C3xC2^4:3C4 - GroupNames

G = C₃×C₂⁴⋊₃C₄ order 192 = 2⁶·3

Direct product of C₃ and C₂⁴⋊₃C₄

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

Aliases: C₃×C₂⁴⋊₃C₄, C₂⁴⋊₇C₁₂, C₂⁵.₅C₆, (C₂³×C₆)⋊₅C₄, (C₂⁴×C₆).₁C₂, C₆.₈₇C₂²≀C₂, C₂⁴.₂₉(C₂×C₆), C₂³.₃₈(C₃×D₄), C₂².₂₉(C₆×D₄), C₂³.₂₉(C₂×C₁₂), (C₂²×C₁₂)⋊₃C₂², (C₂²×C₆).₁₅₃D₄, (C₂³×C₆).₈₃C₂², C₂³.₅₆(C₂²×C₆), (C₂²×C₆).₄₄₃C₂³, C₂².₂₈(C₂²×C₁₂), (C₂×C₂²⋊C₄)⋊₁C₆, (C₆×C₂²⋊C₄)⋊₅C₂, (C₂²×C₄)⋊₂(C₂×C₆), C₂.₄(C₆×C₂²⋊C₄), (C₂×C₆)⋊₇(C₂²⋊C₄), C₂.₁(C₃×C₂²≀C₂), (C₂×C₆).₅₉₆(C₂×D₄), C₆.₉₁(C₂×C₂²⋊C₄), C₂²⋊₃(C₃×C₂²⋊C₄), (C₂²×C₆).₁₁₀(C₂×C₄), (C₂×C₆).₂₁₅(C₂²×C₄), SmallGroup(192,812)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₃×C₂⁴⋊₃C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₆ — C₂²×C₁₂ — C₆×C₂²⋊C₄ — C₃×C₂⁴⋊₃C₄

Lower central

C₁ — C₂² — C₃×C₂⁴⋊₃C₄

Upper central

C₁ — C₂²×C₆ — C₃×C₂⁴⋊₃C₄

Generators and relations for C₃×C₂⁴⋊₃C₄
G = < a,b,c,d,e,f | a³=b²=c²=d²=e²=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, fcf^-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 898 in 506 conjugacy classes, 130 normal (10 characteristic)
C₁, C₂^[×7], C₂^[×12], C₃, C₄^[×4], C₂², C₂²^[×18], C₂²^[×68], C₆^[×7], C₆^[×12], C₂×C₄^[×12], C₂³, C₂³^[×18], C₂³^[×68], C₁₂^[×4], C₂×C₆, C₂×C₆^[×18], C₂×C₆^[×68], C₂²⋊C₄^[×12], C₂²×C₄^[×4], C₂⁴^[×7], C₂⁴^[×12], C₂×C₁₂^[×12], C₂²×C₆, C₂²×C₆^[×18], C₂²×C₆^[×68], C₂×C₂²⋊C₄^[×6], C₂⁵, C₃×C₂²⋊C₄^[×12], C₂²×C₁₂^[×4], C₂³×C₆^[×7], C₂³×C₆^[×12], C₂⁴⋊₃C₄, C₆×C₂²⋊C₄^[×6], C₂⁴×C₆, C₃×C₂⁴⋊₃C₄
Quotients: C₁, C₂^[×7], C₃, C₄^[×4], C₂²^[×7], C₆^[×7], C₂×C₄^[×6], D₄^[×12], C₂³, C₁₂^[×4], C₂×C₆^[×7], C₂²⋊C₄^[×12], C₂²×C₄, C₂×D₄^[×6], C₂×C₁₂^[×6], C₃×D₄^[×12], C₂²×C₆, C₂×C₂²⋊C₄^[×3], C₂²≀C₂^[×4], C₃×C₂²⋊C₄^[×12], C₂²×C₁₂, C₆×D₄^[×6], C₂⁴⋊₃C₄, C₆×C₂²⋊C₄^[×3], C₃×C₂²≀C₂^[×4], C₃×C₂⁴⋊₃C₄

Smallest permutation representation of C₃×C₂⁴⋊₃C₄
►On 48 points

Generators in S₄₈

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

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

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

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

(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 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)

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

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

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

84 conjugacy classes

class	1	2A	···	2G	2H	···	2S	3A	3B	4A	···	4H	6A	···	6N	6O	···	6AL	12A	···	12P
order	1	2	···	2	2	···	2	3	3	4	···	4	6	···	6	6	···	6	12	···	12
size	1	1	···	1	2	···	2	1	1	4	···	4	1	···	1	2	···	2	4	···	4

84 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2
type	+	+	+						+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	D₄	C₃×D₄
kernel	C₃×C₂⁴⋊₃C₄	C₆×C₂²⋊C₄	C₂⁴×C₆	C₂⁴⋊₃C₄	C₂³×C₆	C₂×C₂²⋊C₄	C₂⁵	C₂⁴	C₂²×C₆	C₂³
# reps	1	6	1	2	8	12	2	16	12	24

Matrix representation of C₃×C₂⁴⋊₃C₄ ►in GL₆(𝔽₁₃)

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

1	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	12	0
0	0	0	0	0	12

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

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

C₃×C₂⁴⋊₃C₄ in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes_3C_4

% in TeX

G:=Group("C3xC2^4:3C4");

// GroupNames label

G:=SmallGroup(192,812);

// by ID

G=gap.SmallGroup(192,812);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094]);

// Polycyclic

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

// generators/relations

G = C3×C24⋊3C4 order 192 = 26·3

Direct product of C3 and C24⋊3C4

G = C₃×C₂⁴⋊₃C₄ order 192 = 2⁶·3

Direct product of C₃ and C₂⁴⋊₃C₄