C2xD4:6D6 - GroupNames

G = C₂×D₄⋊₆D₆ order 192 = 2⁶·3

Direct product of C₂ and D₄⋊₆D₆

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×D₄⋊₆D₆, C₂⁴⋊₁₃D₆, C₆.₇C₂⁵, D₁₂⋊₉C₂³, D₆.₃C₂⁴, C₁₂.₄₂C₂⁴, Dic₆⋊₉C₂³, C₆⋊₁2₊¹⁺⁴, Dic₃.₃C₂⁴, (C₂×D₄)⋊₄₆D₆, (C₄×S₃)⋊₁C₂³, (C₂×C₁₂)⋊₅C₂³, D₄⋊₇(C₂²×S₃), (C₃×D₄)⋊₈C₂³, (C₂²×C₄)⋊₃₅D₆, C₃⋊D₄⋊₃C₂³, C₂.₈(S₃×C₂⁴), (S₃×D₄)⋊₁₁C₂², (C₂²×D₄)⋊₁₆S₃, (C₆×D₄)⋊₅₁C₂², C₄.₄₂(S₃×C₂³), C₂³⋊₅(C₂²×S₃), (C₂²×C₆)⋊₇C₂³, C₃⋊₁(C₂×2₊¹⁺⁴), C₄○D₁₂⋊₂₂C₂², (C₂×D₁₂)⋊₆₁C₂², (C₂×C₆).₃₂₇C₂⁴, (C₂²×S₃)⋊₄C₂³, (C₂³×C₆)⋊₁₆C₂², (C₂×Dic₃)⋊₅C₂³, C₂².₈(S₃×C₂³), D₄⋊₂S₃⋊₁₂C₂², (S₃×C₂³)⋊₁₇C₂², (C₂²×C₁₂)⋊₂₆C₂², (C₂×Dic₆)⋊₇₂C₂², (C₂²×Dic₃)⋊₃₈C₂², (D₄×C₂×C₆)⋊₁₁C₂, (C₂×S₃×D₄)⋊₂₇C₂, (S₃×C₂×C₄)⋊₃₃C₂², (C₂×C₄)⋊₅(C₂²×S₃), (C₂×C₄○D₁₂)⋊₃₄C₂, (C₂×D₄⋊₂S₃)⋊₂₉C₂, (C₂²×C₃⋊D₄)⋊₂₁C₂, (C₂×C₃⋊D₄)⋊₅₂C₂², SmallGroup(192,1516)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂×D₄⋊₆D₆

Chief series

C₁ — C₃ — C₆ — D₆ — C₂²×S₃ — S₃×C₂³ — C₂×S₃×D₄ — C₂×D₄⋊₆D₆

Lower central

C₃ — C₆ — C₂×D₄⋊₆D₆

Upper central

C₁ — C₂² — C₂²×D₄

Generators and relations for C₂×D₄⋊₆D₆
G = < a,b,c,d,e | a²=b⁴=c²=d⁶=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd^-1=b^-1, be=eb, dcd^-1=ece=b²c, ede=d^-1 >

Subgroups: 2056 in 898 conjugacy classes, 447 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, C₄○D₁₂, S₃×D₄, D₄⋊₂S₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×D₄, S₃×C₂³, C₂³×C₆, C₂×2₊¹⁺⁴, C₂×C₄○D₁₂, C₂×S₃×D₄, C₂×D₄⋊₂S₃, D₄⋊₆D₆, C₂²×C₃⋊D₄, D₄×C₂×C₆, C₂×D₄⋊₆D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²×S₃, 2₊¹⁺⁴, C₂⁵, S₃×C₂³, C₂×2₊¹⁺⁴, D₄⋊₆D₆, S₃×C₂⁴, C₂×D₄⋊₆D₆

Smallest permutation representation of C₂×D₄⋊₆D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	2N	···	2U	3	4A	4B	4C	4D	4E	···	4L	6A	···	6G	6H	···	6O	12A	12B	12C	12D
order	1	2	2	2	2	···	2	2	···	2	3	4	4	4	4	4	···	4	6	···	6	6	···	6	12	12	12	12
size	1	1	1	1	2	···	2	6	···	6	2	2	2	2	2	6	···	6	2	···	2	4	···	4	4	4	4	4

54 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

2₊¹⁺⁴

D₄⋊₆D₆

kernel

C₂×D₄⋊₆D₆

C₂×C₄○D₁₂

C₂×S₃×D₄

C₂×D₄⋊₂S₃

D₄⋊₆D₆

C₂²×C₃⋊D₄

D₄×C₂×C₆

C₂²×D₄

C₂²×C₄

C₂×D₄

C₂⁴

C₆

C₂

# reps

Matrix representation of C₂×D₄⋊₆D₆ ►in GL₆(𝔽₁₃)

12	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

12	0	0	0	0	0
0	12	0	0	0	0
0	0	12	0	2	0
0	0	0	12	0	2
0	0	12	0	1	0
0	0	0	12	0	1

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

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

12	0	0	0	0	0
12	1	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

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

C₂×D₄⋊₆D₆ in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_6D_6

% in TeX

G:=Group("C2xD4:6D6");

// GroupNames label

G:=SmallGroup(192,1516);

// by ID

G=gap.SmallGroup(192,1516);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,297,1684,6278]);

// Polycyclic

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

// generators/relations

G = C2×D4⋊6D6 order 192 = 26·3

Direct product of C2 and D4⋊6D6

G = C₂×D₄⋊₆D₆ order 192 = 2⁶·3

Direct product of C₂ and D₄⋊₆D₆