C6.46ES+(2,2) - GroupNames

G = C₆.₄₆2₊¹⁺⁴ order 192 = 2⁶·3

46^th non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂×D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₄₆2₊¹⁺⁴, C₄⋊C₄⋊₈D₆, (C₂×D₄)⋊₁₀D₆, C₄⋊D₄⋊₂₀S₃, C₂²⋊C₄⋊₁₂D₆, C₂³⋊₂D₆⋊₁₃C₂, (D₄×Dic₃)⋊₂₄C₂, (C₆×D₄)⋊₃₁C₂², C₂.₂₉(D₄○D₁₂), (C₂×C₆).₁₆₁C₂⁴, C₄⋊Dic₃⋊₁₂C₂², (C₂×Dic₆)⋊₈C₂², (C₂²×C₄).₂₄₄D₆, C₂³.₁₄D₆⋊₃₂C₂, C₂.₄₈(D₄⋊₆D₆), C₁₂.₄₈D₄⋊₄₄C₂, (C₂×C₁₂).₆₂₆C₂³, Dic₃⋊C₄⋊₃₄C₂², D₆⋊C₄.₁₄₆C₂², C₃⋊₅(C₂².₃₂C₂⁴), (C₄×Dic₃)⋊₂₅C₂², (C₂²×C₆).₂₇C₂³, C₂³.₃₁(C₂²×S₃), C₂³.₁₁D₆⋊₂₂C₂, C₆.D₄⋊₂₆C₂², C₂³.₂₁D₆⋊₁₁C₂, C₂².₇(D₄⋊₂S₃), (S₃×C₂³).₅₀C₂², (C₂²×S₃).₆₈C₂³, C₂².₁₈₂(S₃×C₂³), (C₂²×C₁₂).₃₁₂C₂², (C₂×Dic₃).₂₂₉C₂³, (C₂²×Dic₃)⋊₂₂C₂², (C₂×D₆⋊C₄)⋊₂₆C₂, C₄⋊C₄⋊S₃⋊₁₄C₂, C₆.₈₅(C₂×C₄○D₄), (C₃×C₄⋊D₄)⋊₂₃C₂, (C₃×C₄⋊C₄)⋊₁₅C₂², (C₂×C₆).₂₃(C₄○D₄), C₂.₄₀(C₂×D₄⋊₂S₃), (C₃×C₂²⋊C₄)⋊₁₇C₂², (C₂×C₄).₁₈₀(C₂²×S₃), (C₂×C₃⋊D₄).₃₄C₂², SmallGroup(192,1176)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₆.₄₆2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — C₂³⋊₂D₆ — C₆.₄₆2₊¹⁺⁴

Lower central

C₃ — C₂×C₆ — C₆.₄₆2₊¹⁺⁴

Upper central

C₁ — C₂² — C₄⋊D₄

Generators and relations for C₆.₄₆2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=c²=1, d²=a³b², e²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=b^-1, bd=db, ebe^-1=a³b, cd=dc, ce=ec, ede^-1=b²d >

Subgroups: 720 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, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, Dic₆, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂×Dic₆, C₂²×Dic₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×D₄, C₆×D₄, S₃×C₂³, C₂².₃₂C₂⁴, C₂³.₁₁D₆, C₂³.₂₁D₆, C₄⋊C₄⋊S₃, C₁₂.₄₈D₄, C₂×D₆⋊C₄, D₄×Dic₃, C₂³⋊₂D₆, C₂³.₁₄D₆, C₃×C₄⋊D₄, C₆.₄₆2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, D₄⋊₂S₃, S₃×C₂³, C₂².₃₂C₂⁴, C₂×D₄⋊₂S₃, D₄⋊₆D₆, D₄○D₁₂, C₆.₄₆2₊¹⁺⁴

Smallest permutation representation of C₆.₄₆2₊¹⁺⁴
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	6A	6B	6C	6D	6E	6F	6G	12A	12B	12C	12D	12E	12F
order	1	2	2	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	4	4	4	4	4	6	6	6	6	6	6	6	12	12	12	12	12	12
size	1	1	1	1	2	2	4	4	12	12	2	4	4	4	4	6	6	6	6	12	12	12	12	2	2	2	4	4	8	8	4	4	4	4	8	8

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄○D₄

2₊¹⁺⁴

D₄⋊₂S₃

D₄⋊₆D₆

D₄○D₁₂

kernel

C₆.₄₆2₊¹⁺⁴

C₂³.₁₁D₆

C₂³.₂₁D₆

C₄⋊C₄⋊S₃

C₁₂.₄₈D₄

C₂×D₆⋊C₄

D₄×Dic₃

C₂³⋊₂D₆

C₂³.₁₄D₆

C₃×C₄⋊D₄

C₄⋊D₄

C₂²⋊C₄

C₄⋊C₄

C₂²×C₄

C₂×D₄

C₂×C₆

C₆

C₂²

C₂

# reps

Matrix representation of C₆.₄₆2₊¹⁺⁴ ►in GL₆(𝔽₁₃)

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

12	0	0	0	0	0
8	1	0	0	0	0
0	0	0	0	1	12
0	0	1	1	2	1
0	0	8	8	12	0
0	0	9	8	12	0

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	8	12	0
0	0	8	8	0	12

5	0	0	0	0	0
0	5	0	0	0	0
0	0	4	11	0	0
0	0	2	9	0	0
0	0	9	0	11	9
0	0	11	2	11	2

5	11	0	0	0	0
0	8	0	0	0	0
0	0	2	9	0	0
0	0	4	11	0	0
0	0	9	0	11	9
0	0	0	4	4	2

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

C₆.₄₆2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{46}2_+^{1+4}

% in TeX

G:=Group("C6.46ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1176);

// by ID

G=gap.SmallGroup(192,1176);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,675,297,136,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;

// generators/relations

G = C6.462+ 1+4 order 192 = 26·3

46th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

G = C₆.₄₆2₊¹⁺⁴ order 192 = 2⁶·3

46^th non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₂×D₄=C₂