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G = C3×C8○D8order 192 = 26·3

Direct product of C3 and C8○D8

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

Aliases: C3×C8○D8, D85C12, Q165C12, C24.77D4, SD163C12, C4≀C27C6, (C4×C8)⋊10C6, (C4×C24)⋊21C2, C8○D410C6, (C3×D8)⋊11C4, C4○D8.5C6, C4.82(C6×D4), C8.30(C3×D4), C8.C48C6, 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

C1C4 — C3×C8○D8
C1C2C4C2×C4C2×C12C3×M4(2)C3×C4≀C2 — C3×C8○D8
C1C2C4 — C3×C8○D8
C1C24C2×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 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C6, C6 [×3], C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×2], C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], C24 [×4], C24 [×2], C2×C12, C2×C12 [×3], C3×D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C4×C12, C2×C24 [×2], C2×C24 [×2], C3×M4(2) [×2], C3×M4(2) [×2], C3×D8, C3×SD16 [×2], C3×Q16, C3×C4○D4 [×2], C8○D8, C4×C24, C3×C4≀C2 [×2], C3×C8.C4, C3×C8○D4 [×2], C3×C4○D8, C3×C8○D8
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C2×C12 [×6], C3×D4 [×2], 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 23)(2 28 24)(3 29 17)(4 30 18)(5 31 19)(6 32 20)(7 25 21)(8 26 22)(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 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)

G:=sub<Sym(48)| (1,27,23)(2,28,24)(3,29,17)(4,30,18)(5,31,19)(6,32,20)(7,25,21)(8,26,22)(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)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)>;

G:=Group( (1,27,23)(2,28,24)(3,29,17)(4,30,18)(5,31,19)(6,32,20)(7,25,21)(8,26,22)(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)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35) );

G=PermutationGroup([(1,27,23),(2,28,24),(3,29,17),(4,30,18),(5,31,19),(6,32,20),(7,25,21),(8,26,22),(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),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35)])

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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