C6xES+(2,2)

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C₆×2₊⁽¹⁺⁴⁾, C₆.₂₄C₂⁵, C₁₂.₉₀C₂⁴, C₂⁴⋊₈(C₂×C₆), D₄⋊₄(C₂²×C₆), C₂.₄(C₂⁴×C₆), Q₈⋊₅(C₂²×C₆), (C₂×C₁₂)⋊₁₁C₂³, (C₃×D₄)⋊₁₅C₂³, (C₆×D₄)⋊₆₈C₂², (C₂²×D₄)⋊₁₅C₆, C₄.₁₃(C₂³×C₆), (C₂²×C₆)⋊₄C₂³, C₂³⋊₂(C₂²×C₆), (C₂³×C₆)⋊₆C₂², (C₃×Q₈)⋊₁₄C₂³, (C₆×Q₈)⋊₆₀C₂², (C₂×C₆).₃₈₇C₂⁴, C₂².₂(C₂³×C₆), (C₂²×C₁₂)⋊₅₃C₂², (D₄×C₂×C₆)⋊₂₇C₂, C₄○D₄⋊₈(C₂×C₆), (C₂×C₄○D₄)⋊₁₇C₆, (C₆×C₄○D₄)⋊₂₉C₂, (C₂×D₄)⋊₁₇(C₂×C₆), (C₂×C₄)⋊₂(C₂²×C₆), (C₂×Q₈)⋊₂₂(C₂×C₆), (C₂²×C₄)⋊₁₄(C₂×C₆), (C₃×C₄○D₄)⋊₂₆C₂², SmallGroup(192,1534)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₆×2₊⁽¹⁺⁴⁾

Chief series

C₁ — C₂ — C₆ — C₂×C₆ — C₃×D₄ — C₆×D₄ — C₃×2₊⁽¹⁺⁴⁾ — C₆×2₊⁽¹⁺⁴⁾

Lower central

C₁ — C₂ — C₆×2₊⁽¹⁺⁴⁾

Upper central

C₁ — C₂×C₆ — C₆×2₊⁽¹⁺⁴⁾

Subgroups: 1186 in 898 conjugacy classes, 754 normal (8 characteristic)
C₁, C₂, C₂^[×2], C₂^[×18], C₃, C₄^[×12], C₂², C₂²^[×18], C₂²^[×42], C₆, C₆^[×2], C₆^[×18], C₂×C₄^[×42], D₄^[×72], Q₈^[×8], C₂³^[×33], C₂³^[×12], C₁₂^[×12], C₂×C₆, C₂×C₆^[×18], C₂×C₆^[×42], C₂²×C₄^[×9], C₂×D₄^[×90], C₂×Q₈^[×2], C₄○D₄^[×48], C₂⁴^[×6], C₂×C₁₂^[×42], C₃×D₄^[×72], C₃×Q₈^[×8], C₂²×C₆^[×33], C₂²×C₆^[×12], C₂²×D₄^[×9], C₂×C₄○D₄^[×6], 2₊⁽¹⁺⁴⁾^[×16], C₂²×C₁₂^[×9], C₆×D₄^[×90], C₆×Q₈^[×2], C₃×C₄○D₄^[×48], C₂³×C₆^[×6], C₂×2₊⁽¹⁺⁴⁾, D₄×C₂×C₆^[×9], C₆×C₄○D₄^[×6], C₃×2₊⁽¹⁺⁴⁾^[×16], C₆×2₊⁽¹⁺⁴⁾

Quotients:
C₁, C₂^[×31], C₃, C₂²^[×155], C₆^[×31], C₂³^[×155], C₂×C₆^[×155], C₂⁴^[×31], C₂²×C₆^[×155], 2₊⁽¹⁺⁴⁾^[×2], C₂⁵, C₂³×C₆^[×31], C₂×2₊⁽¹⁺⁴⁾, C₃×2₊⁽¹⁺⁴⁾^[×2], C₂⁴×C₆, C₆×2₊⁽¹⁺⁴⁾

Generators and relations
G = < a,b,c,d,e | a⁶=b⁴=c²=e²=1, d²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=b²d >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₅(𝔽₁₃)

12	0	0	0	0
0	10	0	0	0
0	0	10	0	0
0	0	0	10	0
0	0	0	0	10

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

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

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

1	0	0	0	0
0	0	0	1	0
0	0	0	0	1
0	1	0	0	0
0	0	1	0	0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[12,0,0,0,0,0,0,0,0,1,0,0,0,12,0,0,0,1,0,0,0,12,0,0,0],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,12,0,0,0,1,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0] >;

102 conjugacy classes

class	1	2A	2B	2C	2D	···	2U	3A	3B	4A	···	4L	6A	···	6F	6G	···	6AP	12A	···	12X
order	1	2	2	2	2	···	2	3	3	4	···	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	2	···	2	1	1	2	···	2	1	···	1	2	···	2	2	···	2

102 irreducible representations

dim	1	1	1	1	1	1	1	1	4	4
type	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	2₊⁽¹⁺⁴⁾	C₃×2₊⁽¹⁺⁴⁾
kernel	C₆×2₊⁽¹⁺⁴⁾	D₄×C₂×C₆	C₆×C₄○D₄	C₃×2₊⁽¹⁺⁴⁾	C₂×2₊⁽¹⁺⁴⁾	C₂²×D₄	C₂×C₄○D₄	2₊⁽¹⁺⁴⁾	C₆	C₂
# reps	1	9	6	16	2	18	12	32	2	4

In GAP, Magma, Sage, TeX

C_6\times 2_+^{(1+4)}

% in TeX

G:=Group("C6xES+(2,2)");

// GroupNames label

G:=SmallGroup(192,1534);

// by ID

G=gap.SmallGroup(192,1534);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,1373,1059,2915]);

// Polycyclic

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

// generators/relations

G = C₆×2₊⁽¹⁺⁴⁾ order 192 = 2⁶·3

Direct product of C₆ and 2₊⁽¹⁺⁴⁾

G = C6×2+ (1+4) order 192 = 26·3

Direct product of C6 and 2+ (1+4)

G = C₆×2₊⁽¹⁺⁴⁾ order 192 = 2⁶·3

Direct product of C₆ and 2₊⁽¹⁺⁴⁾