C3xC2^3.D4 - GroupNames

G = C₃×C₂³.D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³.D₄

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₃×C₂³.D₄, C₄.D₄.C₆, C₂²⋊C₄⋊₂C₁₂, (C₂²×C₄)⋊₃C₁₂, (C₂²×C₁₂)⋊₄C₄, (C₂×C₁₂).₁₈D₄, C₂³⋊C₄.₁C₆, C₂³.₂(C₃×D₄), (C₂²×C₆).₂D₄, C₂³.₂(C₂×C₁₂), C₆.₃₃(C₂³⋊C₄), (C₆×D₄).₁₇₅C₂², C₂².D₄.₁C₆, (C₂×C₄).₂(C₃×D₄), (C₃×C₂²⋊C₄)⋊₄C₄, (C₂×D₄).₂(C₂×C₆), C₂.₇(C₃×C₂³⋊C₄), (C₃×C₂³⋊C₄).₃C₂, (C₂²×C₆).₉(C₂×C₄), (C₃×C₄.D₄).₂C₂, (C₂×C₆).₇₄(C₂²⋊C₄), C₂².₁₁(C₃×C₂²⋊C₄), (C₃×C₂².D₄).₄C₂, SmallGroup(192,158)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₃×C₂³.D₄

Chief series

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

Lower central

C₁ — C₂ — C₂² — C₂³ — C₃×C₂³.D₄

Upper central

C₁ — C₆ — C₂×C₆ — C₆×D₄ — C₃×C₂³.D₄

Generators and relations for C₃×C₂³.D₄
G = < a,b,c,d,e,f | a³=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=bcd, bf=fb, ece^-1=fcf^-1=cd=dc, de=ed, df=fd, fef^-1=be³ >

Subgroups: 162 in 68 conjugacy classes, 26 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₂, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂³⋊C₄, C₄.D₄, C₂².D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₃×C₄⋊C₄, C₃×M₄(2), C₂²×C₁₂, C₆×D₄, C₂³.D₄, C₃×C₂³⋊C₄, C₃×C₄.D₄, C₃×C₂².D₄, C₃×C₂³.D₄
Quotients: C₁, C₂, C₃, C₄, C₂², C₆, C₂×C₄, D₄, C₁₂, C₂×C₆, C₂²⋊C₄, C₂×C₁₂, C₃×D₄, C₂³⋊C₄, C₃×C₂²⋊C₄, C₂³.D₄, C₃×C₂³⋊C₄, C₃×C₂³.D₄

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

Generators in S₄₈

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

(2 36)(3 7)(4 34)(6 40)(8 38)(9 13)(10 44)(12 42)(14 48)(16 46)(17 28)(18 22)(19 26)(21 32)(23 30)(25 29)(33 37)(43 47)

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F	6G	6H	8A	8B	12A	···	12F	12G	···	12L	24A	24B	24C	24D
order	1	2	2	2	2	3	3	4	4	4	4	4	4	6	6	6	6	6	6	6	6	8	8	12	···	12	12	···	12	24	24	24	24
size	1	1	2	4	4	1	1	4	4	4	8	8	8	1	1	2	2	4	4	4	4	8	8	4	···	4	8	···	8	8	8	8	8

39 irreducible representations

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

Matrix representation of C₃×C₂³.D₄ ►in GL₄(𝔽₇₃) generated by

64	0	0	0
0	64	0	0
0	0	64	0
0	0	0	64

1	0	0	0
0	1	0	0
0	0	72	0
0	0	0	72

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

72	0	0	0
0	72	0	0
0	0	72	0
0	0	0	72

23	50	50	50
23	50	23	23
23	23	23	50
50	50	23	50

1	0	0	0
0	72	0	0
0	0	0	72
0	0	1	0

G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[23,23,23,50,50,50,23,50,50,23,23,23,50,23,50,50],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0] >;

C₃×C₂³.D₄ in GAP, Magma, Sage, TeX

C_3\times C_2^3.D_4

% in TeX

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

// GroupNames label

G:=SmallGroup(192,158);

// by ID

G=gap.SmallGroup(192,158);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,1271,375,6053]);

// Polycyclic

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

// generators/relations

G = C3×C23.D4 order 192 = 26·3

Direct product of C3 and C23.D4

G = C₃×C₂³.D₄ order 192 = 2⁶·3

Direct product of C₃ and C₂³.D₄