Copied to
clipboard

G = C3xC4oD8order 96 = 25·3

Direct product of C3 and C4oD8

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

Aliases: C3xC4oD8, C12oD8, D8:3C6, C12oQ16, Q16:3C6, C12oSD16, SD16:3C6, C12.69D4, C12.47C23, C24.28C22, C4o(C3xD8), (C2xC8):4C6, C12o(C3xD8), C4o(C3xQ16), (C2xC24):9C2, C4oD4:3C6, (C3xD8):7C2, C8.6(C2xC6), C12o(C3xQ16), C4o(C3xSD16), (C3xQ16):7C2, C12o(C3xSD16), D4.2(C2xC6), C2.14(C6xD4), C4.20(C3xD4), (C2xC6).11D4, C6.77(C2xD4), Q8.5(C2xC6), (C3xSD16):7C2, C4.4(C22xC6), C22.1(C3xD4), (C3xD4).12C22, (C3xQ8).13C22, (C2xC12).132C22, (C3xC4oD4):6C2, (C2xC4).28(C2xC6), SmallGroup(96,182)

Series: Derived Chief Lower central Upper central

C1C4 — C3xC4oD8
C1C2C4C12C3xD4C3xD8 — C3xC4oD8
C1C2C4 — C3xC4oD8
C1C12C2xC12 — C3xC4oD8

Generators and relations for C3xC4oD8
 G = < a,b,c,d | a3=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >

Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C12, C12, C2xC6, C2xC6, C2xC8, D8, SD16, Q16, C4oD4, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C4oD8, C2xC24, C3xD8, C3xSD16, C3xQ16, C3xC4oD4, C3xC4oD8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C4oD8, C6xD4, C3xC4oD8

Smallest permutation representation of C3xC4oD8
On 48 points
Generators in S48
(1 12 19)(2 13 20)(3 14 21)(4 15 22)(5 16 23)(6 9 24)(7 10 17)(8 11 18)(25 36 48)(26 37 41)(27 38 42)(28 39 43)(29 40 44)(30 33 45)(31 34 46)(32 35 47)
(1 41 5 45)(2 42 6 46)(3 43 7 47)(4 44 8 48)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)
(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 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)

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

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

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

C3xC4oD8 is a maximal subgroup of
D8:2Dic3  C24.41D4  Q16:D6  Q16.D6  D8.9D6  D8:5Dic3  D8:4Dic3  SD16:D6  D8:15D6  D8:11D6  D8.10D6
C3xC4oD8 is a maximal quotient of
C12xD8  C12xSD16  C12xQ16

42 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B8C8D12A12B12C12D12E12F12G12H12I12J24A···24H
order1222233444446666666688881212121212121212121224···24
size11244111124411224444222211112244442···2

42 irreducible representations

dim111111111111222222
type++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3xD4C3xD4C4oD8C3xC4oD8
kernelC3xC4oD8C2xC24C3xD8C3xSD16C3xQ16C3xC4oD4C4oD8C2xC8D8SD16Q16C4oD4C12C2xC6C4C22C3C1
# reps111212222424112248

Matrix representation of C3xC4oD8 in GL2(F73) generated by

640
064
,
460
046
,
1657
1616
,
10
072
G:=sub<GL(2,GF(73))| [64,0,0,64],[46,0,0,46],[16,16,57,16],[1,0,0,72] >;

C3xC4oD8 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_8
% in TeX

G:=Group("C3xC4oD8");
// GroupNames label

G:=SmallGroup(96,182);
// by ID

G=gap.SmallGroup(96,182);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,230,2164,1090,88]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<