C3xD4.9D4 - GroupNames

G = C₃×D₄.₉D₄ order 192 = 2⁶·3

Direct product of C₃ and D₄.₉D₄

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

Aliases: C₃×D₄.₉D₄, 2₊¹⁺⁴.₄C₆, C₄≀C₂⋊₃C₆, C₄²⋊₆(C₂×C₆), D₄.₉(C₃×D₄), C₄.₂₉(C₆×D₄), (C₃×D₄).₄₃D₄, C₄.D₄⋊₂C₆, C₄.₄D₄⋊₂C₆, C₈.C₂²⋊₁C₆, (C₃×Q₈).₄₃D₄, Q₈.₁₄(C₃×D₄), (C₂²×C₆).₆D₄, C₂³.₆(C₃×D₄), (C₄×C₁₂)⋊₃₆C₂², C₁₂.₃₉₀(C₂×D₄), M₄(2)⋊₂(C₂×C₆), (C₆×Q₈)⋊₂₇C₂², C₂².₁₆(C₆×D₄), C₆.₁₀₂C₂²≀C₂, (C₂×C₁₂).₆₁₁C₂³, (C₆×D₄).₁₈₂C₂², (C₃×M₄(2))⋊₁₅C₂², (C₃×2₊¹⁺⁴).₃C₂, (C₃×C₄≀C₂)⋊₇C₂, (C₂×Q₈)⋊₃(C₂×C₆), C₄○D₄.₈(C₂×C₆), (C₂×D₄).₇(C₂×C₆), (C₃×C₄.D₄)⋊₆C₂, (C₂×C₆).₄₁₁(C₂×D₄), (C₃×C₈.C₂²)⋊₈C₂, (C₂×C₄).₆(C₂²×C₆), C₂.₁₆(C₃×C₂²≀C₂), (C₃×C₄.₄D₄)⋊₂₂C₂, (C₃×C₄○D₄).₃₃C₂², SmallGroup(192,888)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₄ — C₃×D₄.₉D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂×C₁₂ — C₆×Q₈ — C₃×C₈.C₂² — C₃×D₄.₉D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₃×D₄.₉D₄

Upper central

C₁ — C₆ — C₂×C₁₂ — C₃×D₄.₉D₄

Generators and relations for C₃×D₄.₉D₄
G = < a,b,c,d,e | a³=b⁴=c²=d⁴=1, e²=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b^-1, bd=db, dcd^-1=b^-1c, ece^-1=bc, ede^-1=b²d^-1 >

Subgroups: 306 in 152 conjugacy classes, 54 normal (22 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₁₂, C₁₂, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, M₄(2), SD₁₆, Q₁₆, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₃×D₄, C₃×Q₈, C₃×Q₈, C₂²×C₆, C₂²×C₆, C₄.D₄, C₄≀C₂, C₄.₄D₄, C₈.C₂², 2₊¹⁺⁴, C₄×C₁₂, C₃×C₂²⋊C₄, C₃×M₄(2), C₃×SD₁₆, C₃×Q₁₆, C₆×D₄, C₆×D₄, C₆×Q₈, C₃×C₄○D₄, C₃×C₄○D₄, D₄.₉D₄, C₃×C₄.D₄, C₃×C₄≀C₂, C₃×C₄.₄D₄, C₃×C₈.C₂², C₃×2₊¹⁺⁴, C₃×D₄.₉D₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₂³, C₂×C₆, C₂×D₄, C₃×D₄, C₂²×C₆, C₂²≀C₂, C₆×D₄, D₄.₉D₄, C₃×C₂²≀C₂, C₃×D₄.₉D₄

Smallest permutation representation of C₃×D₄.₉D₄
►On 48 points

Generators in S₄₈

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

(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)(6 8)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(30 32)(33 35)(38 40)(41 43)(46 48)

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

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	+							+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₃	C₆	C₆	C₆	C₆	C₆	D₄	D₄	D₄	C₃×D₄	C₃×D₄	C₃×D₄	D₄.₉D₄	C₃×D₄.₉D₄
kernel	C₃×D₄.₉D₄	C₃×C₄.D₄	C₃×C₄≀C₂	C₃×C₄.₄D₄	C₃×C₈.C₂²	C₃×2₊¹⁺⁴	D₄.₉D₄	C₄.D₄	C₄≀C₂	C₄.₄D₄	C₈.C₂²	2₊¹⁺⁴	C₃×D₄	C₃×Q₈	C₂²×C₆	D₄	Q₈	C₂³	C₃	C₁
# reps	1	1	2	1	2	1	2	2	4	2	4	2	2	2	2	4	4	4	2	4

Matrix representation of C₃×D₄.₉D₄ ►in GL₆(𝔽₇₃)

8	0	0	0	0	0
0	8	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	0	0	1	0
0	0	0	0	0	1
0	0	72	0	0	0
0	0	0	72	0	0

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

55	69	0	0	0	0
63	18	0	0	0	0
0	0	23	23	50	23
0	0	23	23	23	50
0	0	23	50	23	23
0	0	50	23	23	23

18	4	0	0	0	0
47	55	0	0	0	0
0	0	23	23	50	23
0	0	23	23	23	50
0	0	50	23	50	50
0	0	23	50	50	50

G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,72,0,0],[55,63,0,0,0,0,69,18,0,0,0,0,0,0,23,23,23,50,0,0,23,23,50,23,0,0,50,23,23,23,0,0,23,50,23,23],[18,47,0,0,0,0,4,55,0,0,0,0,0,0,23,23,50,23,0,0,23,23,23,50,0,0,50,23,50,50,0,0,23,50,50,50] >;

C₃×D₄.₉D₄ in GAP, Magma, Sage, TeX

C_3\times D_4._9D_4

% in TeX

G:=Group("C3xD4.9D4");

// GroupNames label

G:=SmallGroup(192,888);

// by ID

G=gap.SmallGroup(192,888);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,4204,2111,1068,172,3036]);

// Polycyclic

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

// generators/relations

G = C3×D4.9D4 order 192 = 26·3

Direct product of C3 and D4.9D4

G = C₃×D₄.₉D₄ order 192 = 2⁶·3

Direct product of C₃ and D₄.₉D₄