C3xC2^3:3D4 - GroupNames

G = C₃×C₂³⋊₃D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³⋊₃D₄

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

Aliases: C₃×C₂³⋊₃D₄, C₆.₁₅₁2₊¹⁺⁴, C₄⋊D₄⋊₅C₆, C₂⁴⋊₅(C₂×C₆), C₂³⋊₃(C₃×D₄), (C₂²×C₆)⋊₆D₄, C₂²≀C₂⋊₄C₆, (C₂²×D₄)⋊₉C₆, C₂².₂(C₆×D₄), (C₆×D₄)⋊₃₅C₂², (C₂³×C₆)⋊₃C₂², (C₂×C₆).₃₅₄C₂⁴, C₂².D₄⋊₂C₆, C₆.₁₈₉(C₂²×D₄), (C₂×C₁₂).₆₆₃C₂³, (C₂²×C₁₂)⋊₄₇C₂², C₂².₂₈(C₂³×C₆), C₂³.₄₀(C₂²×C₆), (C₂²×C₆).₉₀C₂³, C₂.₃(C₃×2₊¹⁺⁴), C₄⋊C₄⋊₃(C₂×C₆), (D₄×C₂×C₆)⋊₂₁C₂, C₂.₁₃(D₄×C₂×C₆), (C₂×D₄)⋊₃(C₂×C₆), C₂²⋊C₄⋊₃(C₂×C₆), (C₂²×C₄)⋊₈(C₂×C₆), (C₂×C₆).₉₀(C₂×D₄), (C₃×C₄⋊D₄)⋊₃₂C₂, (C₆×C₂²⋊C₄)⋊₃₂C₂, (C₂×C₂²⋊C₄)⋊₁₂C₆, (C₃×C₄⋊C₄)⋊₃₇C₂², (C₃×C₂²≀C₂)⋊₁₂C₂, (C₂×C₄).₂₁(C₂²×C₆), (C₃×C₂²⋊C₄)⋊₃₈C₂², (C₃×C₂².D₄)⋊₂₁C₂, SmallGroup(192,1423)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₂³⋊₃D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₆ — C₂²×C₆ — C₆×D₄ — C₃×C₄⋊D₄ — C₃×C₂³⋊₃D₄

Lower central

C₁ — C₂² — C₃×C₂³⋊₃D₄

Upper central

C₁ — C₂×C₆ — C₃×C₂³⋊₃D₄

Generators and relations for C₃×C₂³⋊₃D₄
G = < a,b,c,d,e,f | a³=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece^-1=fcf=cd=dc, de=ed, df=fd, fef=e^-1 >

Subgroups: 642 in 346 conjugacy classes, 162 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂²×D₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂³×C₆, C₂³×C₆, C₂³⋊₃D₄, C₆×C₂²⋊C₄, C₃×C₂²≀C₂, C₃×C₄⋊D₄, C₃×C₂².D₄, D₄×C₂×C₆, C₃×C₂³⋊₃D₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂×C₆, C₂×D₄, C₂⁴, C₃×D₄, C₂²×C₆, C₂²×D₄, 2₊¹⁺⁴, C₆×D₄, C₂³×C₆, C₂³⋊₃D₄, D₄×C₂×C₆, C₃×2₊¹⁺⁴, C₃×C₂³⋊₃D₄

Smallest permutation representation of C₃×C₂³⋊₃D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	3A	3B	4A	···	4H	6A	···	6F	6G	···	6R	6S	···	6Z	12A	···	12P
order	1	2	2	2	2	···	2	2	2	2	2	3	3	4	···	4	6	···	6	6	···	6	6	···	6	12	···	12
size	1	1	1	1	2	···	2	4	4	4	4	1	1	4	···	4	1	···	1	2	···	2	4	···	4	4	···	4

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+	+	+							+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	D₄	C₃×D₄	2₊¹⁺⁴	C₃×2₊¹⁺⁴
kernel	C₃×C₂³⋊₃D₄	C₆×C₂²⋊C₄	C₃×C₂²≀C₂	C₃×C₄⋊D₄	C₃×C₂².D₄	D₄×C₂×C₆	C₂³⋊₃D₄	C₂×C₂²⋊C₄	C₂²≀C₂	C₄⋊D₄	C₂².D₄	C₂²×D₄	C₂²×C₆	C₂³	C₆	C₂
# reps	1	1	4	4	4	2	2	2	8	8	8	4	4	8	2	4

Matrix representation of C₃×C₂³⋊₃D₄ ►in GL₆(𝔽₁₃)

9	0	0	0	0	0
0	9	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

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

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	1	0	1	0
0	0	0	0	0	1

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

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

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

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

C₃×C₂³⋊₃D₄ in GAP, Magma, Sage, TeX

C_3\times C_2^3\rtimes_3D_4

% in TeX

G:=Group("C3xC2^3:3D4");

// GroupNames label

G:=SmallGroup(192,1423);

// by ID

G=gap.SmallGroup(192,1423);

# by ID

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

// Polycyclic

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

// generators/relations

G = C3×C23⋊3D4 order 192 = 26·3

Direct product of C3 and C23⋊3D4

G = C₃×C₂³⋊₃D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³⋊₃D₄