C3xC2^2.32C2^4 - GroupNames

G = C₃×C₂².₃₂C₂⁴ order 192 = 2⁶·3

Direct product of C₃ and C₂².₃₂C₂⁴

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

Aliases: C₃×C₂².₃₂C₂⁴, C₆.₁₅₄2₊¹⁺⁴, (C₄×D₄)⋊₁₀C₆, C₄⋊D₄⋊₈C₆, C₂²⋊Q₈⋊₇C₆, C₂²≀C₂⋊₆C₆, (D₄×C₁₂)⋊₃₉C₂, C₄²⋊₁₀(C₂×C₆), C₄.₄D₄⋊₈C₆, C₄²⋊₂C₂⋊₃C₆, (C₄×C₁₂)⋊₄₁C₂², (C₆×D₄)⋊₃₇C₂², C₂⁴.₁₈(C₂×C₆), (C₆×Q₈)⋊₂₈C₂², (C₂×C₆).₃₅₈C₂⁴, C₂².D₄⋊₄C₆, (C₂×C₁₂).₆₆₇C₂³, (C₂²×C₁₂)⋊₄₉C₂², C₂³.₁₁(C₂²×C₆), (C₂³×C₆).₁₇C₂², (C₂²×C₆).₉₃C₂³, C₂².₃₂(C₂³×C₆), C₂.₆(C₃×2₊¹⁺⁴), C₄⋊C₄⋊₁₅(C₂×C₆), (C₂×D₄)⋊₅(C₂×C₆), (C₂×Q₈)⋊₄(C₂×C₆), C₂.₁₅(C₆×C₄○D₄), (C₃×C₄⋊D₄)⋊₃₅C₂, (C₃×C₄⋊C₄)⋊₇₁C₂², (C₂×C₂²⋊C₄)⋊₁₃C₆, C₂²⋊C₄⋊₁₅(C₂×C₆), (C₆×C₂²⋊C₄)⋊₃₃C₂, (C₂²×C₄)⋊₁₀(C₂×C₆), C₆.₂₃₄(C₂×C₄○D₄), (C₃×C₂²⋊Q₈)⋊₃₄C₂, (C₃×C₂²≀C₂)⋊₁₄C₂, C₂².₄(C₃×C₄○D₄), (C₂×C₆).₅₂(C₄○D₄), (C₃×C₄.₄D₄)⋊₂₈C₂, (C₂×C₄).₂₅(C₂²×C₆), (C₃×C₄²⋊₂C₂)⋊₁₂C₂, (C₃×C₂²⋊C₄)⋊₆₉C₂², (C₃×C₂².D₄)⋊₂₃C₂, SmallGroup(192,1427)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₂².₃₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂×C₆ — C₂²×C₆ — C₃×C₂²⋊C₄ — C₃×C₄⋊D₄ — C₃×C₂².₃₂C₂⁴

Lower central

C₁ — C₂² — C₃×C₂².₃₂C₂⁴

Upper central

C₁ — C₂×C₆ — C₃×C₂².₃₂C₂⁴

Generators and relations for C₃×C₂².₃₂C₂⁴
G = < a,b,c,d,e,f,g | a³=b²=c²=d²=f²=g²=1, e²=c, 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: 434 in 250 conjugacy classes, 146 normal (38 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×C₄⋊C₄, C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₆×Q₈, C₂³×C₆, C₂².₃₂C₂⁴, C₆×C₂²⋊C₄, D₄×C₁₂, C₃×C₂²≀C₂, C₃×C₄⋊D₄, C₃×C₄⋊D₄, C₃×C₂²⋊Q₈, C₃×C₂².D₄, C₃×C₄.₄D₄, C₃×C₄²⋊₂C₂, C₃×C₂².₃₂C₂⁴
Quotients: C₁, C₂, C₃, C₂², C₆, C₂³, C₂×C₆, C₄○D₄, C₂⁴, C₂²×C₆, C₂×C₄○D₄, 2₊¹⁺⁴, C₃×C₄○D₄, C₂³×C₆, C₂².₃₂C₂⁴, C₆×C₄○D₄, C₃×2₊¹⁺⁴, C₃×C₂².₃₂C₂⁴

Smallest permutation representation of C₃×C₂².₃₂C₂⁴
►On 48 points

Generators in S₄₈

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

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

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

(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 12)(4 10)(5 22)(7 24)(13 15)(14 41)(16 43)(18 28)(20 26)(29 46)(30 32)(31 48)(33 35)(34 37)(36 39)(38 40)(42 44)(45 47)

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	3A	3B	4A	4B	4C	4D	4E	···	4L	6A	···	6F	6G	6H	6I	6J	6K	···	6R	12A	···	12H	12I	···	12X
order	1	2	2	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	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	1	1	2	2	4	4
type	+	+	+	+	+	+	+	+	+												+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	C₆	C₆	C₆	C₄○D₄	C₃×C₄○D₄	2₊¹⁺⁴	C₃×2₊¹⁺⁴
kernel	C₃×C₂².₃₂C₂⁴	C₆×C₂²⋊C₄	D₄×C₁₂	C₃×C₂²≀C₂	C₃×C₄⋊D₄	C₃×C₂²⋊Q₈	C₃×C₂².D₄	C₃×C₄.₄D₄	C₃×C₄²⋊₂C₂	C₂².₃₂C₂⁴	C₂×C₂²⋊C₄	C₄×D₄	C₂²≀C₂	C₄⋊D₄	C₂²⋊Q₈	C₂².D₄	C₄.₄D₄	C₄²⋊₂C₂	C₂×C₆	C₂²	C₆	C₂
# reps	1	1	2	2	3	1	2	2	2	2	2	4	4	6	2	4	4	4	4	8	2	4

Matrix representation of C₃×C₂².₃₂C₂⁴ ►in GL₆(𝔽₁₃)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	9	0	0	0
0	0	0	9	0	0
0	0	0	0	9	0
0	0	0	0	0	9

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	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	0
0	0	0	0	0	12

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	12	1	0
0	0	0	1	0	0
0	0	1	1	0	0
0	0	0	11	0	12

8	0	0	0	0	0
0	8	0	0	0	0
0	0	0	1	12	0
0	0	12	0	1	1
0	0	0	0	1	1
0	0	0	0	11	12

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

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

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

C₃×C₂².₃₂C₂⁴ in GAP, Magma, Sage, TeX

C_3\times C_2^2._{32}C_2^4

% in TeX

G:=Group("C3xC2^2.32C2^4");

// GroupNames label

G:=SmallGroup(192,1427);

// by ID

G=gap.SmallGroup(192,1427);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,555,1571]);

// 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=c,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 = C3×C22.32C24 order 192 = 26·3

Direct product of C3 and C22.32C24

G = C₃×C₂².₃₂C₂⁴ order 192 = 2⁶·3

Direct product of C₃ and C₂².₃₂C₂⁴