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G = C3×C4○D8order 96 = 25·3

Direct product of C3 and C4○D8

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

Aliases: C3×C4○D8, C12D8, D83C6, C12Q16, Q163C6, C12SD16, SD163C6, C12.69D4, C12.47C23, C24.28C22, C4(C3×D8), (C2×C8)⋊4C6, C12(C3×D8), C4(C3×Q16), (C2×C24)⋊9C2, C4○D43C6, (C3×D8)⋊7C2, C8.6(C2×C6), C12(C3×Q16), C4(C3×SD16), (C3×Q16)⋊7C2, C12(C3×SD16), D4.2(C2×C6), C2.14(C6×D4), C4.20(C3×D4), (C2×C6).11D4, C6.77(C2×D4), Q8.5(C2×C6), (C3×SD16)⋊7C2, C4.4(C22×C6), C22.1(C3×D4), (C3×D4).12C22, (C3×Q8).13C22, (C2×C12).132C22, (C3×C4○D4)⋊6C2, (C2×C4).28(C2×C6), SmallGroup(96,182)

Series: Derived Chief Lower central Upper central

C1C4 — C3×C4○D8
C1C2C4C12C3×D4C3×D8 — C3×C4○D8
C1C2C4 — C3×C4○D8
C1C12C2×C12 — C3×C4○D8

Generators and relations for C3×C4○D8
 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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C6, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], D4 [×2], Q8 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C24 [×2], C2×C12, C2×C12 [×2], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, C3×C4○D4 [×2], C3×C4○D8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C2×C6 [×7], C2×D4, C3×D4 [×2], C22×C6, C4○D8, C6×D4, C3×C4○D8

Smallest permutation representation of C3×C4○D8
On 48 points
Generators in S48
(1 33 19)(2 34 20)(3 35 21)(4 36 22)(5 37 23)(6 38 24)(7 39 17)(8 40 18)(9 41 26)(10 42 27)(11 43 28)(12 44 29)(13 45 30)(14 46 31)(15 47 32)(16 48 25)
(1 41 5 45)(2 42 6 46)(3 43 7 47)(4 44 8 48)(9 23 13 19)(10 24 14 20)(11 17 15 21)(12 18 16 22)(25 36 29 40)(26 37 30 33)(27 38 31 34)(28 39 32 35)
(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)(10 16)(11 15)(12 14)(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)| (1,33,19)(2,34,20)(3,35,21)(4,36,22)(5,37,23)(6,38,24)(7,39,17)(8,40,18)(9,41,26)(10,42,27)(11,43,28)(12,44,29)(13,45,30)(14,46,31)(15,47,32)(16,48,25), (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,23,13,19)(10,24,14,20)(11,17,15,21)(12,18,16,22)(25,36,29,40)(26,37,30,33)(27,38,31,34)(28,39,32,35), (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)(10,16)(11,15)(12,14)(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( (1,33,19)(2,34,20)(3,35,21)(4,36,22)(5,37,23)(6,38,24)(7,39,17)(8,40,18)(9,41,26)(10,42,27)(11,43,28)(12,44,29)(13,45,30)(14,46,31)(15,47,32)(16,48,25), (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,23,13,19)(10,24,14,20)(11,17,15,21)(12,18,16,22)(25,36,29,40)(26,37,30,33)(27,38,31,34)(28,39,32,35), (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)(10,16)(11,15)(12,14)(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([(1,33,19),(2,34,20),(3,35,21),(4,36,22),(5,37,23),(6,38,24),(7,39,17),(8,40,18),(9,41,26),(10,42,27),(11,43,28),(12,44,29),(13,45,30),(14,46,31),(15,47,32),(16,48,25)], [(1,41,5,45),(2,42,6,46),(3,43,7,47),(4,44,8,48),(9,23,13,19),(10,24,14,20),(11,17,15,21),(12,18,16,22),(25,36,29,40),(26,37,30,33),(27,38,31,34),(28,39,32,35)], [(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),(10,16),(11,15),(12,14),(17,21),(18,20),(22,24),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(42,48),(43,47),(44,46)])

C3×C4○D8 is a maximal subgroup of
D82Dic3  C24.41D4  Q16⋊D6  Q16.D6  D8.9D6  D85Dic3  D84Dic3  SD16⋊D6  D815D6  D811D6  D8.10D6
C3×C4○D8 is a maximal quotient of
C12×D8  C12×SD16  C12×Q16

42 conjugacy classes

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

42 irreducible representations

dim111111111111222222
type++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3×D4C3×D4C4○D8C3×C4○D8
kernelC3×C4○D8C2×C24C3×D8C3×SD16C3×Q16C3×C4○D4C4○D8C2×C8D8SD16Q16C4○D4C12C2×C6C4C22C3C1
# reps111212222424112248

Matrix representation of C3×C4○D8 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] >;

C3×C4○D8 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

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