D12:11D4 - GroupNames

G = D₁₂:₁₁D₄ order 192 = 2⁶·3

4^th semidirect product of D₁₂ and D₄ acting via D₄/C₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₂:₁₁D₄, C₄²:₂₉D₆, C₆.₇₇2₊¹⁺⁴, C₃:₄D₄², C₄:₂(S₃xD₄), D₆:₈(C₂xD₄), C₁₂:₃(C₂xD₄), (C₂xD₄):₂₆D₆, C₄:₁D₄:₇S₃, (C₄xD₁₂):₄₉C₂, C₂³:₂D₆:₂₇C₂, D₆:₃D₄:₃₆C₂, (C₄xC₁₂):₂₇C₂², D₆:C₄:₇₀C₂², (C₆xD₄):₃₃C₂², C₆.₉₅(C₂²xD₄), (C₂xC₆).₂₆₁C₂⁴, C₄:Dic₃:₇₄C₂², C₂.₈₁(D₄:₆D₆), (C₂xC₁₂).₅₀₉C₂³, (S₃xC₂³):₁₃C₂², (C₂²xC₆).₇₅C₂³, C₂³.₇₇(C₂²xS₃), (C₂xD₁₂).₂₆₉C₂², C₆.D₄:₃₇C₂², C₂².₂₈₂(S₃xC₂³), (C₂²xS₃).₂₂₉C₂³, (C₂xDic₃).₁₃₆C₂³, (C₂xS₃xD₄):₂₀C₂, C₂.₆₈(C₂xS₃xD₄), (C₃xC₄:₁D₄):₈C₂, (S₃xC₂xC₄):₂₉C₂², (C₂xC₃:D₄):₂₇C₂², (C₂xC₄).₂₁₄(C₂²xS₃), SmallGroup(192,1276)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — D₁₂:₁₁D₄

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — C₂xS₃xD₄ — D₁₂:₁₁D₄

Lower central

C₃ — C₂xC₆ — D₁₂:₁₁D₄

Upper central

C₁ — C₂² — C₄:₁D₄

Generators and relations for D₁₂:₁₁D₄
G = < a,b,c,d | a¹²=b²=c⁴=d²=1, bab=a^-1, ac=ca, dad=a⁷, cbc^-1=a⁶b, bd=db, dcd=c^-1 >

Subgroups: 1376 in 428 conjugacy classes, 115 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₄xD₄, C₂²wrC₂, C₄:D₄, C₄:₁D₄, C₂²xD₄, C₄:Dic₃, D₆:C₄, C₆.D₄, C₄xC₁₂, S₃xC₂xC₄, C₂xD₁₂, S₃xD₄, C₂xC₃:D₄, C₆xD₄, S₃xC₂³, D₄², C₄xD₁₂, C₂³:₂D₆, D₆:₃D₄, C₃xC₄:₁D₄, C₂xS₃xD₄, D₁₂:₁₁D₄
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₂²xS₃, C₂²xD₄, 2₊¹⁺⁴, S₃xD₄, S₃xC₂³, D₄², C₂xS₃xD₄, D₄:₆D₆, D₁₂:₁₁D₄

Smallest permutation representation of D₁₂:₁₁D₄
►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 38)(2 37)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

2₊¹⁺⁴

S₃xD₄

D₄:₆D₆

kernel

D₁₂:₁₁D₄

C₄xD₁₂

C₂³:₂D₆

D₆:₃D₄

C₃xC₄:₁D₄

C₂xS₃xD₄

C₄:₁D₄

D₁₂

C₄²

C₂xD₄

C₆

C₄

C₂

# reps

Matrix representation of D₁₂:₁₁D₄ ►in GL₆(F₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	12	1	0	0
0	0	0	0	7	10
0	0	0	0	8	6

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	1	0	0
0	0	0	1	0	0
0	0	0	0	6	3
0	0	0	0	10	7

12	3	0	0	0	0
8	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	6	3
0	0	0	0	5	7

12	0	0	0	0	0
8	1	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	7	10
0	0	0	0	3	6

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,7,8,0,0,0,0,10,6],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,5,0,0,0,0,3,7],[12,8,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,3,0,0,0,0,10,6] >;

D₁₂:₁₁D₄ in GAP, Magma, Sage, TeX

D_{12}\rtimes_{11}D_4

% in TeX

G:=Group("D12:11D4");

// GroupNames label

G:=SmallGroup(192,1276);

// by ID

G=gap.SmallGroup(192,1276);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,570,297,136,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = D12:11D4 order 192 = 26·3

4th semidirect product of D12 and D4 acting via D4/C4=C2

G = D₁₂:₁₁D₄ order 192 = 2⁶·3

4^th semidirect product of D₁₂ and D₄ acting via D₄/C₄=C₂