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

Aliases: C3×Q8○M4(2), C24.81C23, C12.94C24, C8○D4⋊12C6, C4○D4.5C12, D4.9(C2×C12), (C6×D4).22C4, (C6×Q8).18C4, (C2×C24)⋊39C22, (C2×D4).10C12, C8.14(C22×C6), C6.64(C23×C4), C4.18(C23×C6), Q8.15(C2×C12), (C2×Q8).11C12, (C2×M4(2))⋊16C6, (C6×M4(2))⋊34C2, M4(2)⋊12(C2×C6), C2.12(C23×C12), C4.23(C22×C12), C23.13(C2×C12), (C2×C12).970C23, C12.168(C22×C4), C22.5(C22×C12), (C3×M4(2))⋊41C22, (C22×C12).464C22, (C2×C8)⋊9(C2×C6), (C3×C8○D4)⋊17C2, (C3×C4○D4).6C4, (C2×C4).32(C2×C12), (C6×C4○D4).25C2, (C2×C4○D4).17C6, C4○D4.22(C2×C6), (C3×D4).31(C2×C4), (C3×Q8).34(C2×C4), (C2×C12).204(C2×C4), (C22×C6).25(C2×C4), (C22×C4).80(C2×C6), (C2×C6).36(C22×C4), (C2×C4).140(C22×C6), (C3×C4○D4).60C22, SmallGroup(192,1457)

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 2A2B···2H3A3B4A4B4C···4I6A6B6C···6P8A···8P12A12B12C12D12E···12R24A···24AF
order122···233444···4666···68···81212121212···1224···24
size112···211112···2112···22···211112···22···2

102 irreducible representations

dim1111111111111144
type++++
imageC1C2C2C2C3C4C4C4C6C6C6C12C12C12Q8○M4(2)C3×Q8○M4(2)
kernelC3×Q8○M4(2)C6×M4(2)C3×C8○D4C6×C4○D4Q8○M4(2)C6×D4C6×Q8C3×C4○D4C2×M4(2)C8○D4C2×C4○D4C2×D4C2×Q8C4○D4C3C1
# reps16812628121621241624

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

8000
0800
0080
0008
,
46000
19272647
00027
00270
,
2746470
04600
00460
00027
,
2603835
0010
02700
19272647
,
104528
0100
00720
00072
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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁