C6xC2^3:C4 - GroupNames

G = C₆×C₂³⋊C₄ order 192 = 2⁶·3

Direct product of C₆ and C₂³⋊C₄

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C₆×C₂³⋊C₄, C₂⁴⋊₄C₁₂, (C₂×D₄)⋊₅C₁₂, (C₆×D₄)⋊₁₇C₄, (C₂³×C₆)⋊₂C₄, (C₂²×C₄)⋊₄C₁₂, C₂³⋊₁(C₂×C₁₂), (C₂²×C₁₂)⋊₇C₄, C₂⁴.₁₁(C₂×C₆), (C₂²×D₄).₅C₆, C₂².₁₀(C₆×D₄), C₂³.₁₁(C₃×D₄), (C₂²×C₆).₁₅₇D₄, C₂³.₁(C₂²×C₆), (C₆×D₄).₂₈₂C₂², (C₂²×C₆).₈₀C₂³, (C₂³×C₆).₁₀C₂², C₂².₆(C₂²×C₁₂), (C₂×C₄)⋊₁(C₂×C₁₂), (C₂×C₁₂)⋊₃(C₂×C₄), (D₄×C₂×C₆).₁₆C₂, C₂²⋊C₄⋊₉(C₂×C₆), (C₂×C₂²⋊C₄)⋊₄C₆, (C₆×C₂²⋊C₄)⋊₉C₂, (C₂²×C₆)⋊₂(C₂×C₄), (C₂×D₄).₄₀(C₂×C₆), (C₂×C₆).₄₀₅(C₂×D₄), C₂.₁₂(C₆×C₂²⋊C₄), C₆.₁₀₀(C₂×C₂²⋊C₄), C₂².₂(C₃×C₂²⋊C₄), (C₃×C₂²⋊C₄)⋊₄₅C₂², (C₂×C₆).₁₅₉(C₂²×C₄), (C₂×C₆).₁₃₅(C₂²⋊C₄), SmallGroup(192,842)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₆×C₂³⋊C₄

Chief series

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

Lower central

C₁ — C₂ — C₂² — C₆×C₂³⋊C₄

Upper central

C₁ — C₂×C₆ — C₂³×C₆ — C₆×C₂³⋊C₄

Generators and relations for C₆×C₂³⋊C₄
G = < a,b,c,d,e | a⁶=b²=c²=d²=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 434 in 210 conjugacy classes, 82 normal (26 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, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²×D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂²×C₁₂, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂³×C₆, C₂×C₂³⋊C₄, C₃×C₂³⋊C₄, C₆×C₂²⋊C₄, D₄×C₂×C₆, C₆×C₂³⋊C₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₂³, C₁₂, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂³⋊C₄, C₂×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂²×C₁₂, C₆×D₄, C₂×C₂³⋊C₄, C₃×C₂³⋊C₄, C₆×C₂²⋊C₄, C₆×C₂³⋊C₄

Smallest permutation representation of C₆×C₂³⋊C₄
►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 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 33)(8 34)(9 35)(10 36)(11 31)(12 32)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)

(7 48)(8 43)(9 44)(10 45)(11 46)(12 47)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)

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

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

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

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

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

66 conjugacy classes

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

66 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+											+		+
image	C₁	C₂	C₂	C₂	C₃	C₄	C₄	C₄	C₆	C₆	C₆	C₁₂	C₁₂	C₁₂	D₄	C₃×D₄	C₂³⋊C₄	C₃×C₂³⋊C₄
kernel	C₆×C₂³⋊C₄	C₃×C₂³⋊C₄	C₆×C₂²⋊C₄	D₄×C₂×C₆	C₂×C₂³⋊C₄	C₂²×C₁₂	C₆×D₄	C₂³×C₆	C₂³⋊C₄	C₂×C₂²⋊C₄	C₂²×D₄	C₂²×C₄	C₂×D₄	C₂⁴	C₂²×C₆	C₂³	C₆	C₂
# reps	1	4	2	1	2	2	4	2	8	4	2	4	8	4	4	8	2	4

Matrix representation of C₆×C₂³⋊C₄ ►in GL₆(𝔽₁₃)

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

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	12	0
0	0	0	0	0	12

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

G:=sub<GL(6,GF(13))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0] >;

C₆×C₂³⋊C₄ in GAP, Magma, Sage, TeX

C_6\times C_2^3\rtimes C_4

% in TeX

G:=Group("C6xC2^3:C4");

// GroupNames label

G:=SmallGroup(192,842);

// by ID

G=gap.SmallGroup(192,842);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,4204,3036]);

// Polycyclic

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

// generators/relations

G = C6×C23⋊C4 order 192 = 26·3

Direct product of C6 and C23⋊C4

G = C₆×C₂³⋊C₄ order 192 = 2⁶·3

Direct product of C₆ and C₂³⋊C₄