Copied to
clipboard

## G = C3×Q8○M4(2)  order 192 = 26·3

### Direct product of C3 and Q8○M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8○M4(2)
 Chief series C1 — C2 — C4 — C12 — C24 — C2×C24 — C3×C8○D4 — C3×Q8○M4(2)
 Lower central C1 — C2 — C3×Q8○M4(2)
 Upper central C1 — C12 — C3×Q8○M4(2)

Generators and relations for C3×Q8○M4(2)
G = < a,b,c,d,e | a3=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 290 in 258 conjugacy classes, 238 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×M4(2), C8○D4, C2×C4○D4, C2×C24, C3×M4(2), C22×C12, C6×D4, C6×Q8, C3×C4○D4, Q8○M4(2), C6×M4(2), C3×C8○D4, C6×C4○D4, C3×Q8○M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C24, C2×C12, C22×C6, C23×C4, C22×C12, C23×C6, Q8○M4(2), C23×C12, C3×Q8○M4(2)

Smallest permutation representation of C3×Q8○M4(2)
On 48 points
Generators in S48
(1 34 45)(2 35 46)(3 36 47)(4 37 48)(5 38 41)(6 39 42)(7 40 43)(8 33 44)(9 24 28)(10 17 29)(11 18 30)(12 19 31)(13 20 32)(14 21 25)(15 22 26)(16 23 27)
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 41 13 45)(10 42 14 46)(11 43 15 47)(12 44 16 48)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 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 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)

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

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

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

102 conjugacy classes

 class 1 2A 2B ··· 2H 3A 3B 4A 4B 4C ··· 4I 6A 6B 6C ··· 6P 8A ··· 8P 12A 12B 12C 12D 12E ··· 12R 24A ··· 24AF order 1 2 2 ··· 2 3 3 4 4 4 ··· 4 6 6 6 ··· 6 8 ··· 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 2 ··· 2 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + image C1 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C12 C12 C12 Q8○M4(2) C3×Q8○M4(2) kernel C3×Q8○M4(2) C6×M4(2) C3×C8○D4 C6×C4○D4 Q8○M4(2) C6×D4 C6×Q8 C3×C4○D4 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C3 C1 # reps 1 6 8 1 2 6 2 8 12 16 2 12 4 16 2 4

Matrix representation of C3×Q8○M4(2) in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 46 0 0 0 19 27 26 47 0 0 0 27 0 0 27 0
,
 27 46 47 0 0 46 0 0 0 0 46 0 0 0 0 27
,
 26 0 38 35 0 0 1 0 0 27 0 0 19 27 26 47
,
 1 0 45 28 0 1 0 0 0 0 72 0 0 0 0 72
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[46,19,0,0,0,27,0,0,0,26,0,27,0,47,27,0],[27,0,0,0,46,46,0,0,47,0,46,0,0,0,0,27],[26,0,0,19,0,0,27,27,38,1,0,26,35,0,0,47],[1,0,0,0,0,1,0,0,45,0,72,0,28,0,0,72] >;

C3×Q8○M4(2) in GAP, Magma, Sage, TeX

C_3\times Q_8\circ M_4(2)
% in TeX

G:=Group("C3xQ8oM4(2)");
// GroupNames label

G:=SmallGroup(192,1457);
// by ID

G=gap.SmallGroup(192,1457);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,1059,2915,124]);
// Polycyclic

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

׿
×
𝔽