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G = C3×C8○D8  order 192 = 26·3

Direct product of C3 and C8○D8

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

Aliases: C3×C8○D8, D8⋊5C12, Q16⋊5C12, C24.77D4, SD16⋊3C12, C4≀C2⋊7C6, (C4×C8)⋊10C6, (C4×C24)⋊21C2, C8○D4⋊10C6, (C3×D8)⋊11C4, C4○D8.5C6, C4.82(C6×D4), C8.30(C3×D4), C8.C4⋊8C6, C24.67(C2×C4), C8.11(C2×C12), (C3×Q16)⋊11C4, (C3×SD16)⋊7C4, D4.3(C2×C12), C6.120(C4×D4), C2.18(D4×C12), Q8.8(C2×C12), C12.487(C2×D4), C42.73(C2×C6), C4.15(C22×C12), (C4×C12).358C22, C12.160(C22×C4), (C2×C24).411C22, (C2×C12).910C23, M4(2).11(C2×C6), (C3×M4(2)).45C22, (C3×C4≀C2)⋊15C2, (C3×C8○D4)⋊15C2, (C2×C8).101(C2×C6), C4○D4.15(C2×C6), (C3×C4○D8).10C2, (C3×D4).20(C2×C4), (C3×Q8).21(C2×C4), (C3×C8.C4)⋊17C2, C22.1(C3×C4○D4), (C2×C6).49(C4○D4), (C2×C4).85(C22×C6), (C3×C4○D4).53C22, SmallGroup(192,876)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C4 — C3×C8○D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C12 — C3×M4(2) — C3×C4≀C2 — C3×C8○D8
Lower central
C1 — C2 — C4 — C3×C8○D8
Upper central
C1 — C24 — C2×C24 — C3×C8○D8

Generators and relations for C3×C8○D8
 G = < a,b,c,d | a3=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >

Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C4×C12, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C8○D8, C4×C24, C3×C4≀C2, C3×C8.C4, C3×C8○D4, C3×C4○D8, C3×C8○D8
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C22×C12, C6×D4, C3×C4○D4, C8○D8, D4×C12, C3×C8○D8

Smallest permutation representation of C3×C8○D8
►On 48 points
Generators in S48
(1 27 21)(2 28 22)(3 29 23)(4 30 24)(5 31 17)(6 32 18)(7 25 19)(8 26 20)(9 36 41)(10 37 42)(11 38 43)(12 39 44)(13 40 45)(14 33 46)(15 34 47)(16 35 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)

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

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

84 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C···4G4H4I6A6B6C6D6E6F6G6H8A8B8C8D8E···8J8K8L8M8N12A12B12C12D12E···12N12O12P12Q12R24A···24H24I···24T24U···24AB
order1222233444···4446666666688888···888881212121212···121212121224···2424···2424···24
size1124411112···2441122444411112···2444411112···244441···12···24···4

84 irreducible representations

dim111111111111111111222222
type+++++++
imageC1C2C2C2C2C2C3C4C4C4C6C6C6C6C6C12C12C12D4C4○D4C3×D4C3×C4○D4C8○D8C3×C8○D8
kernelC3×C8○D8C4×C24C3×C4≀C2C3×C8.C4C3×C8○D4C3×C4○D8C8○D8C3×D8C3×SD16C3×Q16C4×C8C4≀C2C8.C4C8○D4C4○D8D8SD16Q16C24C2×C6C8C22C3C1
# reps1121212242242424842244816

Matrix representation of C3×C8○D8 ►in GL3(𝔽73) generated by

6400
010
001
,
100
0100
0010
,
7200
0630
0051
,
100
001
010
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[1,0,0,0,10,0,0,0,10],[72,0,0,0,63,0,0,0,51],[1,0,0,0,0,1,0,1,0] >;
 

C3×C8○D8 in GAP, Magma, Sage, TeX 

C_3\times C_8\circ D_8
 
% in TeX
 
G:=Group("C3xC8oD8");
 
// GroupNames label
 
G:=SmallGroup(192,876);
 
// by ID
 
G=gap.SmallGroup(192,876);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,268,4204,2111,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
 
// generators/relations
 

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