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G = C3×D4○D8  order 192 = 26·3

Direct product of C3 and D4○D8

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

Aliases: C3×D4○D8, C24.48C23, C12.85C24, 2+ 1+4⋊8C6, C8○D4⋊7C6, C4○D8⋊4C6, D8⋊7(C2×C6), (C2×D8)⋊12C6, (C6×D8)⋊26C2, C8⋊C22⋊4C6, Q16⋊7(C2×C6), C4.45(C6×D4), SD16⋊4(C2×C6), D4.11(C3×D4), (C3×D4).45D4, C4.8(C23×C6), Q8.16(C3×D4), (C3×Q8).45D4, C22.7(C6×D4), (C2×C24)⋊23C22, C12.406(C2×D4), (C6×D4)⋊40C22, (C3×D8)⋊21C22, M4(2)⋊6(C2×C6), C8.10(C22×C6), D4.5(C22×C6), Q8.9(C22×C6), (C3×Q16)⋊21C22, (C3×D4).38C23, C6.206(C22×D4), (C3×Q8).39C23, (C2×C12).687C23, (C3×SD16)⋊20C22, (C3×2+ 1+4)⋊9C2, (C3×M4(2))⋊27C22, (C2×C8)⋊4(C2×C6), C2.30(D4×C2×C6), C4○D4⋊2(C2×C6), (C3×C8○D4)⋊8C2, (C2×D4)⋊7(C2×C6), (C3×C4○D8)⋊11C2, (C3×C8⋊C22)⋊11C2, (C2×C6).184(C2×D4), (C3×C4○D4)⋊14C22, (C2×C4).48(C22×C6), SmallGroup(192,1465)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C4 — C3×D4○D8
Chief series
C1 — C2 — C4 — C12 — C3×D4 — C3×D8 — C6×D8 — C3×D4○D8
Lower central
C1 — C2 — C4 — C3×D4○D8
Upper central
C1 — C6 — C3×C4○D4 — C3×D4○D8

Generators and relations for C3×D4○D8
 G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C2×C24, C3×M4(2), C3×D8, C3×SD16, C3×Q16, C6×D4, C6×D4, C3×C4○D4, C3×C4○D4, C3×C4○D4, D4○D8, C3×C8○D4, C6×D8, C3×C4○D8, C3×C8⋊C22, C3×2+ 1+4, C3×D4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, D4○D8, D4×C2×C6, C3×D4○D8

Smallest permutation representation of C3×D4○D8
►On 48 points
Generators in S48
(1 35 19)(2 36 20)(3 37 21)(4 38 22)(5 39 23)(6 40 24)(7 33 17)(8 34 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 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)

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

66 conjugacy classes

class 1 2A2B2C2D2E···2J3A3B4A4B4C4D4E4F6A6B6C···6H6I···6T8A8B8C8D8E12A···12H12I12J12K12L24A24B24C24D24E···24J
order122222···233444444666···66···68888812···12121212122424242424···24
size112224···411222244112···24···4224442···2444422224···4

66 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4D4C3×D4C3×D4D4○D8C3×D4○D8
kernelC3×D4○D8C3×C8○D4C6×D8C3×C4○D8C3×C8⋊C22C3×2+ 1+4D4○D8C8○D4C2×D8C4○D8C8⋊C222+ 1+4C3×D4C3×Q8D4Q8C3C1
# reps1133622266124316224

Matrix representation of C3×D4○D8 ►in GL4(𝔽7) generated by

4000
0400
0040
0004
,
6635
3651
1001
1132
,
6014
0545
0455
0665
,
5051
1521
1625
5511
,
0450
4610
4520
6146
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,3,1,1,6,6,0,1,3,5,0,3,5,1,1,2],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5],[5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[0,4,4,6,4,6,5,1,5,1,2,4,0,0,0,6] >;
 

C3×D4○D8 in GAP, Magma, Sage, TeX 

C_3\times D_4\circ D_8
 
% in TeX
 
G:=Group("C3xD4oD8");
 
// GroupNames label
 
G:=SmallGroup(192,1465);
 
// by ID
 
G=gap.SmallGroup(192,1465);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,745,6053,3036,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
 
// generators/relations
 

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