Copied to
clipboard

G = C3×D44D4order 192 = 26·3

Direct product of C3 and D44D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×D44D4, 2+ 1+46C6, C4≀C21C6, D44(C3×D4), Q85(C3×D4), (C3×D4)⋊22D4, C41D43C6, C8⋊C221C6, C425(C2×C6), (C3×Q8)⋊22D4, C4.27(C6×D4), C4.D41C6, C23.5(C3×D4), (C22×C6).5D4, (C4×C12)⋊35C22, C12.388(C2×D4), (C6×D4)⋊28C22, M4(2)⋊1(C2×C6), C22.14(C6×D4), C6.100C22≀C2, (C2×C12).609C23, (C3×2+ 1+4)⋊7C2, (C3×M4(2))⋊14C22, (C3×C4≀C2)⋊5C2, (C2×D4)⋊2(C2×C6), (C3×C8⋊C22)⋊8C2, C4○D4.6(C2×C6), (C3×C41D4)⋊11C2, (C2×C6).409(C2×D4), (C3×C4.D4)⋊5C2, (C2×C4).4(C22×C6), C2.14(C3×C22≀C2), (C3×C4○D4).31C22, SmallGroup(192,886)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C3×D44D4
C1C2C22C2×C4C2×C12C6×D4C3×C8⋊C22 — C3×D44D4
C1C2C2×C4 — C3×D44D4
C1C6C2×C12 — C3×D44D4

Generators and relations for C3×D44D4
 G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×11], C6, C6 [×6], C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×14], Q8 [×2], C23 [×2], C23 [×3], C12 [×2], C12 [×4], C2×C6, C2×C6 [×11], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×2], C2×D4 [×6], C4○D4 [×2], C4○D4 [×2], C24 [×2], C2×C12, C2×C12 [×5], C3×D4 [×2], C3×D4 [×14], C3×Q8 [×2], C22×C6 [×2], C22×C6 [×3], C4.D4, C4≀C2 [×2], C41D4, C8⋊C22 [×2], 2+ 1+4, C4×C12, C3×M4(2) [×2], C3×D8 [×2], C3×SD16 [×2], C6×D4 [×2], C6×D4 [×6], C3×C4○D4 [×2], C3×C4○D4 [×2], D44D4, C3×C4.D4, C3×C4≀C2 [×2], C3×C41D4, C3×C8⋊C22 [×2], C3×2+ 1+4, C3×D44D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, C2×C6 [×7], C2×D4 [×3], C3×D4 [×6], C22×C6, C22≀C2, C6×D4 [×3], D44D4, C3×C22≀C2, C3×D44D4

Permutation representations of C3×D44D4
On 24 points - transitive group 24T350
Generators in S24
(1 11 7)(2 12 8)(3 9 5)(4 10 6)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 20)(8 19)(9 22)(10 21)(11 24)(12 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3)(5 7)(9 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)

G:=sub<Sym(24)| (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,22)(10,21)(11,24)(12,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(5,7)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)>;

G:=Group( (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,22)(10,21)(11,24)(12,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(5,7)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23) );

G=PermutationGroup([(1,11,7),(2,12,8),(3,9,5),(4,10,6),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,20),(8,19),(9,22),(10,21),(11,24),(12,23)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3),(5,7),(9,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23)])

G:=TransitiveGroup(24,350);

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E4F6A6B6C6D6E···6L6M6N8A8B12A12B12C12D12E···12L24A24B24C24D
order122222223344444466666···666881212121212···1224242424
size112444481122444411224···4888822224···48888

48 irreducible representations

dim11111111111122222244
type++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4D4C3×D4C3×D4C3×D4D44D4C3×D44D4
kernelC3×D44D4C3×C4.D4C3×C4≀C2C3×C41D4C3×C8⋊C22C3×2+ 1+4D44D4C4.D4C4≀C2C41D4C8⋊C222+ 1+4C3×D4C3×Q8C22×C6D4Q8C23C3C1
# reps11212122424222244424

Matrix representation of C3×D44D4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
3231
1010
4446
4350
,
1454
3440
5214
3341
,
0140
0105
0010
3420
,
1045
0155
0060
0006
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,4,4,2,0,4,3,3,1,4,5,1,0,6,0],[1,3,5,3,4,4,2,3,5,4,1,4,4,0,4,1],[0,0,0,3,1,1,0,4,4,0,1,2,0,5,0,0],[1,0,0,0,0,1,0,0,4,5,6,0,5,5,0,6] >;

C3×D44D4 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_4D_4
% in TeX

G:=Group("C3xD4:4D4");
// GroupNames label

G:=SmallGroup(192,886);
// by ID

G=gap.SmallGroup(192,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,4204,2111,1068,172,3036]);
// Polycyclic

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

׿
×
𝔽