C3xC2^2.11C2^4 - GroupNames

G = C₃×C₂².₁₁C₂⁴ order 192 = 2⁶·3

Direct product of C₃ and C₂².₁₁C₂⁴

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

Aliases: C₃×C₂².₁₁C₂⁴, C₆.₁₄₉2₊¹⁺⁴, (C₄×D₄)⋊₅C₆, D₄⋊₇(C₂×C₁₂), (C₆×D₄)⋊₂₃C₄, C₄²⋊₇(C₂×C₆), (C₂×D₄)⋊₁₁C₁₂, (D₄×C₁₂)⋊₃₄C₂, C₂³⋊₃(C₂×C₁₂), C₄²⋊C₂⋊₆C₆, (C₄×C₁₂)⋊₃₈C₂², C₆.₅₉(C₂³×C₄), C₂.₇(C₂³×C₁₂), C₂⁴.₁₂(C₂×C₆), (C₂×C₆).₃₃₈C₂⁴, C₄.₁₉(C₂²×C₁₂), (C₂²×C₁₂)⋊₅C₂², (C₂²×D₄).₁₀C₆, (C₂×C₁₂).₇₀₉C₂³, C₁₂.₁₆₄(C₂²×C₄), (C₆×D₄).₃₃₂C₂², (C₂³×C₆).₁₁C₂², C₂².₂(C₂²×C₁₂), C₂³.₃₄(C₂²×C₆), C₂².₁₁(C₂³×C₆), C₂.₁(C₃×2₊¹⁺⁴), (C₂²×C₆).₂₅₄C₂³, C₄⋊C₄⋊₂₀(C₂×C₆), (C₂×C₄)⋊₄(C₂×C₁₂), (D₄×C₂×C₆).₂₂C₂, (C₂×C₁₂)⋊₂₅(C₂×C₄), (C₃×D₄)⋊₂₇(C₂×C₄), (C₂×C₂²⋊C₄)⋊₅C₆, (C₂²×C₆)⋊₅(C₂×C₄), (C₂²×C₄)⋊₄(C₂×C₆), (C₃×C₄⋊C₄)⋊₇₇C₂², C₂²⋊C₄⋊₁₈(C₂×C₆), (C₆×C₂²⋊C₄)⋊₁₀C₂, (C₂×D₄).₇₈(C₂×C₆), (C₂×C₆).₃₃(C₂²×C₄), (C₂×C₄).₅₆(C₂²×C₆), (C₃×C₄²⋊C₂)⋊₂₇C₂, (C₃×C₂²⋊C₄)⋊₇₂C₂², SmallGroup(192,1407)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₃×C₂².₁₁C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂×C₆ — C₂×C₁₂ — C₃×C₂²⋊C₄ — D₄×C₁₂ — C₃×C₂².₁₁C₂⁴

Lower central

C₁ — C₂ — C₃×C₂².₁₁C₂⁴

Upper central

C₁ — C₂×C₆ — C₃×C₂².₁₁C₂⁴

Generators and relations for C₃×C₂².₁₁C₂⁴
G = < a,b,c,d,e,f,g | a³=b²=c²=e²=f²=g²=1, d²=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >

Subgroups: 514 in 338 conjugacy classes, 242 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²×D₄, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₂³×C₆, C₂².₁₁C₂⁴, C₆×C₂²⋊C₄, C₃×C₄²⋊C₂, D₄×C₁₂, D₄×C₂×C₆, C₃×C₂².₁₁C₂⁴
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, C₂³, C₁₂, C₂×C₆, C₂²×C₄, C₂⁴, C₂×C₁₂, C₂²×C₆, C₂³×C₄, 2₊¹⁺⁴, C₂²×C₁₂, C₂³×C₆, C₂².₁₁C₂⁴, C₂³×C₁₂, C₃×2₊¹⁺⁴, C₃×C₂².₁₁C₂⁴

Smallest permutation representation of C₃×C₂².₁₁C₂⁴
►On 48 points

Generators in S₄₈

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

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

(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 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 40)(4 38)(6 25)(8 27)(10 18)(12 20)(13 21)(15 23)(29 45)(31 47)(34 42)(36 44)

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

(2 40)(4 38)(5 28)(7 26)(10 18)(12 20)(14 22)(16 24)(30 46)(32 48)(34 42)(36 44)

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

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

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

102 conjugacy classes

class	1	2A	2B	2C	2D	···	2M	3A	3B	4A	···	4T	6A	···	6F	6G	···	6Z	12A	···	12AN
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	1	1	1	1	4	4
type	+	+	+	+	+								+
image	C₁	C₂	C₂	C₂	C₂	C₃	C₄	C₆	C₆	C₆	C₆	C₁₂	2₊¹⁺⁴	C₃×2₊¹⁺⁴
kernel	C₃×C₂².₁₁C₂⁴	C₆×C₂²⋊C₄	C₃×C₄²⋊C₂	D₄×C₁₂	D₄×C₂×C₆	C₂².₁₁C₂⁴	C₆×D₄	C₂×C₂²⋊C₄	C₄²⋊C₂	C₄×D₄	C₂²×D₄	C₂×D₄	C₆	C₂
# reps	1	4	2	8	1	2	16	8	4	16	2	32	2	4

Matrix representation of C₃×C₂².₁₁C₂⁴ ►in GL₅(𝔽₁₃)

3	0	0	0	0
0	9	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

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

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

5	0	0	0	0
0	0	0	1	0
0	0	0	0	1
0	1	0	0	0
0	0	1	0	0

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

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

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

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

C₃×C₂².₁₁C₂⁴ in GAP, Magma, Sage, TeX

C_3\times C_2^2._{11}C_2^4

% in TeX

G:=Group("C3xC2^2.11C2^4");

// GroupNames label

G:=SmallGroup(192,1407);

// by ID

G=gap.SmallGroup(192,1407);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,555,1571]);

// Polycyclic

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

// generators/relations

G = C3×C22.11C24 order 192 = 26·3

Direct product of C3 and C22.11C24

G = C₃×C₂².₁₁C₂⁴ order 192 = 2⁶·3

Direct product of C₃ and C₂².₁₁C₂⁴