy classes, 63 normal (41 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²×D₄, C₆.D₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂²×Dic₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂×C₃⋊D₄, C₂²×C₁₂, S₃×C₂³, C₂³×C₆, C₂×C₂³⋊C₄, C₂³.₆D₆, C₂×C₆.D₄, C₆×C₂²⋊C₄, C₂²×C₃⋊D₄, C₂×C₂³.₆D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, D₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄×S₃, D₁₂, C₃⋊D₄, C₂²×S₃, C₂³⋊C₄, C₂×C₂²⋊C₄, D₆⋊C₄, S₃×C₂×C₄, C₂×D₁₂, C₂×C₃⋊D₄, C₂×C₂³⋊C₄, C₂³.₆D₆, C₂×D₆⋊C₄, C₂×C₂³.₆D₆

Smallest permutation representation of C₂×C₂³.₆D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+				+	+	+	+		+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₄	D₆	D₆	C₄×S₃	D₁₂	C₃⋊D₄	C₂³⋊C₄	C₂³.₆D₆
kernel	C₂×C₂³.₆D₆	C₂³.₆D₆	C₂×C₆.D₄	C₆×C₂²⋊C₄	C₂²×C₃⋊D₄	C₂²×Dic₃	C₂×C₃⋊D₄	S₃×C₂³	C₂×C₂²⋊C₄	C₂²×C₆	C₂²⋊C₄	C₂⁴	C₂³	C₂³	C₂³	C₆	C₂
# reps	1	4	1	1	1	2	4	2	1	4	2	1	4	4	4	2	4

Matrix representation of C₂×C₂³.₆D₆ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	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	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	2	4	0	0
0	0	9	11	0	0
0	0	0	0	2	4
0	0	0	0	9	11

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	0	0	12	0
0	0	0	0	0	12

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

1	12	0	0	0	0
1	0	0	0	0	0
0	0	0	0	0	1
0	0	0	0	12	12
0	0	9	11	0	0
0	0	2	11	0	0

0	12	0	0	0	0
12	0	0	0	0	0
0	0	0	0	12	12
0	0	0	0	0	1
0	0	11	2	0	0
0	0	4	2	0	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,9,2,0,0,0,0,11,11,0,0,0,12,0,0,0,0,1,12,0,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,11,4,0,0,0,0,2,2,0,0,12,0,0,0,0,0,12,1,0,0] >;

C₂×C₂³.₆D₆ in GAP, Magma, Sage, TeX

C_2\times C_2^3._6D_6

% in TeX

G:=Group("C2xC2^3.6D6");

// GroupNames label

G:=SmallGroup(192,513);

// by ID

G=gap.SmallGroup(192,513);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,438,6278]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;

// generators/relations

G = C2×C23.6D6 order 192 = 26·3

Direct product of C2 and C23.6D6

G = C₂×C₂³.₆D₆ order 192 = 2⁶·3

Direct product of C₂ and C₂³.₆D₆