C2^4.41D6 - GroupNames

G = C₂⁴.₄₁D₆ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₄₁D₆, C₆.₅2₊¹⁺⁴, Dic₃⋊D₄⋊₂C₂, C₁₂⋊₇D₄⋊₄C₂, D₆⋊C₄⋊₂C₂², C₂⁴⋊₄S₃⋊₃C₂, C₂³.₉D₆⋊₂C₂, (C₂×D₁₂)⋊₄C₂², (C₂×C₆).₃₈C₂⁴, C₄⋊Dic₃⋊₆C₂², C₂²⋊C₄.₈₇D₆, (C₂²×C₄).₆₁D₆, C₂.₉(D₄⋊₆D₆), C₁₂.₄₈D₄⋊₄C₂, Dic₃⋊C₄⋊₂C₂², (C₂×Dic₆)⋊₃C₂², C₂³.₈D₆⋊₁C₂, (C₂×C₁₂).₁₃₁C₂³, C₂³.₁₁D₆⋊₂C₂, C₃⋊₁(C₂².₃₂C₂⁴), (C₄×Dic₃)⋊₄₈C₂², C₂³.₉₂(C₂²×S₃), C₂².₇₇(S₃×C₂³), (C₂³×C₆).₆₄C₂², C₂².₂₃(C₄○D₁₂), (C₂²×S₃).₁₀C₂³, (C₂²×C₆).₁₂₈C₂³, (C₂×Dic₃).₁₁C₂³, C₆.D₄.₂C₂², (C₂²×C₁₂).₃₅₅C₂², (C₄×C₃⋊D₄)⋊₃₄C₂, (S₃×C₂×C₄)⋊₄₁C₂², C₆.₁₆(C₂×C₄○D₄), (C₆×C₂²⋊C₄)⋊₂₀C₂, (C₂×C₂²⋊C₄)⋊₁₇S₃, C₂.₁₈(C₂×C₄○D₁₂), (C₂×C₃⋊D₄).₇C₂², (C₂×C₆).₁₀₄(C₄○D₄), (C₂×C₄).₂₆₁(C₂²×S₃), (C₃×C₂²⋊C₄).₁₀₉C₂², SmallGroup(192,1053)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁴.₄₁D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — C₂×C₃⋊D₄ — C₄×C₃⋊D₄ — C₂⁴.₄₁D₆

Lower central

C₃ — C₂×C₆ — C₂⁴.₄₁D₆

Upper central

C₁ — C₂² — C₂×C₂²⋊C₄

Generators and relations for C₂⁴.₄₁D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁶=f²=c, ab=ba, ac=ca, faf^-1=ad=da, ae=ea, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁵ >

Subgroups: 680 in 250 conjugacy classes, 95 normal (31 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, Dic₆, C₄×S₃, D₁₂, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₂×Dic₆, S₃×C₂×C₄, C₂×D₁₂, C₂×C₃⋊D₄, C₂²×C₁₂, C₂³×C₆, C₂².₃₂C₂⁴, C₂³.₈D₆, C₂³.₉D₆, Dic₃⋊D₄, C₂³.₁₁D₆, C₁₂.₄₈D₄, C₄×C₃⋊D₄, C₁₂⋊₇D₄, C₂⁴⋊₄S₃, C₆×C₂²⋊C₄, C₂⁴.₄₁D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, C₄○D₁₂, S₃×C₂³, C₂².₃₂C₂⁴, C₂×C₄○D₁₂, D₄⋊₆D₆, C₂⁴.₄₁D₆

Smallest permutation representation of C₂⁴.₄₁D₆
►On 48 points

Generators in S₄₈

(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)

(2 30)(4 32)(6 34)(8 36)(10 26)(12 28)(13 19)(14 37)(15 21)(16 39)(17 23)(18 41)(20 43)(22 45)(24 47)(38 44)(40 46)(42 48)

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄○D₄

C₄○D₁₂

2₊¹⁺⁴

D₄⋊₆D₆

kernel

C₂⁴.₄₁D₆

C₂³.₈D₆

C₂³.₉D₆

Dic₃⋊D₄

C₂³.₁₁D₆

C₁₂.₄₈D₄

C₄×C₃⋊D₄

C₁₂⋊₇D₄

C₂⁴⋊₄S₃

C₆×C₂²⋊C₄

C₂×C₂²⋊C₄

C₂²⋊C₄

C₂²×C₄

C₂⁴

C₂×C₆

C₂²

C₆

C₂

# reps

Matrix representation of C₂⁴.₄₁D₆ ►in GL₆(𝔽₁₃)

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	1	0
0	0	0	0	0	1

12	0	0	0	0	0
0	1	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
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

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

7	0	0	0	0	0
0	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

0	2	0	0	0	0
6	0	0	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0
0	0	0	1	0	0
0	0	1	0	0	0

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

C₂⁴.₄₁D₆ in GAP, Magma, Sage, TeX

C_2^4._{41}D_6

% in TeX

G:=Group("C2^4.41D6");

// GroupNames label

G:=SmallGroup(192,1053);

// by ID

G=gap.SmallGroup(192,1053);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,297,6278]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*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=e^5>;

// generators/relations

G = C24.41D6 order 192 = 26·3

30th non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₄₁D₆ order 192 = 2⁶·3

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