C6.42ES+(2,2) - GroupNames

G = C₆.₄₂2₊¹⁺⁴ order 192 = 2⁶·3

42^nd non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂xD₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₄₂2₊¹⁺⁴, C₄:C₄:₇D₆, (C₂xD₄):₉D₆, C₄:D₄:₁₆S₃, C₂²:C₄:₂₉D₆, (C₂²xC₄):₂₃D₆, C₂³:₂D₆:₁₂C₂, D₆:₃D₄:₂₂C₂, D₆:Q₈:₁₅C₂, D₆.₄(C₄oD₄), (C₆xD₄):₁₅C₂², (C₂xC₆).₁₅₇C₂⁴, (C₂xC₁₂).₄₂C₂³, D₆:C₄.₇₀C₂², C₄:Dic₃:₃₃C₂², Dic₃:₄D₄:₁₀C₂, C₂³.₁₄D₆:₁₄C₂, C₂³.₁₂D₆:₁₈C₂, C₂.₄₄(D₄:₆D₆), Dic₃:C₄:₂₉C₂², (C₂²xC₁₂):₄₁C₂², C₃:₄(C₂².₃₂C₂⁴), (C₄xDic₃):₂₄C₂², (C₂xDic₆):₂₆C₂², C₂³.₈D₆:₁₈C₂, (C₂²xC₆).₂₄C₂³, C₂³.₂₇(C₂²xS₃), C₂³.₁₁D₆:₂₀C₂, C₂³.₂₃D₆:₂₂C₂, C₆.D₄:₂₅C₂², C₂³.₂₈D₆:₂₂C₂, (C₂²xS₃).₆₅C₂³, (S₃xC₂³).₄₉C₂², C₂².₁₇₈(S₃xC₂³), (C₂xDic₃).₇₆C₂³, (C₂²xDic₃):₂₁C₂², (C₄xC₃:D₄):₅₅C₂, (S₃xC₂²:C₄):₇C₂, C₄:C₄:S₃:₁₃C₂, C₂.₄₁(S₃xC₄oD₄), (C₃xC₄:D₄):₁₉C₂, (C₃xC₄:C₄):₁₄C₂², C₆.₁₅₄(C₂xC₄oD₄), (S₃xC₂xC₄).₈₅C₂², (C₃xC₂²:C₄):₁₆C₂², (C₂xC₄).₁₇₈(C₂²xS₃), (C₂xC₃:D₄).₃₀C₂², SmallGroup(192,1172)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₆.₄₂2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — S₃xC₂²:C₄ — C₆.₄₂2₊¹⁺⁴

Lower central

C₃ — C₂xC₆ — C₆.₄₂2₊¹⁺⁴

Upper central

C₁ — C₂² — C₄:D₄

Generators and relations for C₆.₄₂2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=c²=e²=1, d²=a³b², ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=a³b^-1, bd=db, be=eb, dcd^-1=ece=a³c, ede=a³b²d >

Subgroups: 720 in 250 conjugacy classes, 93 normal (91 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂⁴, Dic₆, C₄xS₃, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂xC₂²:C₄, C₄xD₄, C₂²wrC₂, C₄:D₄, C₄:D₄, C₂²:Q₈, C₂².D₄, C₄.₄D₄, C₄²:₂C₂, C₄xDic₃, Dic₃:C₄, C₄:Dic₃, D₆:C₄, C₆.D₄, C₃xC₂²:C₄, C₃xC₄:C₄, C₂xDic₆, S₃xC₂xC₄, C₂²xDic₃, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, S₃xC₂³, C₂².₃₂C₂⁴, C₂³.₈D₆, S₃xC₂²:C₄, Dic₃:₄D₄, C₂³.₁₁D₆, D₆:Q₈, C₄:C₄:S₃, C₄xC₃:D₄, C₂³.₂₈D₆, C₂³.₂₃D₆, C₂³.₁₂D₆, C₂³:₂D₆, D₆:₃D₄, C₂³.₁₄D₆, C₃xC₄:D₄, C₆.₄₂2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄oD₄, C₂⁴, C₂²xS₃, C₂xC₄oD₄, 2₊¹⁺⁴, S₃xC₂³, C₂².₃₂C₂⁴, D₄:₆D₆, S₃xC₄oD₄, C₆.₄₂2₊¹⁺⁴

Smallest permutation representation of C₆.₄₂2₊¹⁺⁴
►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 25 10 34)(2 26 11 35)(3 27 12 36)(4 28 7 31)(5 29 8 32)(6 30 9 33)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 48)

(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(25 34)(26 35)(27 36)(28 31)(29 32)(30 33)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)

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

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄oD₄

2₊¹⁺⁴

D₄:₆D₆

S₃xC₄oD₄

kernel

C₆.₄₂2₊¹⁺⁴

C₂³.₈D₆

S₃xC₂²:C₄

Dic₃:₄D₄

C₂³.₁₁D₆

D₆:Q₈

C₄:C₄:S₃

C₄xC₃:D₄

C₂³.₂₈D₆

C₂³.₂₃D₆

C₂³.₁₂D₆

C₂³:₂D₆

D₆:₃D₄

C₂³.₁₄D₆

C₃xC₄:D₄

C₄:D₄

C₂²:C₄

C₄:C₄

C₂²xC₄

C₂xD₄

D₆

C₆

C₂

# reps

Matrix representation of C₆.₄₂2₊¹⁺⁴ ►in GL₆(F₁₃)

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

5	0	0	0	0	0
0	5	0	0	0	0
0	0	0	0	11	9
0	0	0	0	4	2
0	0	11	9	0	0
0	0	4	2	0	0

12	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	1	0
0	0	0	0	0	1

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	0	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,11,4,0,0,0,0,9,2,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,12,0,0,0,0,0,12,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₆.₄₂2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{42}2_+^{1+4}

% in TeX

G:=Group("C6.42ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1172);

// by ID

G=gap.SmallGroup(192,1172);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,297,6278]);

// Polycyclic

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

// generators/relations

G = C6.422+ 1+4 order 192 = 26·3

42nd non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2xD4=C2

G = C₆.₄₂2₊¹⁺⁴ order 192 = 2⁶·3

42^nd non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂xD₄=C₂