C2xC12.46D4 - GroupNames

G = C₂×C₁₂.₄₆D₄ order 192 = 2⁶·3

Direct product of C₂ and C₁₂.₄₆D₄

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂×C₁₂.₄₆D₄, M₄(2)⋊₂₂D₆, (C₂×C₄).₄₉D₁₂, C₄.₆₅(C₂×D₁₂), C₄.₂₈(D₆⋊C₄), (C₂×D₁₂).₁₅C₄, C₁₂.₄₁₆(C₂×D₄), (C₂×C₁₂).₁₇₂D₄, C₆⋊₁(C₄.D₄), (S₃×C₂³).₃C₄, C₂³.₆₀(C₄×S₃), (C₂×M₄(2))⋊₁₀S₃, (C₆×M₄(2))⋊₁₈C₂, (C₂²×C₄).₁₅₄D₆, C₁₂.₅₃(C₂²⋊C₄), (C₂×C₁₂).₄₁₆C₂³, C₂².₅₀(D₆⋊C₄), (C₂²×D₁₂).₁₄C₂, C₄.Dic₃⋊₂₁C₂², (C₂×D₁₂).₂₅₀C₂², (C₃×M₄(2))⋊₃₄C₂², (C₂²×C₁₂).₁₈₇C₂², C₃⋊₂(C₂×C₄.D₄), (C₂×C₄).₅₂(C₄×S₃), C₂.₂₉(C₂×D₆⋊C₄), C₂².₂₀(S₃×C₂×C₄), C₄.₁₀₉(C₂×C₃⋊D₄), C₆.₅₇(C₂×C₂²⋊C₄), (C₂×C₁₂).₁₀₇(C₂×C₄), (C₂²×S₃).₅(C₂×C₄), (C₂×C₄.Dic₃)⋊₁₅C₂, (C₂²×C₆).₇₀(C₂×C₄), (C₂×C₆).₁₄(C₂²×C₄), (C₂×C₄).₂₅₆(C₃⋊D₄), (C₂×C₆).₆₅(C₂²⋊C₄), (C₂×C₄).₁₂₀(C₂²×S₃), SmallGroup(192,689)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂×C₁₂.₄₆D₄

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂×C₁₂ — C₂×D₁₂ — C₂²×D₁₂ — C₂×C₁₂.₄₆D₄

Lower central

C₃ — C₆ — C₂×C₆ — C₂×C₁₂.₄₆D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂×M₄(2)

Generators and relations for C₂×C₁₂.₄₆D₄
G = < a,b,c,d,e | a²=b⁸=c²=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b⁵, bd=db, ebe=bc, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 664 in 186 conjugacy classes, 63 normal (39 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₆, C₈, C₂×C₄, D₄, C₂³, C₂³, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂×D₄, C₂⁴, C₃⋊C₈, C₂₄, D₁₂, C₂×C₁₂, C₂²×S₃, C₂²×S₃, C₂²×C₆, C₄.D₄, C₂×M₄(2), C₂×M₄(2), C₂²×D₄, C₂×C₃⋊C₈, C₄.Dic₃, C₄.Dic₃, C₂×C₂₄, C₃×M₄(2), C₃×M₄(2), C₂×D₁₂, C₂×D₁₂, C₂²×C₁₂, S₃×C₂³, C₂×C₄.D₄, C₁₂.₄₆D₄, C₂×C₄.Dic₃, C₆×M₄(2), C₂²×D₁₂, C₂×C₁₂.₄₆D₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, D₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄×S₃, D₁₂, C₃⋊D₄, C₂²×S₃, C₄.D₄, C₂×C₂²⋊C₄, D₆⋊C₄, S₃×C₂×C₄, C₂×D₁₂, C₂×C₃⋊D₄, C₂×C₄.D₄, C₁₂.₄₆D₄, C₂×D₆⋊C₄, C₂×C₁₂.₄₆D₄

Smallest permutation representation of C₂×C₁₂.₄₆D₄
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+			+	+	+	+		+			+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	S₃	D₄	D₆	D₆	C₄×S₃	D₁₂	C₃⋊D₄	C₄×S₃	C₄.D₄	C₁₂.₄₆D₄
kernel	C₂×C₁₂.₄₆D₄	C₁₂.₄₆D₄	C₂×C₄.Dic₃	C₆×M₄(2)	C₂²×D₁₂	C₂×D₁₂	S₃×C₂³	C₂×M₄(2)	C₂×C₁₂	M₄(2)	C₂²×C₄	C₂×C₄	C₂×C₄	C₂×C₄	C₂³	C₆	C₂
# reps	1	4	1	1	1	4	4	1	4	2	1	2	4	4	2	2	4

Matrix representation of C₂×C₁₂.₄₆D₄ ►in GL₆(𝔽₇₃)

72	0	0	0	0	0
0	72	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	66	14	0	0
0	0	59	7	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	0	72	0
0	0	0	0	0	72

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

0	1	0	0	0	0
1	0	0	0	0	0
0	0	59	7	0	0
0	0	66	14	0	0
0	0	0	0	59	7
0	0	0	0	66	14

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,59,66,0,0,0,0,7,14,0,0,0,0,0,0,59,66,0,0,0,0,7,14] >;

C₂×C₁₂.₄₆D₄ in GAP, Magma, Sage, TeX

C_2\times C_{12}._{46}D_4

% in TeX

G:=Group("C2xC12.46D4");

// GroupNames label

G:=SmallGroup(192,689);

// by ID

G=gap.SmallGroup(192,689);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,1123,136,438,6278]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=b*c,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C2×C12.46D4 order 192 = 26·3

Direct product of C2 and C12.46D4

G = C₂×C₁₂.₄₆D₄ order 192 = 2⁶·3

Direct product of C₂ and C₁₂.₄₆D₄