C2^4.35D6 - GroupNames

G = C₂⁴.₃₅D₆ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₃₅D₆, C₆.₂2₊¹⁺⁴, C₂³⋊₄(C₄×S₃), (C₂²×C₄)⋊₆D₆, C₂²⋊C₄⋊₅₁D₆, C₆.₉(C₂³×C₄), D₆⋊C₄⋊₅₇C₂², (C₂×C₆).₃₀C₂⁴, D₆.₂(C₂²×C₄), C₂.₁(D₄⋊₆D₆), Dic₃⋊₄D₄⋊₃₉C₂, (C₂×C₁₂).₅₇₁C₂³, Dic₃⋊C₄⋊₅₈C₂², (C₂²×C₁₂)⋊₃₄C₂², C₃⋊₁(C₂².₁₁C₂⁴), (C₄×Dic₃)⋊₄₆C₂², (C₂³×C₆).₅₆C₂², C₂².₁₉(S₃×C₂³), Dic₃.₃(C₂²×C₄), C₆.D₄⋊₆₇C₂², C₂³.₁₆D₆⋊₂₄C₂, (S₃×C₂³).₃₀C₂², C₂³.₂₃₀(C₂²×S₃), (C₂²×C₆).₁₂₂C₂³, (C₂²×Dic₃)⋊₅C₂², (C₂²×S₃).₁₅₀C₂³, (C₂×Dic₃).₁₇₇C₂³, C₃⋊D₄⋊₉(C₂×C₄), (C₂×C₃⋊D₄)⋊₉C₄, (C₄×C₃⋊D₄)⋊₃₃C₂, (C₂×C₂²⋊C₄)⋊₇S₃, (S₃×C₂×C₄)⋊₃₉C₂², C₂.₁₁(S₃×C₂²×C₄), C₂².₂₄(S₃×C₂×C₄), (C₆×C₂²⋊C₄)⋊₂₆C₂, (S₃×C₂²⋊C₄)⋊₂₃C₂, (C₂²×S₃)⋊₆(C₂×C₄), (C₂²×C₆)⋊₁₀(C₂×C₄), (C₂×Dic₃)⋊₁₀(C₂×C₄), (C₂×C₆).₁₈(C₂²×C₄), (C₂²×C₃⋊D₄).₉C₂, (C₂×C₆.D₄)⋊₁₅C₂, (C₃×C₂²⋊C₄)⋊₆₁C₂², (C₂×C₄).₂₅₇(C₂²×S₃), (C₂×C₃⋊D₄).₈₈C₂², SmallGroup(192,1045)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆ — C₂⁴.₃₅D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — C₂²×C₃⋊D₄ — C₂⁴.₃₅D₆

Lower central

C₃ — C₆ — C₂⁴.₃₅D₆

Upper central

C₁ — C₂² — C₂×C₂²⋊C₄

Generators and relations for C₂⁴.₃₅D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁶=f²=c, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e⁵ >

Subgroups: 840 in 338 conjugacy classes, 151 normal (21 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, Dic₃, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²×D₄, C₄×Dic₃, Dic₃⋊C₄, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, S₃×C₂×C₄, C₂²×Dic₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×C₁₂, S₃×C₂³, C₂³×C₆, C₂².₁₁C₂⁴, C₂³.₁₆D₆, S₃×C₂²⋊C₄, Dic₃⋊₄D₄, C₄×C₃⋊D₄, C₂×C₆.D₄, C₆×C₂²⋊C₄, C₂²×C₃⋊D₄, C₂⁴.₃₅D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, C₂³, D₆, C₂²×C₄, C₂⁴, C₄×S₃, C₂²×S₃, C₂³×C₄, 2₊¹⁺⁴, S₃×C₂×C₄, S₃×C₂³, C₂².₁₁C₂⁴, S₃×C₂²×C₄, D₄⋊₆D₆, C₂⁴.₃₅D₆

Smallest permutation representation of C₂⁴.₃₅D₆
►On 48 points

Generators in S₄₈

(2 47)(4 37)(6 39)(8 41)(10 43)(12 45)(14 34)(16 36)(18 26)(20 28)(22 30)(24 32)

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

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim

type

image

C₁

C₂

C₄

S₃

D₆

C₄×S₃

2₊¹⁺⁴

D₄⋊₆D₆

kernel

C₂⁴.₃₅D₆

C₂³.₁₆D₆

S₃×C₂²⋊C₄

Dic₃⋊₄D₄

C₄×C₃⋊D₄

C₂×C₆.D₄

C₆×C₂²⋊C₄

C₂²×C₃⋊D₄

C₂×C₃⋊D₄

C₂×C₂²⋊C₄

C₂²⋊C₄

C₂²×C₄

C₂⁴

C₂³

C₆

C₂

# reps

Matrix representation of C₂⁴.₃₅D₆ ►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	8	0	12	0
0	0	0	8	0	12

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

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

8	8	0	0	0	0
5	0	0	0	0	0
0	0	5	5	2	2
0	0	8	0	11	0
0	0	0	0	8	8
0	0	0	0	5	0

5	5	0	0	0	0
0	8	0	0	0	0
0	0	5	5	2	2
0	0	0	8	0	11
0	0	0	0	8	8
0	0	0	0	0	5

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,8,0,0,0,0,1,0,8,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2],[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],[8,5,0,0,0,0,8,0,0,0,0,0,0,0,5,8,0,0,0,0,5,0,0,0,0,0,2,11,8,5,0,0,2,0,8,0],[5,0,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,5,8,0,0,0,0,2,0,8,0,0,0,2,11,8,5] >;

C₂⁴.₃₅D₆ in GAP, Magma, Sage, TeX

C_2^4._{35}D_6

% in TeX

G:=Group("C2^4.35D6");

// GroupNames label

G:=SmallGroup(192,1045);

// by ID

G=gap.SmallGroup(192,1045);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,570,80,6278]);

// 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,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,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 = C24.35D6 order 192 = 26·3

24th non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₃₅D₆ order 192 = 2⁶·3

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