C2^4.45D6 - GroupNames

G = C₂⁴.₄₅D₆ order 192 = 2⁶·3

34^th non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₄₅D₆, C₆.₂₈2₊¹⁺⁴, (C₂×D₄)⋊₆D₆, C₂²⋊C₄⋊₇D₆, C₂²≀C₂⋊₆S₃, C₂³⋊₂D₆⋊₆C₂, (C₂×Dic₃)⋊₈D₄, (C₆×D₄)⋊₉C₂², Dic₃⋊D₄⋊₁₄C₂, D₆⋊D₄⋊₁₀C₂, C₁₂⋊₃D₄⋊₁₂C₂, C₂⁴⋊₄S₃⋊₇C₂, D₆⋊C₄⋊₁₃C₂², Dic₃.₄(C₂×D₄), C₆.₅₈(C₂²×D₄), C₂².₄₁(S₃×D₄), C₂³.₁₄D₆⋊₄C₂, (C₂×D₁₂)⋊₂₀C₂², (C₂×C₁₂).₃₀C₂³, (C₂×C₆).₁₃₆C₂⁴, C₂.₃₀(D₄⋊₆D₆), Dic₃⋊C₄⋊₁₁C₂², C₃⋊₂(C₂².₂₉C₂⁴), (C₄×Dic₃)⋊₁₆C₂², (C₂×Dic₆)⋊₂₁C₂², C₂³.₁₆D₆⋊₃C₂, (C₂³×C₆).₆₉C₂², (C₂²×C₆).₁₀C₂³, C₂³.₁₁D₆⋊₁₃C₂, C₆.D₄⋊₁₆C₂², (S₃×C₂³).₄₄C₂², (C₂²×S₃).₅₅C₂³, C₂².₁₅₇(S₃×C₂³), C₂³.₁₁₉(C₂²×S₃), (C₂×Dic₃).₆₁C₂³, (C₂²×Dic₃)⋊₁₅C₂², C₂.₃₁(C₂×S₃×D₄), (S₃×C₂×C₄)⋊₉C₂², (C₂×C₆).₅₅(C₂×D₄), (C₃×C₂²≀C₂)⋊₇C₂, (C₂×D₄⋊₂S₃)⋊₇C₂, (C₂×C₃⋊D₄)⋊₉C₂², (C₂²×C₃⋊D₄)⋊₁₀C₂, (C₃×C₂²⋊C₄)⋊₇C₂², (C₂×C₄).₃₀(C₂²×S₃), SmallGroup(192,1151)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁴.₄₅D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — C₂³⋊₂D₆ — C₂⁴.₄₅D₆

Lower central

C₃ — C₂×C₆ — C₂⁴.₄₅D₆

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for C₂⁴.₄₅D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=1, f²=d, ab=ba, eae^-1=faf^-1=ac=ca, ad=da, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de^-1 >

Subgroups: 1008 in 334 conjugacy classes, 103 normal (39 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₄²⋊C₂, C₂²≀C₂, C₂²≀C₂, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₂²×D₄, C₂×C₄○D₄, C₄×Dic₃, Dic₃⋊C₄, D₆⋊C₄, C₆.D₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, D₄⋊₂S₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂×C₃⋊D₄, C₂×C₃⋊D₄, C₆×D₄, C₆×D₄, S₃×C₂³, C₂³×C₆, C₂².₂₉C₂⁴, C₂³.₁₆D₆, D₆⋊D₄, Dic₃⋊D₄, C₂³.₁₁D₆, C₂³⋊₂D₆, C₂³.₁₄D₆, C₁₂⋊₃D₄, C₂⁴⋊₄S₃, C₃×C₂²≀C₂, C₂×D₄⋊₂S₃, C₂²×C₃⋊D₄, C₂⁴.₄₅D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₂⁴, C₂²×S₃, C₂²×D₄, 2₊¹⁺⁴, S₃×D₄, S₃×C₂³, C₂².₂₉C₂⁴, C₂×S₃×D₄, D₄⋊₆D₆, C₂⁴.₄₅D₆

Smallest permutation representation of C₂⁴.₄₅D₆
►On 48 points

Generators in S₄₈

(2 32)(4 34)(6 36)(8 18)(10 14)(12 16)(20 40)(22 42)(24 38)(26 48)(28 44)(30 46)

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

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

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

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

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

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

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

36 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	6J	12A	12B	12C
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	6	12	12	12
size	1	1	1	1	2	2	4	4	4	12	12	12	2	4	4	4	6	6	6	6	12	12	12	2	2	2	4	···	4	8	8	8	8

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

2₊¹⁺⁴

S₃×D₄

D₄⋊₆D₆

kernel

C₂⁴.₄₅D₆

C₂³.₁₆D₆

D₆⋊D₄

Dic₃⋊D₄

C₂³.₁₁D₆

C₂³⋊₂D₆

C₂³.₁₄D₆

C₁₂⋊₃D₄

C₂⁴⋊₄S₃

C₃×C₂²≀C₂

C₂×D₄⋊₂S₃

C₂²×C₃⋊D₄

C₂²≀C₂

C₂×Dic₃

C₂²⋊C₄

C₂×D₄

C₂⁴

C₆

C₂²

C₂

# reps

Matrix representation of C₂⁴.₄₅D₆ ►in GL₆(𝔽₁₃)

12	0	0	0	0	0
0	12	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

1	0	0	0	0	0
0	12	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	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

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

0	12	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

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

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

C₂⁴.₄₅D₆ in GAP, Magma, Sage, TeX

C_2^4._{45}D_6

% in TeX

G:=Group("C2^4.45D6");

// GroupNames label

G:=SmallGroup(192,1151);

// by ID

G=gap.SmallGroup(192,1151);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,675,297,6278]);

// Polycyclic

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

// generators/relations

G = C24.45D6 order 192 = 26·3

34th non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₄₅D₆ order 192 = 2⁶·3

34^th non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²