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## G = C3×C8○D12order 288 = 25·32

### Direct product of C3 and C8○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C8○D12
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×C4○D12 — C3×C8○D12
 Lower central C3 — C6 — C3×C8○D12
 Upper central C1 — C24 — C2×C24

Generators and relations for C3×C8○D12
G = < a,b,c,d | a3=b8=d2=1, c6=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c5 >

Subgroups: 250 in 135 conjugacy classes, 74 normal (46 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8○D4, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C2×C24, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C8○D12, C3×C8○D4, S3×C24, C3×C8⋊S3, C3×C4.Dic3, C6×C24, C3×C4○D12, C3×C8○D12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C8○D4, S3×C6, S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, C8○D12, C3×C8○D4, S3×C2×C12, C3×C8○D12

Smallest permutation representation of C3×C8○D12
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 22 10 19 7 16 4 13)(2 23 11 20 8 17 5 14)(3 24 12 21 9 18 6 15)(25 43 28 46 31 37 34 40)(26 44 29 47 32 38 35 41)(27 45 30 48 33 39 36 42)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6M 6N 6O 6P 6Q 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 12E ··· 12R 12S 12T 12U 12V 24A ··· 24H 24I ··· 24AJ 24AK ··· 24AR order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 6 6 6 8 8 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 2 6 6 1 1 2 2 2 1 1 2 6 6 1 1 2 ··· 2 6 6 6 6 1 1 1 1 2 2 6 6 6 6 1 1 1 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C6 C12 C12 C12 S3 D6 D6 C3×S3 C4×S3 C4×S3 C8○D4 S3×C6 S3×C6 S3×C12 S3×C12 C8○D12 C3×C8○D4 C3×C8○D12 kernel C3×C8○D12 S3×C24 C3×C8⋊S3 C3×C4.Dic3 C6×C24 C3×C4○D12 C8○D12 C3×Dic6 C3×D12 C3×C3⋊D4 S3×C8 C8⋊S3 C4.Dic3 C2×C24 C4○D12 Dic6 D12 C3⋊D4 C2×C24 C24 C2×C12 C2×C8 C12 C2×C6 C32 C8 C2×C4 C4 C22 C3 C3 C1 # reps 1 2 2 1 1 1 2 2 2 4 4 4 2 2 2 4 4 8 1 2 1 2 2 2 4 4 2 4 4 8 8 16

Matrix representation of C3×C8○D12 in GL2(𝔽73) generated by

 8 0 0 8
,
 22 0 0 22
,
 3 0 0 49
,
 0 49 3 0
G:=sub<GL(2,GF(73))| [8,0,0,8],[22,0,0,22],[3,0,0,49],[0,3,49,0] >;

C3×C8○D12 in GAP, Magma, Sage, TeX

C_3\times C_8\circ D_{12}
% in TeX

G:=Group("C3xC8oD12");
// GroupNames label

G:=SmallGroup(288,672);
// by ID

G=gap.SmallGroup(288,672);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,142,102,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^6=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^5>;
// generators/relations

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