C6xC2^2wrC2 - GroupNames

G = C₆×C₂²≀C₂ order 192 = 2⁶·3

Direct product of C₆ and C₂²≀C₂

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

Aliases: C₆×C₂²≀C₂, C₂⁵⋊₅C₆, (C₂⁴×C₆)⋊₁C₂, C₂³⋊₆(C₃×D₄), C₂²⋊₄(C₆×D₄), C₂⁴⋊₁₁(C₂×C₆), (C₂²×D₄)⋊₆C₆, (C₂²×C₆)⋊₁₇D₄, (C₂×C₁₂)⋊₁₀C₂³, (C₆×D₄)⋊₆₀C₂², (C₂³×C₆)⋊₂C₂², C₂³⋊₁(C₂²×C₆), (C₂²×C₆)⋊₃C₂³, (C₂×C₆).₃₄₁C₂⁴, C₆.₁₈₀(C₂²×D₄), (C₂²×C₁₂)⋊₄₅C₂², C₂².₁₅(C₂³×C₆), C₂.₄(D₄×C₂×C₆), (D₄×C₂×C₆)⋊₁₈C₂, (C₂×D₄)⋊₈(C₂×C₆), (C₂×C₆)⋊₁₅(C₂×D₄), (C₂×C₂²⋊C₄)⋊₈C₆, (C₂×C₄)⋊₁(C₂²×C₆), (C₂²×C₄)⋊₆(C₂×C₆), (C₆×C₂²⋊C₄)⋊₂₈C₂, C₂²⋊C₄⋊₁₀(C₂×C₆), (C₃×C₂²⋊C₄)⋊₆₄C₂², SmallGroup(192,1410)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₆×C₂²≀C₂

Chief series

C₁ — C₂ — C₂² — C₂×C₆ — C₂²×C₆ — C₆×D₄ — C₃×C₂²≀C₂ — C₆×C₂²≀C₂

Lower central

C₁ — C₂² — C₆×C₂²≀C₂

Upper central

C₁ — C₂²×C₆ — C₆×C₂²≀C₂

Generators and relations for C₆×C₂²≀C₂
G = < a,b,c,d,e,f | a⁶=b²=c²=d²=e²=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1138 in 662 conjugacy classes, 210 normal (12 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²×D₄, C₂⁵, C₃×C₂²⋊C₄, C₂²×C₁₂, C₆×D₄, C₆×D₄, C₂³×C₆, C₂³×C₆, C₂³×C₆, C₂×C₂²≀C₂, C₆×C₂²⋊C₄, C₃×C₂²≀C₂, D₄×C₂×C₆, C₂⁴×C₆, C₆×C₂²≀C₂
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂×C₆, C₂×D₄, C₂⁴, C₃×D₄, C₂²×C₆, C₂²≀C₂, C₂²×D₄, C₆×D₄, C₂³×C₆, C₂×C₂²≀C₂, C₃×C₂²≀C₂, D₄×C₂×C₆, C₆×C₂²≀C₂

Smallest permutation representation of C₆×C₂²≀C₂
►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 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)(19 45)(20 46)(21 47)(22 48)(23 43)(24 44)(25 41)(26 42)(27 37)(28 38)(29 39)(30 40)

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

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

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

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

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

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

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

84 conjugacy classes

class	1	2A	···	2G	2H	···	2S	2T	2U	3A	3B	4A	···	4F	6A	···	6N	6O	···	6AL	6AM	6AN	6AO	6AP	12A	···	12L
order	1	2	···	2	2	···	2	2	2	3	3	4	···	4	6	···	6	6	···	6	6	6	6	6	12	···	12
size	1	1	···	1	2	···	2	4	4	1	1	4	···	4	1	···	1	2	···	2	4	4	4	4	4	···	4

84 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+	+						+
image	C₁	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	D₄	C₃×D₄
kernel	C₆×C₂²≀C₂	C₆×C₂²⋊C₄	C₃×C₂²≀C₂	D₄×C₂×C₆	C₂⁴×C₆	C₂×C₂²≀C₂	C₂×C₂²⋊C₄	C₂²≀C₂	C₂²×D₄	C₂⁵	C₂²×C₆	C₂³
# reps	1	3	8	3	1	2	6	16	6	2	12	24

Matrix representation of C₆×C₂²≀C₂ ►in GL₆(𝔽₁₃)

9	0	0	0	0	0
0	9	0	0	0	0
0	0	4	0	0	0
0	0	0	4	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	12	0	0	0
0	0	12	1	0	0
0	0	0	0	12	0
0	0	0	0	0	12

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	2	0	0
0	0	0	1	0	0
0	0	0	0	0	12
0	0	0	0	12	0

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C₆×C₂²≀C₂ in GAP, Magma, Sage, TeX

C_6\times C_2^2\wr C_2

% in TeX

G:=Group("C6xC2^2wrC2");

// GroupNames label

G:=SmallGroup(192,1410);

// by ID

G=gap.SmallGroup(192,1410);

# by ID

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

// Polycyclic

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

// generators/relations

G = C6×C22≀C2 order 192 = 26·3

Direct product of C6 and C22≀C2

G = C₆×C₂²≀C₂ order 192 = 2⁶·3

Direct product of C₆ and C₂²≀C₂