M4(2):24D6 - GroupNames

G = M₄(2):₂₄D₆ order 192 = 2⁶·3

8^th semidirect product of M₄(2) and D₆ acting via D₆/C₆=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2):₂₄D₆, C₂³.₂₅D₁₂, C₄oD₁₂:₈C₄, (C₂xD₁₂):₁₆C₄, (C₂xC₄).₅₅D₁₂, C₄.₃₀(D₆:C₄), (C₂xDic₆):₁₆C₄, D₁₂.₂₇(C₂xC₄), (C₂xC₁₂).₁₇₈D₄, C₁₂.₄₄₆(C₂xD₄), D₁₂:C₄:₁₃C₂, (C₆xM₄(2)):₂₃C₂, (C₂xM₄(2)):₁₅S₃, C₁₂.₇₀(C₂²xC₄), Dic₆.₂₈(C₂xC₄), (C₄xDic₃):₄C₂², C₂².₁₇(C₂xD₁₂), (C₂²xC₆).₁₀₇D₄, (C₂²xC₄).₁₆₂D₆, C₁₂.₂₉(C₂²:C₄), (C₂xC₁₂).₄₂₀C₂³, C₂².₂₉(D₆:C₄), C₃:₃(C₄²:C₂²), C₄oD₁₂.₄₃C₂², C₂³.₂₆D₆:₁₇C₂, (C₃xM₄(2)):₃₆C₂², (C₂²xC₁₂).₁₉₃C₂², C₄.₅₅(S₃xC₂xC₄), (C₂xC₄).₅₆(C₄xS₃), C₂.₃₅(C₂xD₆:C₄), (C₂xC₆).₃₃(C₂xD₄), C₄.₁₃₇(C₂xC₃:D₄), C₆.₆₃(C₂xC₂²:C₄), (C₂xC₁₂).₁₁₃(C₂xC₄), (C₂xC₄oD₁₂).₁₅C₂, (C₂xC₄).₂₅₈(C₃:D₄), (C₂xC₆).₂₃(C₂²:C₄), (C₂xC₄).₅₁₃(C₂²xS₃), SmallGroup(192,698)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — M₄(2):₂₄D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₄oD₁₂ — C₂xC₄oD₁₂ — M₄(2):₂₄D₆

Lower central

C₃ — C₆ — C₁₂ — M₄(2):₂₄D₆

Upper central

C₁ — C₄ — C₂²xC₄ — C₂xM₄(2)

Generators and relations for M₄(2):₂₄D₆
G = < a,b,c,d | a⁸=b²=c⁶=d²=1, bab=cac^-1=a⁵, dad=a³b, bc=cb, dbd=a⁴b, dcd=c^-1 >

Subgroups: 440 in 154 conjugacy classes, 59 normal (41 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₄:C₄, C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂₄, Dic₆, Dic₆, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₄wrC₂, C₄²:C₂, C₂xM₄(2), C₂xC₄oD₄, C₄xDic₃, C₄:Dic₃, C₆.D₄, C₂xC₂₄, C₃xM₄(2), C₃xM₄(2), C₂xDic₆, S₃xC₂xC₄, C₂xD₁₂, C₄oD₁₂, C₄oD₁₂, C₂xC₃:D₄, C₂²xC₁₂, C₄²:C₂², D₁₂:C₄, C₂³.₂₆D₆, C₆xM₄(2), C₂xC₄oD₁₂, M₄(2):₂₄D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, D₆, C₂²:C₄, C₂²xC₄, C₂xD₄, C₄xS₃, D₁₂, C₃:D₄, C₂²xS₃, C₂xC₂²:C₄, D₆:C₄, S₃xC₂xC₄, C₂xD₁₂, C₂xC₃:D₄, C₄²:C₂², C₂xD₆:C₄, M₄(2):₂₄D₆

Smallest permutation representation of M₄(2):₂₄D₆
►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 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 45)(18 42)(19 47)(20 44)(21 41)(22 46)(23 43)(24 48)(25 35)(26 40)(27 37)(28 34)(29 39)(30 36)(31 33)(32 38)

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4
type	+	+	+	+	+				+	+	+	+	+		+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	D₄	D₄	D₆	D₆	C₄xS₃	D₁₂	C₃:D₄	D₁₂	C₄²:C₂²	M₄(2):₂₄D₆
kernel	M₄(2):₂₄D₆	D₁₂:C₄	C₂³.₂₆D₆	C₆xM₄(2)	C₂xC₄oD₁₂	C₂xDic₆	C₂xD₁₂	C₄oD₁₂	C₂xM₄(2)	C₂xC₁₂	C₂²xC₆	M₄(2)	C₂²xC₄	C₂xC₄	C₂xC₄	C₂xC₄	C₂³	C₃	C₁
# reps	1	4	1	1	1	2	2	4	1	3	1	2	1	4	2	4	2	2	4

Matrix representation of M₄(2):₂₄D₆ ►in GL₆(F₇₃)

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	0	27
0	0	0	0	46	0
0	0	0	72	0	0
0	0	1	0	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	46	0	0
0	0	27	0	0	0
0	0	0	0	0	27
0	0	0	0	46	0

1	1	0	0	0	0
72	0	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

72	72	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	0	72
0	0	0	0	72	0

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,46,0,0,0,0,27,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

M₄(2):₂₄D₆ in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{24}D_6

% in TeX

G:=Group("M4(2):24D6");

// GroupNames label

G:=SmallGroup(192,698);

// by ID

G=gap.SmallGroup(192,698);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,136,1684,438,102,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^3*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;

// generators/relations

G = M4(2):24D6 order 192 = 26·3

8th semidirect product of M4(2) and D6 acting via D6/C6=C2

G = M₄(2):₂₄D₆ order 192 = 2⁶·3

8^th semidirect product of M₄(2) and D₆ acting via D₆/C₆=C₂