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G = C3×C22⋊C8order 96 = 25·3

Direct product of C3 and C22⋊C8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×C22⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C12 — C2×C24 — C3×C22⋊C8
 Lower central C1 — C2 — C3×C22⋊C8
 Upper central C1 — C2×C12 — C3×C22⋊C8

Generators and relations for C3×C22⋊C8
G = < a,b,c,d | a3=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C3×C22⋊C8
On 48 points
Generators in S48
(1 14 39)(2 15 40)(3 16 33)(4 9 34)(5 10 35)(6 11 36)(7 12 37)(8 13 38)(17 29 41)(18 30 42)(19 31 43)(20 32 44)(21 25 45)(22 26 46)(23 27 47)(24 28 48)
(2 24)(4 18)(6 20)(8 22)(9 30)(11 32)(13 26)(15 28)(34 42)(36 44)(38 46)(40 48)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 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)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 D4 M4(2) C3×D4 C3×M4(2) kernel C3×C22⋊C8 C2×C24 C22×C12 C22⋊C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C6 C2×C4 C23 C22 C12 C6 C4 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 4 4

Matrix representation of C3×C22⋊C8 in GL3(𝔽73) generated by

 8 0 0 0 1 0 0 0 1
,
 72 0 0 0 1 0 0 46 72
,
 1 0 0 0 72 0 0 0 72
,
 63 0 0 0 46 71 0 0 27
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[72,0,0,0,1,46,0,0,72],[1,0,0,0,72,0,0,0,72],[63,0,0,0,46,0,0,71,27] >;

C3×C22⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes C_8
% in TeX

G:=Group("C3xC2^2:C8");
// GroupNames label

G:=SmallGroup(96,48);
// by ID

G=gap.SmallGroup(96,48);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,88]);
// Polycyclic

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

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