C6.56ES+(2,2) - GroupNames

G = C₆.₅₆2₊¹⁺⁴ order 192 = 2⁶·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₅₆2₊¹⁺⁴, C₄⋊C₄⋊₁₃D₆, (C₂×Q₈)⋊₁₀D₆, C₂²⋊Q₈⋊₂₁S₃, D₆⋊D₄⋊₁₇C₂, C₁₂⋊D₄⋊₂₇C₂, C₁₂⋊₇D₄⋊₄₆C₂, D₆⋊C₄⋊₃₃C₂², (C₂×D₁₂)⋊₈C₂², C₂²⋊C₄.₆₄D₆, (C₆×Q₈)⋊₁₀C₂², D₆.D₄⋊₂₂C₂, C₂.₃₈(D₄○D₁₂), (C₂×C₆).₁₈₈C₂⁴, (C₂×C₁₂).₆₄C₂³, C₄⋊Dic₃⋊₁₄C₂², (C₂²×C₄).₂₆₆D₆, Dic₃⋊₄D₄⋊₁₅C₂, C₁₂.₂₃D₄⋊₁₇C₂, C₂.₅₈(D₄⋊₆D₆), Dic₃⋊C₄⋊₂₀C₂², C₃⋊₆(C₂².₃₂C₂⁴), (C₄×Dic₃)⋊₃₀C₂², (S₃×C₂³).₅₅C₂², (C₂²×S₃).₇₉C₂³, C₂².₂₀₉(S₃×C₂³), (C₂²×C₆).₂₁₆C₂³, C₂³.₂₀₆(C₂²×S₃), C₂².₄(Q₈⋊₃S₃), (C₂²×C₁₂).₃₁₆C₂², (C₂×Dic₃).₂₄₁C₂³, (C₂²×Dic₃).₁₂₄C₂², (C₂×D₆⋊C₄)⋊₂₇C₂, (S₃×C₂×C₄)⋊₁₉C₂², C₄⋊C₄⋊S₃⋊₂₃C₂, (C₃×C₄⋊C₄)⋊₂₂C₂², C₆.₁₁₆(C₂×C₄○D₄), (C₃×C₂²⋊Q₈)⋊₂₄C₂, (C₂×C₆).₂₈(C₄○D₄), C₂.₂₀(C₂×Q₈⋊₃S₃), (C₂×C₄).₁₈₅(C₂²×S₃), (C₂×C₃⋊D₄).₄₀C₂², (C₃×C₂²⋊C₄).₄₃C₂², SmallGroup(192,1203)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₆.₅₆2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — D₆⋊D₄ — C₆.₅₆2₊¹⁺⁴

Lower central

C₃ — C₂×C₆ — C₆.₅₆2₊¹⁺⁴

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for C₆.₅₆2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=c²=1, d²=b², e²=a³, ab=ba, ac=ca, dad^-1=a^-1, ae=ea, cbc=a³b^-1, bd=db, ebe^-1=a³b, cd=dc, ce=ec, ede^-1=a³b²d >

Subgroups: 768 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, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₄×S₃, D₁₂, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×Q₈, C₂²×S₃, C₂²×S₃, 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, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×C₄⋊C₄, S₃×C₂×C₄, C₂×D₁₂, C₂×D₁₂, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×Q₈, S₃×C₂³, C₂².₃₂C₂⁴, Dic₃⋊₄D₄, D₆⋊D₄, D₆.D₄, C₁₂⋊D₄, C₄⋊C₄⋊S₃, C₂×D₆⋊C₄, C₁₂⋊₇D₄, C₁₂.₂₃D₄, C₃×C₂²⋊Q₈, C₆.₅₆2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, Q₈⋊₃S₃, S₃×C₂³, C₂².₃₂C₂⁴, D₄⋊₆D₆, C₂×Q₈⋊₃S₃, D₄○D₁₂, C₆.₅₆2₊¹⁺⁴

Smallest permutation representation of C₆.₅₆2₊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 28 10 31)(2 29 11 32)(3 30 12 33)(4 25 7 34)(5 26 8 35)(6 27 9 36)(13 40 22 43)(14 41 23 44)(15 42 24 45)(16 37 19 46)(17 38 20 47)(18 39 21 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 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 38)

(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), (1,28,10,31)(2,29,11,32)(3,30,12,33)(4,25,7,34)(5,26,8,35)(6,27,9,36)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38), (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), (1,28,10,31)(2,29,11,32)(3,30,12,33)(4,25,7,34)(5,26,8,35)(6,27,9,36)(13,40,22,43)(14,41,23,44)(15,42,24,45)(16,37,19,46)(17,38,20,47)(18,39,21,48), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38), (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)], [(1,28,10,31),(2,29,11,32),(3,30,12,33),(4,25,7,34),(5,26,8,35),(6,27,9,36),(13,40,22,43),(14,41,23,44),(15,42,24,45),(16,37,19,46),(17,38,20,47),(18,39,21,48)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(25,46,34,37),(26,45,35,42),(27,44,36,41),(28,43,31,40),(29,48,32,39),(30,47,33,38)], [(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	···	4F	4G	4H	4I	4J	4K	4L	6A	6B	6C	6D	6E	12A	12B	12C	12D	12E	12F	12G	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	12	12	12	12	12	12	12	12
size	1	1	1	1	2	2	12	12	12	12	2	4	···	4	6	6	6	6	12	12	2	2	2	4	4	4	4	4	4	8	8	8	8

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄○D₄

2₊¹⁺⁴

Q₈⋊₃S₃

D₄⋊₆D₆

D₄○D₁₂

kernel

C₆.₅₆2₊¹⁺⁴

Dic₃⋊₄D₄

D₆⋊D₄

D₆.D₄

C₁₂⋊D₄

C₄⋊C₄⋊S₃

C₂×D₆⋊C₄

C₁₂⋊₇D₄

C₁₂.₂₃D₄

C₃×C₂²⋊Q₈

C₂²⋊Q₈

C₂²⋊C₄

C₄⋊C₄

C₂²×C₄

C₂×Q₈

C₂×C₆

C₆

C₂²

C₂

# reps

Matrix representation of C₆.₅₆2₊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	0	0	0	12
0	0	0	0	1	12

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

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

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

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

C₆.₅₆2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{56}2_+^{1+4}

% in TeX

G:=Group("C6.56ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1203);

// by ID

G=gap.SmallGroup(192,1203);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,184,675,297,136,6278]);

// Polycyclic

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

// generators/relations

G = C6.562+ 1+4 order 192 = 26·3

56th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

G = C₆.₅₆2₊¹⁺⁴ order 192 = 2⁶·3

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