C3xC2^2.29C2^4 - GroupNames

G = C₃xC₂².₂₉C₂⁴ order 192 = 2⁶·3

Direct product of C₃ and C₂².₂₉C₂⁴

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

Aliases: C₃xC₂².₂₉C₂⁴, C₆.₁₅₂2₊¹⁺⁴, (C₂xC₁₂):₂₆D₄, C₄:D₄:₆C₆, C₄:₁D₄:₇C₆, C₄²:₉(C₂xC₆), C₄.₁₆(C₆xD₄), C₂²wrC₂:₅C₆, C₄.₄D₄:₆C₆, (C₄xC₁₂):₄₀C₂², C₁₂.₃₂₃(C₂xD₄), (C₂²xD₄):₁₀C₆, (C₆xD₄):₃₆C₂², C₂⁴.₁₇(C₂xC₆), (C₆xQ₈):₅₁C₂², C₂².₂₁(C₆xD₄), C₄²:C₂:₁₀C₆, (C₂xC₆).₃₅₅C₂⁴, C₆.₁₉₀(C₂²xD₄), C₂³.₉(C₂²xC₆), (C₂xC₁₂).₆₆₄C₂³, (C₂²xC₁₂):₄₈C₂², C₂².₂₉(C₂³xC₆), (C₂³xC₆).₁₆C₂², (C₂²xC₆).₉₁C₂³, C₂.₄(C₃x2₊¹⁺⁴), (D₄xC₂xC₆):₂₂C₂, (C₂xC₄):₄(C₃xD₄), C₂.₁₄(D₄xC₂xC₆), C₄:C₄:₁₄(C₂xC₆), (C₂xC₄oD₄):₈C₆, (C₂xD₄):₄(C₂xC₆), (C₆xC₄oD₄):₂₀C₂, C₂²:C₄:₄(C₂xC₆), (C₂²xC₄):₉(C₂xC₆), (C₂xQ₈):₁₃(C₂xC₆), (C₃xC₄:D₄):₃₃C₂, (C₃xC₄:₁D₄):₁₆C₂, (C₃xC₄:C₄):₇₀C₂², (C₂xC₆).₄₁₇(C₂xD₄), (C₃xC₂²wrC₂):₁₃C₂, (C₃xC₄.₄D₄):₂₆C₂, (C₂xC₄).₂₂(C₂²xC₆), (C₃xC₄²:C₂):₃₁C₂, (C₃xC₂²:C₄):₃₉C₂², SmallGroup(192,1424)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃xC₂².₂₉C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂xC₆ — C₂²xC₆ — C₆xD₄ — C₃xC₄:₁D₄ — C₃xC₂².₂₉C₂⁴

Lower central

C₁ — C₂² — C₃xC₂².₂₉C₂⁴

Upper central

C₁ — C₂xC₆ — C₃xC₂².₂₉C₂⁴

Generators and relations for C₃xC₂².₂₉C₂⁴
G = < a,b,c,d,e,f,g | a³=b²=c²=d²=f²=g²=1, e²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede^-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 610 in 334 conjugacy classes, 162 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, C₁₂, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₂xC₁₂, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₄²:C₂, C₂²wrC₂, C₄:D₄, C₄.₄D₄, C₄:₁D₄, C₂²xD₄, C₂xC₄oD₄, C₄xC₁₂, C₃xC₂²:C₄, C₃xC₄:C₄, C₂²xC₁₂, C₂²xC₁₂, C₆xD₄, C₆xD₄, C₆xD₄, C₆xQ₈, C₃xC₄oD₄, C₂³xC₆, C₂².₂₉C₂⁴, C₃xC₄²:C₂, C₃xC₂²wrC₂, C₃xC₄:D₄, C₃xC₄.₄D₄, C₃xC₄:₁D₄, D₄xC₂xC₆, C₆xC₄oD₄, C₃xC₂².₂₉C₂⁴
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂xC₆, C₂xD₄, C₂⁴, C₃xD₄, C₂²xC₆, C₂²xD₄, 2₊¹⁺⁴, C₆xD₄, C₂³xC₆, C₂².₂₉C₂⁴, D₄xC₂xC₆, C₃x2₊¹⁺⁴, C₃xC₂².₂₉C₂⁴

Smallest permutation representation of C₃xC₂².₂₉C₂⁴
►On 48 points

Generators in S₄₈

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

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

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

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

(2 4)(6 8)(10 12)(13 29)(14 32)(15 31)(16 30)(18 20)(22 24)(26 28)(33 41)(34 44)(35 43)(36 42)(37 45)(38 48)(39 47)(40 46)

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+	+	+									+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	C₆	D₄	C₃xD₄	2₊¹⁺⁴	C₃x2₊¹⁺⁴
kernel	C₃xC₂².₂₉C₂⁴	C₃xC₄²:C₂	C₃xC₂²wrC₂	C₃xC₄:D₄	C₃xC₄.₄D₄	C₃xC₄:₁D₄	D₄xC₂xC₆	C₆xC₄oD₄	C₂².₂₉C₂⁴	C₄²:C₂	C₂²wrC₂	C₄:D₄	C₄.₄D₄	C₄:₁D₄	C₂²xD₄	C₂xC₄oD₄	C₂xC₁₂	C₂xC₄	C₆	C₂
# reps	1	1	4	4	2	2	1	1	2	2	8	8	4	4	2	2	4	8	2	4

Matrix representation of C₃xC₂².₂₉C₂⁴ ►in GL₆(F₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	3

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

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	1	0
0	0	0	0	0	1

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

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

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

1	0	0	0	0	0
0	1	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

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

C₃xC₂².₂₉C₂⁴ in GAP, Magma, Sage, TeX

C_3\times C_2^2._{29}C_2^4

% in TeX

G:=Group("C3xC2^2.29C2^4");

// GroupNames label

G:=SmallGroup(192,1424);

// by ID

G=gap.SmallGroup(192,1424);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,344,2102,555,1571]);

// Polycyclic

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

// generators/relations

G = C3xC22.29C24 order 192 = 26·3

Direct product of C3 and C22.29C24

G = C₃xC₂².₂₉C₂⁴ order 192 = 2⁶·3

Direct product of C₃ and C₂².₂₉C₂⁴