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## G = C3×C8.C8order 192 = 26·3

### Direct product of C3 and C8.C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×C8.C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C24 — C3×M5(2) — C3×C8.C8
 Lower central C1 — C2 — C4 — C3×C8.C8
 Upper central C1 — C24 — C2×C24 — C3×C8.C8

Generators and relations for C3×C8.C8
G = < a,b,c | a3=b8=1, c8=b4, ab=ba, ac=ca, cbc-1=b3 >

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

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 Q8 M4(2) C3×D4 C3×Q8 C3×M4(2) C8.C8 C3×C8.C8 kernel C3×C8.C8 C4×C24 C3×M5(2) C8.C8 C4×C12 C2×C24 C4×C8 M5(2) C24 C42 C2×C8 C8 C24 C24 C2×C6 C8 C8 C22 C3 C1 # reps 1 1 2 2 2 2 2 4 8 4 4 16 1 1 2 2 2 4 8 16

Matrix representation of C3×C8.C8 in GL3(𝔽97) generated by

 61 0 0 0 1 0 0 0 1
,
 1 0 0 0 50 0 0 0 64
,
 47 0 0 0 0 1 0 64 0
G:=sub<GL(3,GF(97))| [61,0,0,0,1,0,0,0,1],[1,0,0,0,50,0,0,0,64],[47,0,0,0,0,64,0,1,0] >;

C3×C8.C8 in GAP, Magma, Sage, TeX

C_3\times C_8.C_8
% in TeX

G:=Group("C3xC8.C8");
// GroupNames label

G:=SmallGroup(192,170);
// by ID

G=gap.SmallGroup(192,170);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,3027,248,102,124]);
// Polycyclic

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

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