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## G = C2×C6×C3⋊C8order 288 = 25·32

### Direct product of C2×C6 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C6×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×C3⋊C8 — C6×C3⋊C8 — C2×C6×C3⋊C8
 Lower central C3 — C2×C6×C3⋊C8
 Upper central C1 — C22×C12

Generators and relations for C2×C6×C3⋊C8
G = < a,b,c,d | a2=b6=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 250 in 179 conjugacy classes, 130 normal (22 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4 [×3], C22 [×7], C6 [×2], C6 [×12], C6 [×7], C8 [×4], C2×C4 [×6], C23, C32, C12 [×2], C12 [×6], C12 [×4], C2×C6 [×14], C2×C6 [×7], C2×C8 [×6], C22×C4, C3×C6, C3×C6 [×6], C3⋊C8 [×4], C24 [×4], C2×C12 [×12], C2×C12 [×6], C22×C6 [×2], C22×C6, C22×C8, C3×C12, C3×C12 [×3], C62 [×7], C2×C3⋊C8 [×6], C2×C24 [×6], C22×C12 [×2], C22×C12, C3×C3⋊C8 [×4], C6×C12 [×6], C2×C62, C22×C3⋊C8, C22×C24, C6×C3⋊C8 [×6], C2×C6×C12, C2×C6×C3⋊C8
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C8 [×4], C2×C4 [×6], C23, Dic3 [×4], C12 [×4], D6 [×3], C2×C6 [×7], C2×C8 [×6], C22×C4, C3×S3, C3⋊C8 [×4], C24 [×4], C2×Dic3 [×6], C2×C12 [×6], C22×S3, C22×C6, C22×C8, C3×Dic3 [×4], S3×C6 [×3], C2×C3⋊C8 [×6], C2×C24 [×6], C22×Dic3, C22×C12, C3×C3⋊C8 [×4], C6×Dic3 [×6], S3×C2×C6, C22×C3⋊C8, C22×C24, C6×C3⋊C8 [×6], Dic3×C2×C6, C2×C6×C3⋊C8

Smallest permutation representation of C2×C6×C3⋊C8
On 96 points
Generators in S96
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 81)(48 82)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 73)(64 74)
(1 85 13 80 55 27)(2 86 14 73 56 28)(3 87 15 74 49 29)(4 88 16 75 50 30)(5 81 9 76 51 31)(6 82 10 77 52 32)(7 83 11 78 53 25)(8 84 12 79 54 26)(17 47 33 58 91 67)(18 48 34 59 92 68)(19 41 35 60 93 69)(20 42 36 61 94 70)(21 43 37 62 95 71)(22 44 38 63 96 72)(23 45 39 64 89 65)(24 46 40 57 90 66)
(1 13 55)(2 56 14)(3 15 49)(4 50 16)(5 9 51)(6 52 10)(7 11 53)(8 54 12)(17 33 91)(18 92 34)(19 35 93)(20 94 36)(21 37 95)(22 96 38)(23 39 89)(24 90 40)(25 83 78)(26 79 84)(27 85 80)(28 73 86)(29 87 74)(30 75 88)(31 81 76)(32 77 82)(41 60 69)(42 70 61)(43 62 71)(44 72 63)(45 64 65)(46 66 57)(47 58 67)(48 68 59)
(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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,73)(64,74), (1,85,13,80,55,27)(2,86,14,73,56,28)(3,87,15,74,49,29)(4,88,16,75,50,30)(5,81,9,76,51,31)(6,82,10,77,52,32)(7,83,11,78,53,25)(8,84,12,79,54,26)(17,47,33,58,91,67)(18,48,34,59,92,68)(19,41,35,60,93,69)(20,42,36,61,94,70)(21,43,37,62,95,71)(22,44,38,63,96,72)(23,45,39,64,89,65)(24,46,40,57,90,66), (1,13,55)(2,56,14)(3,15,49)(4,50,16)(5,9,51)(6,52,10)(7,11,53)(8,54,12)(17,33,91)(18,92,34)(19,35,93)(20,94,36)(21,37,95)(22,96,38)(23,39,89)(24,90,40)(25,83,78)(26,79,84)(27,85,80)(28,73,86)(29,87,74)(30,75,88)(31,81,76)(32,77,82)(41,60,69)(42,70,61)(43,62,71)(44,72,63)(45,64,65)(46,66,57)(47,58,67)(48,68,59), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,96)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,73)(64,74), (1,85,13,80,55,27)(2,86,14,73,56,28)(3,87,15,74,49,29)(4,88,16,75,50,30)(5,81,9,76,51,31)(6,82,10,77,52,32)(7,83,11,78,53,25)(8,84,12,79,54,26)(17,47,33,58,91,67)(18,48,34,59,92,68)(19,41,35,60,93,69)(20,42,36,61,94,70)(21,43,37,62,95,71)(22,44,38,63,96,72)(23,45,39,64,89,65)(24,46,40,57,90,66), (1,13,55)(2,56,14)(3,15,49)(4,50,16)(5,9,51)(6,52,10)(7,11,53)(8,54,12)(17,33,91)(18,92,34)(19,35,93)(20,94,36)(21,37,95)(22,96,38)(23,39,89)(24,90,40)(25,83,78)(26,79,84)(27,85,80)(28,73,86)(29,87,74)(30,75,88)(31,81,76)(32,77,82)(41,60,69)(42,70,61)(43,62,71)(44,72,63)(45,64,65)(46,66,57)(47,58,67)(48,68,59), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );

G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(25,69),(26,70),(27,71),(28,72),(29,65),(30,66),(31,67),(32,68),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,96),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,73),(64,74)], [(1,85,13,80,55,27),(2,86,14,73,56,28),(3,87,15,74,49,29),(4,88,16,75,50,30),(5,81,9,76,51,31),(6,82,10,77,52,32),(7,83,11,78,53,25),(8,84,12,79,54,26),(17,47,33,58,91,67),(18,48,34,59,92,68),(19,41,35,60,93,69),(20,42,36,61,94,70),(21,43,37,62,95,71),(22,44,38,63,96,72),(23,45,39,64,89,65),(24,46,40,57,90,66)], [(1,13,55),(2,56,14),(3,15,49),(4,50,16),(5,9,51),(6,52,10),(7,11,53),(8,54,12),(17,33,91),(18,92,34),(19,35,93),(20,94,36),(21,37,95),(22,96,38),(23,39,89),(24,90,40),(25,83,78),(26,79,84),(27,85,80),(28,73,86),(29,87,74),(30,75,88),(31,81,76),(32,77,82),(41,60,69),(42,70,61),(43,62,71),(44,72,63),(45,64,65),(46,66,57),(47,58,67),(48,68,59)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)])

144 conjugacy classes

 class 1 2A ··· 2G 3A 3B 3C 3D 3E 4A ··· 4H 6A ··· 6N 6O ··· 6AI 8A ··· 8P 12A ··· 12P 12Q ··· 12AN 24A ··· 24AF order 1 2 ··· 2 3 3 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 ··· 1 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 C3×C3⋊C8 kernel C2×C6×C3⋊C8 C6×C3⋊C8 C2×C6×C12 C22×C3⋊C8 C6×C12 C2×C62 C2×C3⋊C8 C22×C12 C62 C2×C12 C22×C6 C2×C6 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 6 1 2 6 2 12 2 16 12 4 32 1 3 3 1 2 8 6 6 2 16

Matrix representation of C2×C6×C3⋊C8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 9 0 0 0 0 65 0 0 0 0 65
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 0 8
,
 51 0 0 0 0 46 0 0 0 0 0 72 0 0 72 0
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,65,0,0,0,0,65],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[51,0,0,0,0,46,0,0,0,0,0,72,0,0,72,0] >;

C2×C6×C3⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_3\rtimes C_8
% in TeX

G:=Group("C2xC6xC3:C8");
// GroupNames label

G:=SmallGroup(288,691);
// by ID

G=gap.SmallGroup(288,691);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,102,9414]);
// Polycyclic

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

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