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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 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×7], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×4], C24 [×4], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24 [×2], C2×C24 [×3], C3×M4(2) [×3], C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C3×C24 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C8○D12, C3×C8○D4, S3×C24 [×2], C3×C8⋊S3 [×2], C3×C4.Dic3, C6×C24, C3×C4○D12, C3×C8○D12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C8○D4, S3×C6 [×3], S3×C2×C4, C22×C12, S3×C12 [×2], 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 13 10 22 7 19 4 16)(2 14 11 23 8 20 5 17)(3 15 12 24 9 21 6 18)(25 40 28 43 31 46 34 37)(26 41 29 44 32 47 35 38)(27 42 30 45 33 48 36 39)
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)

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

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

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

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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