C6.122ES+(2,2) - GroupNames

G = C₆.₁₂₂2₊¹⁺⁴ order 192 = 2⁶·3

31^st non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₄○D₄=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆.₁₂₂2₊¹⁺⁴, C₄⋊C₄⋊₃₀D₆, C₂²⋊C₄⋊₁₇D₆, C₄.D₁₂⋊₃₂C₂, (D₄×Dic₃)⋊₂₈C₂, (C₂×D₄).₁₆₆D₆, C₂³⋊₂D₆.₂C₂, D₆.₂₁(C₄○D₄), C₂³.₉D₆⋊₃₁C₂, C₂.₄₃(D₄○D₁₂), (C₂×C₁₂).₇₄C₂³, (C₂×C₆).₂₀₂C₂⁴, D₆⋊C₄.₃₃C₂², C₄⋊Dic₃⋊₃₉C₂², (C₂²×C₄).₂₇₄D₆, C₂².D₄⋊₇S₃, Dic₃⋊₄D₄⋊₂₁C₂, Dic₃⋊C₄⋊₂₄C₂², (C₄×Dic₃)⋊₅₅C₂², (C₂×Dic₆)⋊₃₀C₂², (C₆×D₄).₁₄₀C₂², Dic₃.D₄⋊₃₂C₂, C₂³.₁₁D₆⋊₃₃C₂, C₃⋊₆(C₂².₄₅C₂⁴), C₂³.₂₁D₆⋊₂₁C₂, C₆.D₄⋊₃₁C₂², C₂³.₂₆D₆⋊₁₂C₂, (C₂²×S₃).₈₆C₂³, (S₃×C₂³).₅₉C₂², C₂³.₂₁₂(C₂²×S₃), C₂².₂₂₃(S₃×C₂³), (C₂²×C₆).₂₂₂C₂³, C₂².₁₉(D₄⋊₂S₃), (C₂²×C₁₂).₁₁₄C₂², (C₂×Dic₃).₂₄₅C₂³, (C₂²×Dic₃)⋊₂₆C₂², (C₂×D₆⋊C₄)⋊₂₄C₂, C₄⋊C₄⋊S₃⋊₂₈C₂, C₄⋊C₄⋊₇S₃⋊₃₂C₂, C₂.₆₄(S₃×C₄○D₄), (S₃×C₂²⋊C₄)⋊₁₄C₂, (C₃×C₄⋊C₄)⋊₂₈C₂², C₆.₁₇₆(C₂×C₄○D₄), (C₂×C₆).₄₇(C₄○D₄), C₂.₅₃(C₂×D₄⋊₂S₃), (S₃×C₂×C₄).₁₁₁C₂², (C₃×C₂²⋊C₄)⋊₂₄C₂², (C₂×C₄).₂₉₇(C₂²×S₃), (C₂×C₃⋊D₄).₄₇C₂², (C₃×C₂².D₄)⋊₁₀C₂, SmallGroup(192,1217)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₆.₁₂₂2₊¹⁺⁴

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×S₃ — S₃×C₂³ — C₂×D₆⋊C₄ — C₆.₁₂₂2₊¹⁺⁴

Lower central

C₃ — C₂×C₆ — C₆.₁₂₂2₊¹⁺⁴

Upper central

C₁ — C₂² — C₂².D₄

Generators and relations for C₆.₁₂₂2₊¹⁺⁴
G = < a,b,c,d,e | a⁶=b⁴=c²=1, d²=b², e²=a³, bab^-1=eae^-1=a^-1, ac=ca, ad=da, cbc=b^-1, dbd^-1=a³b, be=eb, dcd^-1=a³c, ce=ec, ede^-1=a³b²d >

Subgroups: 672 in 248 conjugacy classes, 97 normal (91 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, D₆, 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×Q₈, C₂⁴, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂×Dic₆, S₃×C₂×C₄, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×C₁₂, C₆×D₄, S₃×C₂³, C₂².₄₅C₂⁴, Dic₃.D₄, S₃×C₂²⋊C₄, Dic₃⋊₄D₄, C₂³.₉D₆, C₂³.₁₁D₆, C₂³.₂₁D₆, C₄⋊C₄⋊₇S₃, C₄.D₁₂, C₄⋊C₄⋊S₃, C₂³.₂₆D₆, C₂×D₆⋊C₄, D₄×Dic₃, C₂³⋊₂D₆, C₃×C₂².D₄, C₆.₁₂₂2₊¹⁺⁴
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, D₄⋊₂S₃, S₃×C₂³, C₂².₄₅C₂⁴, C₂×D₄⋊₂S₃, S₃×C₄○D₄, D₄○D₁₂, C₆.₁₂₂2₊¹⁺⁴

Smallest permutation representation of C₆.₁₂₂2₊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 43 18 9)(2 48 13 8)(3 47 14 7)(4 46 15 12)(5 45 16 11)(6 44 17 10)(19 41 29 31)(20 40 30 36)(21 39 25 35)(22 38 26 34)(23 37 27 33)(24 42 28 32)

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

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3	4A	4B	4C	4D	4E	4F	4G	···	4L	4M	4N	4O	6A	6B	6C	6D	6E	6F	12A	12B	12C	12D	12E	12F	12G
order	1	2	2	2	2	2	2	2	2	2	3	4	4	4	4	4	4	4	···	4	4	4	4	6	6	6	6	6	6	12	12	12	12	12	12	12
size	1	1	1	1	2	2	4	6	6	12	2	2	2	4	4	4	4	6	···	6	12	12	12	2	2	2	4	4	8	4	4	4	4	8	8	8

39 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+			+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₆	D₆	C₄○D₄	C₄○D₄	2₊¹⁺⁴	D₄⋊₂S₃	S₃×C₄○D₄	D₄○D₁₂
kernel	C₆.₁₂₂2₊¹⁺⁴	Dic₃.D₄	S₃×C₂²⋊C₄	Dic₃⋊₄D₄	C₂³.₉D₆	C₂³.₁₁D₆	C₂³.₂₁D₆	C₄⋊C₄⋊₇S₃	C₄.D₁₂	C₄⋊C₄⋊S₃	C₂³.₂₆D₆	C₂×D₆⋊C₄	D₄×Dic₃	C₂³⋊₂D₆	C₃×C₂².D₄	C₂².D₄	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×D₄	D₆	C₂×C₆	C₆	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	1	2	1	1	1	1	1	1	3	2	1	1	4	4	1	2	2	2

Matrix representation of C₆.₁₂₂2₊¹⁺⁴ ►in GL₆(𝔽₁₃)

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

0	8	0	0	0	0
5	0	0	0	0	0
0	0	1	0	0	0
0	0	12	12	0	0
0	0	0	0	4	9
0	0	0	0	1	9

0	5	0	0	0	0
8	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	4	9
0	0	0	0	7	9

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	8	0
0	0	0	0	3	5

0	1	0	0	0	0
12	0	0	0	0	0
0	0	12	0	0	0
0	0	1	1	0	0
0	0	0	0	8	0
0	0	0	0	0	8

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,4,1,0,0,0,0,9,9],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,7,0,0,0,0,9,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,3,0,0,0,0,0,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C₆.₁₂₂2₊¹⁺⁴ in GAP, Magma, Sage, TeX

C_6._{122}2_+^{1+4}

% in TeX

G:=Group("C6.122ES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1217);

// by ID

G=gap.SmallGroup(192,1217);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,100,346,297,136,6278]);

// Polycyclic

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

// generators/relations

G = C6.1222+ 1+4 order 192 = 26·3

31st non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

G = C₆.₁₂₂2₊¹⁺⁴ order 192 = 2⁶·3

31^st non-split extension by C₆ of 2₊¹⁺⁴ acting via 2₊¹⁺⁴/C₄○D₄=C₂