M4(2).21D6 - GroupNames

G = M₄(2).₂₁D₆ order 192 = 2⁶·3

4^th non-split extension by M₄(2) of D₆ acting via D₆/S₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M₄(2).₂₁D₆, (C₄xS₃).₃₆D₄, (C₂xD₁₂).₄C₄, C₄.₁₅₁(S₃xD₄), C₁₂.₉₆(C₂xD₄), C₄.₁₀D₄:₆S₃, (S₃xM₄(2)):₇C₂, (C₂xC₁₂).₈C₂³, (C₂xQ₈).₁₁₉D₆, (C₆xQ₈).₆C₂², C₁₂.₁₀D₄:₄C₂, D₆.₃(C₂²:C₄), C₁₂.₄₆D₄:₁₁C₂, (C₂xD₁₂).₄₀C₂², C₄.Dic₃.₅C₂², Dic₃.₁₇(C₂²:C₄), (C₃xM₄(2)).₂₁C₂², C₃:₂(M₄(2).₈C₂²), (S₃xC₂xC₄).₃C₄, (C₂xC₄).₇(C₄xS₃), (C₂xC₁₂).₇(C₂xC₄), (S₃xC₂xC₄).₄C₂², C₂².₁₇(S₃xC₂xC₄), C₂.₁₆(S₃xC₂²:C₄), C₆.₁₅(C₂xC₂²:C₄), (C₂xC₄).₈(C₂²xS₃), (C₂²xS₃).₃(C₂xC₄), (C₂xC₆).₁₁(C₂²xC₄), (C₂xQ₈:₃S₃).₁C₂, (C₃xC₄.₁₀D₄):₁₀C₂, (C₂xDic₃).₈₆(C₂xC₄), SmallGroup(192,310)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — M₄(2).₂₁D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — S₃xC₂xC₄ — C₂xQ₈:₃S₃ — M₄(2).₂₁D₆

Lower central

C₃ — C₆ — C₂xC₆ — M₄(2).₂₁D₆

Upper central

C₁ — C₂ — C₂xC₄ — C₄.₁₀D₄

Generators and relations for M₄(2).₂₁D₆
G = < a,b,c,d | a⁸=b²=d²=1, c⁶=a⁴, bab=a⁵, cac^-1=ab, dad=a⁵b, bc=cb, bd=db, dcd=a⁴c⁵ >

Subgroups: 432 in 150 conjugacy classes, 51 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₂xC₈, M₄(2), M₄(2), C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₃:C₈, C₂₄, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xQ₈, C₂²xS₃, C₂²xS₃, C₄.D₄, C₄.₁₀D₄, C₄.₁₀D₄, C₂xM₄(2), C₂xC₄oD₄, S₃xC₈, C₈:S₃, C₄.Dic₃, C₃xM₄(2), S₃xC₂xC₄, S₃xC₂xC₄, C₂xD₁₂, C₂xD₁₂, Q₈:₃S₃, C₆xQ₈, M₄(2).₈C₂², C₁₂.₄₆D₄, C₁₂.₁₀D₄, C₃xC₄.₁₀D₄, S₃xM₄(2), C₂xQ₈:₃S₃, M₄(2).₂₁D₆
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, C₂³, D₆, C₂²:C₄, C₂²xC₄, C₂xD₄, C₄xS₃, C₂²xS₃, C₂xC₂²:C₄, S₃xC₂xC₄, S₃xD₄, M₄(2).₈C₂², S₃xC₂²:C₄, M₄(2).₂₁D₆

Smallest permutation representation of M₄(2).₂₁D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim

type

image

C₁

C₂

C₄

S₃

D₄

D₆

C₄xS₃

S₃xD₄

M₄(2).₈C₂²

M₄(2).₂₁D₆

kernel

M₄(2).₂₁D₆

C₁₂.₄₆D₄

C₁₂.₁₀D₄

C₃xC₄.₁₀D₄

S₃xM₄(2)

C₂xQ₈:₃S₃

S₃xC₂xC₄

C₂xD₁₂

C₄.₁₀D₄

C₄xS₃

M₄(2)

C₂xQ₈

C₂xC₄

C₄

C₃

C₁

# reps

Matrix representation of M₄(2).₂₁D₆ ►in GL₆(F₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	18	18	18	71
0	0	5	1	47	8
0	0	23	26	27	70
0	0	26	40	25	27

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	65	72	0	0
0	0	0	0	1	0
0	0	19	0	18	72

0	72	0	0	0	0
1	72	0	0	0	0
0	0	1	0	48	0
0	0	42	46	0	3
0	0	3	0	72	0
0	0	72	0	58	27

1	72	0	0	0	0
0	72	0	0	0	0
0	0	72	0	25	0
0	0	31	27	0	70
0	0	0	0	1	0
0	0	1	48	15	46

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,5,23,26,0,0,18,1,26,40,0,0,18,47,27,25,0,0,71,8,70,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,65,0,19,0,0,0,72,0,0,0,0,0,0,1,18,0,0,0,0,0,72],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,42,3,72,0,0,0,46,0,0,0,0,48,0,72,58,0,0,0,3,0,27],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,31,0,1,0,0,0,27,0,48,0,0,25,0,1,15,0,0,0,70,0,46] >;

M₄(2).₂₁D₆ in GAP, Magma, Sage, TeX

M_4(2)._{21}D_6

% in TeX

G:=Group("M4(2).21D6");

// GroupNames label

G:=SmallGroup(192,310);

// by ID

G=gap.SmallGroup(192,310);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,219,58,570,136,438,6278]);

// Polycyclic

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

// generators/relations

G = M4(2).21D6 order 192 = 26·3

4th non-split extension by M4(2) of D6 acting via D6/S3=C2

G = M₄(2).₂₁D₆ order 192 = 2⁶·3

4^th non-split extension by M₄(2) of D₆ acting via D₆/S₃=C₂