C6.146ES+(2,2) - GroupNames

G = C₆.₁₄₆2₊¹⁺⁴ order 192 = 2⁶·3

55^th non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₄oD₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₁₄₆2₊¹⁺⁴, (C₂xD₄):₄₅D₆, (C₂xC₁₂):₁₆D₄, (C₂xQ₈):₃₇D₆, (C₂²xC₄):₃₄D₆, C₂³:₂D₆:₃₂C₂, C₁₂:₃D₄:₃₀C₂, C₁₂:₇D₄:₄₈C₂, D₆:C₄:₃₈C₂², C₁₂.₄₃₀(C₂xD₄), (C₆xD₄):₄₈C₂², (C₆xQ₈):₄₁C₂², C₂.₇₀(D₄oD₁₂), (C₂²xD₁₂):₂₁C₂, (C₂xC₆).₃₁₆C₂⁴, C₄:Dic₃:₆₆C₂², C₆.₁₆₆(C₂²xD₄), C₁₂.₂₃D₄:₃₂C₂, (C₂xC₁₂).₆₅₃C₂³, (C₂²xC₁₂):₃₂C₂², C₃:₇(C₂².₂₉C₂⁴), (C₄xDic₃):₄₅C₂², (C₂xD₁₂).₂₈₁C₂², C₂³.₂₆D₆:₃₉C₂, (S₃xC₂³).₈₀C₂², C₂³.₂₂₂(C₂²xS₃), (C₂²xC₆).₂₄₂C₂³, C₂².₃₂₅(S₃xC₂³), (C₂²xS₃).₁₃₈C₂³, (C₂xDic₃).₁₆₃C₂³, C₆.D₄.₁₃₆C₂², (C₆xC₄oD₄):₈C₂, (C₂xC₄oD₄):₁₂S₃, (C₂xC₄):₇(C₃:D₄), (C₂xC₆).₈₂(C₂xD₄), C₄.₃₃(C₂xC₃:D₄), (C₂xC₃:D₄):₃₁C₂², C₂.₃₉(C₂²xC₃:D₄), C₂².₂₄(C₂xC₃:D₄), (C₂xC₄).₂₅₁(C₂²xS₃), SmallGroup(192,1389)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₆.₁₄₆2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — C₂²xD₁₂ — C₆.₁₄₆2₊¹⁺⁴

Lower central

C₃ — C₂xC₆ — C₆.₁₄₆2₊¹⁺⁴

Upper central

C₁ — C₂² — C₂xC₄oD₄

Generators and relations for C₆.₁₄₆2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=e²=1, c²=a³, d²=a³b², ab=ba, cac^-1=dad^-1=a^-1, ae=ea, cbc^-1=a³b^-1, dbd^-1=a³b, be=eb, cd=dc, ece=a³c, ede=a³b²d >

Subgroups: 1032 in 334 conjugacy classes, 111 normal (23 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₄²:C₂, C₂²wrC₂, C₄:D₄, C₄.₄D₄, C₄:₁D₄, C₂²xD₄, C₂xC₄oD₄, C₄xDic₃, C₄:Dic₃, D₆:C₄, C₆.D₄, C₂xD₁₂, C₂xD₁₂, C₂xC₃:D₄, C₂²xC₁₂, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, S₃xC₂³, C₂².₂₉C₂⁴, C₂³.₂₆D₆, C₁₂:₇D₄, C₂³:₂D₆, C₁₂:₃D₄, C₁₂.₂₃D₄, C₂²xD₁₂, C₆xC₄oD₄, C₆.₁₄₆2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₃:D₄, C₂²xS₃, C₂²xD₄, 2₊¹⁺⁴, C₂xC₃:D₄, S₃xC₂³, C₂².₂₉C₂⁴, D₄oD₁₂, C₂²xC₃:D₄, C₆.₁₄₆2₊¹⁺⁴

Smallest permutation representation of C₆.₁₄₆2₊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 12 17 43)(2 7 18 44)(3 8 13 45)(4 9 14 46)(5 10 15 47)(6 11 16 48)(19 41 26 34)(20 42 27 35)(21 37 28 36)(22 38 29 31)(23 39 30 32)(24 40 25 33)

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃:D₄

2₊¹⁺⁴

D₄oD₁₂

kernel

C₆.₁₄₆2₊¹⁺⁴

C₂³.₂₆D₆

C₁₂:₇D₄

C₂³:₂D₆

C₁₂:₃D₄

C₁₂.₂₃D₄

C₂²xD₁₂

C₆xC₄oD₄

C₂xC₄oD₄

C₂xC₁₂

C₂²xC₄

C₂xD₄

C₂xQ₈

C₂xC₄

C₆

C₂

# reps

Matrix representation of C₆.₁₄₆2₊¹⁺⁴ ►in GL₆(F₁₃)

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

2	4	0	0	0	0
9	11	0	0	0	0
0	0	10	6	6	1
0	0	7	3	12	7
0	0	0	0	3	7
0	0	0	0	6	10

2	4	0	0	0	0
2	11	0	0	0	0
0	0	6	10	1	6
0	0	3	7	7	12
0	0	0	0	7	3
0	0	0	0	10	6

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

2	4	0	0	0	0
9	11	0	0	0	0
0	0	1	0	11	0
0	0	0	1	0	11
0	0	0	0	12	0
0	0	0	0	0	12

G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,10,7,0,0,0,0,6,3,0,0,0,0,6,12,3,6,0,0,1,7,7,10],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,6,3,0,0,0,0,10,7,0,0,0,0,1,7,7,10,0,0,6,12,3,6],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,0,12,0,12,0,0,12,0,12,0,0,0,0,2,0,1,0,0,2,0,1,0],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,11,0,12,0,0,0,0,11,0,12] >;

C₆.₁₄₆2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{146}2_+^{1+4}

% in TeX

G:=Group("C6.146ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1389);

// by ID

G=gap.SmallGroup(192,1389);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,570,6278]);

// Polycyclic

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

// generators/relations

G = C6.1462+ 1+4 order 192 = 26·3

55th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4oD4=C2

G = C₆.₁₄₆2₊¹⁺⁴ order 192 = 2⁶·3

55^th non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₄oD₄=C₂