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## G = C3×C8.Q8order 192 = 26·3

### Direct product of C3 and C8.Q8

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C3×C8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C24 — C3×C4.Q8 — C3×C8.Q8
 Lower central C1 — C2 — C4 — C8 — C3×C8.Q8
 Upper central C1 — C6 — C2×C12 — C2×C24 — C3×C8.Q8

Generators and relations for C3×C8.Q8
G = < a,b,c,d | a3=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >

Smallest permutation representation of C3×C8.Q8
On 48 points
Generators in S48
(1 30 36)(2 31 37)(3 32 38)(4 17 39)(5 18 40)(6 19 41)(7 20 42)(8 21 43)(9 22 44)(10 23 45)(11 24 46)(12 25 47)(13 26 48)(14 27 33)(15 28 34)(16 29 35)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 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 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 23 25 31)(18 26)(19 29 27 21)(20 32)(24 28)(33 43 41 35)(34 46)(37 39 45 47)(38 42)(40 48)

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

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

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

48 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 8E 12A 12B 12C 12D 12E 12F 12G 12H 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 24G 24H 24I 24J 48A ··· 48H order 1 2 2 3 3 4 4 4 4 6 6 6 6 8 8 8 8 8 12 12 12 12 12 12 12 12 16 16 16 16 24 24 24 24 24 24 24 24 24 24 48 ··· 48 size 1 1 2 1 1 2 2 8 8 1 1 2 2 2 2 4 8 8 2 2 2 2 8 8 8 8 4 4 4 4 2 2 2 2 4 4 8 8 8 8 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + - + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 Q8 D4 SD16 SD16 C3×Q8 C3×D4 C3×SD16 C3×SD16 C8.Q8 C3×C8.Q8 kernel C3×C8.Q8 C3×C4.Q8 C3×C8.C4 C3×M5(2) C8.Q8 C48 C4.Q8 C8.C4 M5(2) C16 C24 C2×C12 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 4 4 2 4

Matrix representation of C3×C8.Q8 in GL4(𝔽97) generated by

 61 0 0 0 0 61 0 0 0 0 61 0 0 0 0 61
,
 57 40 0 0 57 57 0 0 0 0 40 57 0 0 40 40
,
 0 0 1 0 0 0 0 1 57 40 0 0 57 57 0 0
,
 1 0 0 0 0 96 0 0 0 0 40 57 0 0 57 57
G:=sub<GL(4,GF(97))| [61,0,0,0,0,61,0,0,0,0,61,0,0,0,0,61],[57,57,0,0,40,57,0,0,0,0,40,40,0,0,57,40],[0,0,57,57,0,0,40,57,1,0,0,0,0,1,0,0],[1,0,0,0,0,96,0,0,0,0,40,57,0,0,57,57] >;

C3×C8.Q8 in GAP, Magma, Sage, TeX

C_3\times C_8.Q_8
% in TeX

G:=Group("C3xC8.Q8");
// GroupNames label

G:=SmallGroup(192,171);
// by ID

G=gap.SmallGroup(192,171);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,1683,136,2111,6053,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^3,d*c*d^-1=b^4*c^3>;
// generators/relations

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