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## G = C8×3- 1+2order 216 = 23·33

### Direct product of C8 and 3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C8×3- 1+2
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C4×3- 1+2 — C8×3- 1+2
 Lower central C1 — C3 — C8×3- 1+2
 Upper central C1 — C24 — C8×3- 1+2

Generators and relations for C8×3- 1+2
G = < a,b,c | a8=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C8×3- 1+2
On 72 points
Generators in S72
(1 21 63 36 42 66 53 10)(2 22 55 28 43 67 54 11)(3 23 56 29 44 68 46 12)(4 24 57 30 45 69 47 13)(5 25 58 31 37 70 48 14)(6 26 59 32 38 71 49 15)(7 27 60 33 39 72 50 16)(8 19 61 34 40 64 51 17)(9 20 62 35 41 65 52 18)
(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(2 8 5)(3 6 9)(11 17 14)(12 15 18)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(37 43 40)(38 41 44)(46 49 52)(48 54 51)(55 61 58)(56 59 62)(64 70 67)(65 68 71)

G:=sub<Sym(72)| (1,21,63,36,42,66,53,10)(2,22,55,28,43,67,54,11)(3,23,56,29,44,68,46,12)(4,24,57,30,45,69,47,13)(5,25,58,31,37,70,48,14)(6,26,59,32,38,71,49,15)(7,27,60,33,39,72,50,16)(8,19,61,34,40,64,51,17)(9,20,62,35,41,65,52,18), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51)(55,61,58)(56,59,62)(64,70,67)(65,68,71)>;

G:=Group( (1,21,63,36,42,66,53,10)(2,22,55,28,43,67,54,11)(3,23,56,29,44,68,46,12)(4,24,57,30,45,69,47,13)(5,25,58,31,37,70,48,14)(6,26,59,32,38,71,49,15)(7,27,60,33,39,72,50,16)(8,19,61,34,40,64,51,17)(9,20,62,35,41,65,52,18), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (2,8,5)(3,6,9)(11,17,14)(12,15,18)(19,25,22)(20,23,26)(28,34,31)(29,32,35)(37,43,40)(38,41,44)(46,49,52)(48,54,51)(55,61,58)(56,59,62)(64,70,67)(65,68,71) );

G=PermutationGroup([[(1,21,63,36,42,66,53,10),(2,22,55,28,43,67,54,11),(3,23,56,29,44,68,46,12),(4,24,57,30,45,69,47,13),(5,25,58,31,37,70,48,14),(6,26,59,32,38,71,49,15),(7,27,60,33,39,72,50,16),(8,19,61,34,40,64,51,17),(9,20,62,35,41,65,52,18)], [(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),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(2,8,5),(3,6,9),(11,17,14),(12,15,18),(19,25,22),(20,23,26),(28,34,31),(29,32,35),(37,43,40),(38,41,44),(46,49,52),(48,54,51),(55,61,58),(56,59,62),(64,70,67),(65,68,71)]])

C8×3- 1+2 is a maximal subgroup of   C9⋊C48  C72.C6  C72⋊C6  C722C6  D72⋊C3

88 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 8A 8B 8C 8D 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 24A ··· 24H 24I ··· 24P 36A ··· 36L 72A ··· 72X order 1 2 3 3 3 3 4 4 6 6 6 6 8 8 8 8 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 24 ··· 24 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 3 3 1 1 1 1 3 3 1 1 1 1 3 ··· 3 1 1 1 1 3 3 3 3 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C4 C6 C6 C8 C12 C12 C24 C24 3- 1+2 C2×3- 1+2 C4×3- 1+2 C8×3- 1+2 kernel C8×3- 1+2 C4×3- 1+2 C72 C3×C24 C2×3- 1+2 C36 C3×C12 3- 1+2 C18 C3×C6 C9 C32 C8 C4 C2 C1 # reps 1 1 6 2 2 6 2 4 12 4 24 8 2 2 4 8

Matrix representation of C8×3- 1+2 in GL4(𝔽73) generated by

 10 0 0 0 0 27 0 0 0 0 27 0 0 0 0 27
,
 8 0 0 0 0 0 1 0 0 0 0 8 0 1 0 0
,
 64 0 0 0 0 1 0 0 0 0 8 0 0 0 0 64
G:=sub<GL(4,GF(73))| [10,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[8,0,0,0,0,0,0,1,0,1,0,0,0,0,8,0],[64,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64] >;

C8×3- 1+2 in GAP, Magma, Sage, TeX

C_8\times 3_-^{1+2}
% in TeX

G:=Group("C8xES-(3,1)");
// GroupNames label

G:=SmallGroup(216,20);
// by ID

G=gap.SmallGroup(216,20);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-2,108,223,386,165]);
// Polycyclic

G:=Group<a,b,c|a^8=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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