C6.40ES+(2,2) - GroupNames

G = C₆.₄₀2₊¹⁺⁴ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₄₀2₊¹⁺⁴, C₄:C₄:₆D₆, C₃:D₄:₂D₄, (C₂xD₄):₂₄D₆, D₆:₈(C₄oD₄), C₄:D₄:₁₃S₃, C₃:₅(D₄:₅D₄), C₂²:C₄:₂₇D₆, D₆.₁₇(C₂xD₄), (C₂²xC₄):₂₁D₆, C₂³:₂D₆:₂₃C₂, Dic₃:D₄:₂₁C₂, C₂².₆(S₃xD₄), D₆:C₄:₆₇C₂², D₆:Q₈:₁₄C₂, (C₆xD₄):₃₀C₂², C₆.₆₉(C₂²xD₄), Dic₃:₄D₄:₉C₂, D₆.D₄:₁₂C₂, (C₂xC₆).₁₅₄C₂⁴, (C₂xC₁₂).₄₁C₂³, Dic₃.₂₂(C₂xD₄), C₂³.₁₄D₆:₃₀C₂, C₂.₄₂(D₄:₆D₆), Dic₃:C₄:₁₆C₂², (C₂²xC₁₂):₄₀C₂², (C₂xDic₆):₂₅C₂², (C₄xDic₃):₅₄C₂², C₂³.₂₃D₆:₉C₂, (C₂²xC₆).₂₁C₂³, C₂³.₁₁D₆:₁₉C₂, (C₂xD₁₂).₁₄₃C₂², C₆.D₄:₅₂C₂², (S₃xC₂³).₄₇C₂², C₂².₁₇₅(S₃xC₂³), C₂³.₁₉₂(C₂²xS₃), (C₂xDic₃).₇₄C₂³, (C₂²xS₃).₁₈₈C₂³, (C₂²xDic₃):₂₀C₂², (C₂xS₃xD₄):₁₂C₂, C₂.₄₂(C₂xS₃xD₄), (C₂xC₆).₆(C₂xD₄), (C₂xD₆:C₄):₃₆C₂, (C₄xC₃:D₄):₅₄C₂, (S₃xC₂²:C₄):₅C₂, (S₃xC₂xC₄):₅₀C₂², C₂.₃₉(S₃xC₄oD₄), (C₃xC₄:D₄):₁₆C₂, (C₃xC₄:C₄):₁₂C₂², C₆.₁₅₂(C₂xC₄oD₄), (C₂xD₄:₂S₃):₁₄C₂, (C₂xC₃:D₄):₁₆C₂², (C₃xC₂²:C₄):₁₄C₂², (C₂xC₄).₁₇₇(C₂²xS₃), SmallGroup(192,1169)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₆.₄₀2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — C₂xS₃xD₄ — C₆.₄₀2₊¹⁺⁴

Lower central

C₃ — C₂xC₆ — C₆.₄₀2₊¹⁺⁴

Upper central

C₁ — C₂² — C₄:D₄

Generators and relations for C₆.₄₀2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=c²=1, d²=b², e²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=b^-1, dbd^-1=ebe^-1=a³b, cd=dc, ce=ec, ede^-1=a³b²d >

Subgroups: 992 in 334 conjugacy classes, 105 normal (91 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, Dic₃, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂xC₂²:C₄, C₄xD₄, C₂²wrC₂, C₄:D₄, C₄:D₄, C₂²:Q₈, C₂².D₄, C₄.₄D₄, C₂²xD₄, C₂xC₄oD₄, C₄xDic₃, Dic₃:C₄, D₆:C₄, C₆.D₄, C₃xC₂²:C₄, C₃xC₄:C₄, C₂xDic₆, S₃xC₂xC₄, C₂xD₁₂, S₃xD₄, D₄:₂S₃, C₂²xDic₃, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, S₃xC₂³, D₄:₅D₄, S₃xC₂²:C₄, Dic₃:₄D₄, Dic₃:D₄, C₂³.₁₁D₆, D₆.D₄, D₆:Q₈, C₂xD₆:C₄, C₄xC₃:D₄, C₂³.₂₃D₆, C₂³:₂D₆, C₂³.₁₄D₆, C₃xC₄:D₄, C₂xS₃xD₄, C₂xD₄:₂S₃, C₆.₄₀2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₄oD₄, C₂⁴, C₂²xS₃, C₂²xD₄, C₂xC₄oD₄, 2₊¹⁺⁴, S₃xD₄, S₃xC₂³, D₄:₅D₄, C₂xS₃xD₄, D₄:₆D₆, S₃xC₄oD₄, C₆.₄₀2₊¹⁺⁴

Smallest permutation representation of C₆.₄₀2₊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 28 7 34)(2 29 8 35)(3 30 9 36)(4 25 10 31)(5 26 11 32)(6 27 12 33)(13 40 19 46)(14 41 20 47)(15 42 21 48)(16 37 22 43)(17 38 23 44)(18 39 24 45)

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	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	2	2	2	3	4	4	4	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	2	2	4	4	6	6	6	6	12	2	2	2	4	4	4	6	6	6	6	12	12	12	2	2	2	4	4	8	8	4	4	4	4	8	8

39 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₄oD₄

2₊¹⁺⁴

S₃xD₄

D₄:₆D₆

S₃xC₄oD₄

kernel

C₆.₄₀2₊¹⁺⁴

S₃xC₂²:C₄

Dic₃:₄D₄

Dic₃:D₄

C₂³.₁₁D₆

D₆.D₄

D₆:Q₈

C₂xD₆:C₄

C₄xC₃:D₄

C₂³.₂₃D₆

C₂³:₂D₆

C₂³.₁₄D₆

C₃xC₄:D₄

C₂xS₃xD₄

C₂xD₄:₂S₃

C₄:D₄

C₃:D₄

C₂²:C₄

C₄:C₄

C₂²xC₄

C₂xD₄

D₆

C₆

C₂²

C₂

# reps

Matrix representation of C₆.₄₀2₊¹⁺⁴ ►in GL₆(F₁₃)

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

0	1	0	0	0	0
12	0	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	3	1

1	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

8	0	0	0	0	0
0	5	0	0	0	0
0	0	1	0	0	0
0	0	12	12	0	0
0	0	0	0	8	1
0	0	0	0	2	5

5	0	0	0	0	0
0	8	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	5	12
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,3,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,8,2,0,0,0,0,1,5],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,12,8] >;

C₆.₄₀2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{40}2_+^{1+4}

% in TeX

G:=Group("C6.40ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1169);

// by ID

G=gap.SmallGroup(192,1169);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,136,6278]);

// Polycyclic

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

// generators/relations

G = C6.402+ 1+4 order 192 = 26·3

40th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2xD4=C2

G = C₆.₄₀2₊¹⁺⁴ order 192 = 2⁶·3

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