C2^4:12D6 - GroupNames

G = C₂⁴:₁₂D₆ order 192 = 2⁶·3

7^th semidirect product of C₂⁴ and D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴:₁₂D₆, C₆.₈₉2₊¹⁺⁴, (C₂xD₄):₄₀D₆, (C₂²xC₄):₃₁D₆, (C₂²xC₆):₁₃D₄, C₂³:₂D₆:₃₀C₂, D₆:C₄:₃₆C₂², (C₂²xD₄):₁₃S₃, (C₆xD₄):₅₈C₂², C₂³:₅(C₃:D₄), C₃:₅(C₂³:₃D₄), C₂⁴:₄S₃:₁₂C₂, (C₂xC₆).₂₉₉C₂⁴, (C₂³xC₆):₁₄C₂², C₆.₁₄₆(C₂²xD₄), C₂³.₁₄D₆:₄₁C₂, C₂.₉₂(D₄:₆D₆), (C₂xC₁₂).₆₄₄C₂³, Dic₃:C₄:₃₈C₂², (S₃xC₂³):₁₅C₂², (C₂²xC₁₂):₄₄C₂², C₆.D₄:₆₄C₂², C₂³.₂₃D₆:₂₉C₂, C₂³.₂₈D₆:₂₈C₂, C₂³.₂₁₆(C₂²xS₃), (C₂²xC₆).₂₃₃C₂³, C₂².₃₁₂(S₃xC₂³), (C₂²xS₃).₁₃₀C₂³, (C₂xDic₃).₁₅₄C₂³, (C₂²xDic₃):₃₄C₂², (D₄xC₂xC₆):₁₇C₂, (C₂xC₆).₅₈₂(C₂xD₄), (C₂²xC₃:D₄):₁₇C₂, (C₂xC₃:D₄):₄₈C₂², C₂.₁₉(C₂²xC₃:D₄), C₂².₂₀(C₂xC₃:D₄), (C₂xC₆.D₄):₃₀C₂, (C₂xC₄).₂₃₈(C₂²xS₃), SmallGroup(192,1363)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂⁴:₁₂D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — C₂²xC₃:D₄ — C₂⁴:₁₂D₆

Lower central

C₃ — C₂xC₆ — C₂⁴:₁₂D₆

Upper central

C₁ — C₂² — C₂²xD₄

Generators and relations for C₂⁴:₁₂D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=f²=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe^-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 968 in 346 conjugacy classes, 111 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₂xDic₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₂xC₂²:C₄, C₂²wrC₂, C₄:D₄, C₂².D₄, C₂²xD₄, C₂²xD₄, Dic₃:C₄, D₆:C₄, C₆.D₄, C₂²xDic₃, C₂²xDic₃, C₂xC₃:D₄, C₂xC₃:D₄, C₂²xC₁₂, C₆xD₄, C₆xD₄, S₃xC₂³, C₂³xC₆, C₂³:₃D₄, C₂³.₂₈D₆, C₂³.₂₃D₆, C₂³:₂D₆, C₂³.₁₄D₆, C₂xC₆.D₄, C₂⁴:₄S₃, C₂²xC₃:D₄, D₄xC₂xC₆, C₂⁴:₁₂D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₃:D₄, C₂²xS₃, C₂²xD₄, 2₊¹⁺⁴, C₂xC₃:D₄, S₃xC₂³, C₂³:₃D₄, D₄:₆D₆, C₂²xC₃:D₄, C₂⁴:₁₂D₆

Smallest permutation representation of C₂⁴:₁₂D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₃:D₄

2₊¹⁺⁴

D₄:₆D₆

kernel

C₂⁴:₁₂D₆

C₂³.₂₈D₆

C₂³.₂₃D₆

C₂³:₂D₆

C₂³.₁₄D₆

C₂xC₆.D₄

C₂⁴:₄S₃

C₂²xC₃:D₄

D₄xC₂xC₆

C₂²xD₄

C₂²xC₆

C₂²xC₄

C₂xD₄

C₂⁴

C₂³

C₆

C₂

# reps

Matrix representation of C₂⁴:₁₂D₆ ►in GL₆(F₁₃)

1	0	0	0	0	0
0	1	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

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

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

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

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

C₂⁴:₁₂D₆ in GAP, Magma, Sage, TeX

C_2^4\rtimes_{12}D_6

% in TeX

G:=Group("C2^4:12D6");

// GroupNames label

G:=SmallGroup(192,1363);

// by ID

G=gap.SmallGroup(192,1363);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24:12D6 order 192 = 26·3

7th semidirect product of C24 and D6 acting via D6/C3=C22

G = C₂⁴:₁₂D₆ order 192 = 2⁶·3

7^th semidirect product of C₂⁴ and D₆ acting via D₆/C₃=C₂²