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## G = C2×S3×C3⋊C8order 288 = 25·32

### Direct product of C2, S3 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×S3×C3⋊C8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×C3⋊C8 — C2×S3×C3⋊C8
 Lower central C32 — C2×S3×C3⋊C8
 Upper central C1 — C2×C4

Generators and relations for C2×S3×C3⋊C8
G = < a,b,c,d,e | a2=b3=c2=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 370 in 163 conjugacy classes, 84 normal (34 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C22×C4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22×C8, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C2×C3⋊C8, C2×C3⋊C8, C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8, C324C8, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C8, C22×C3⋊C8, S3×C3⋊C8, C6×C3⋊C8, C2×C324C8, S3×C2×C12, C2×S3×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C2×C8, C22×C4, C3⋊C8, C4×S3, C2×Dic3, C22×S3, C22×C8, S32, S3×C8, C2×C3⋊C8, S3×C2×C4, C22×Dic3, S3×Dic3, C2×S32, S3×C2×C8, C22×C3⋊C8, S3×C3⋊C8, C2×S3×Dic3, C2×S3×C3⋊C8

Smallest permutation representation of C2×S3×C3⋊C8
On 96 points
Generators in S96
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 89)(65 77)(66 78)(67 79)(68 80)(69 73)(70 74)(71 75)(72 76)
(1 75 86)(2 76 87)(3 77 88)(4 78 81)(5 79 82)(6 80 83)(7 73 84)(8 74 85)(9 47 50)(10 48 51)(11 41 52)(12 42 53)(13 43 54)(14 44 55)(15 45 56)(16 46 49)(17 72 31)(18 65 32)(19 66 25)(20 67 26)(21 68 27)(22 69 28)(23 70 29)(24 71 30)(33 62 92)(34 63 93)(35 64 94)(36 57 95)(37 58 96)(38 59 89)(39 60 90)(40 61 91)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 86)(42 87)(43 88)(44 81)(45 82)(46 83)(47 84)(48 85)(49 80)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)
(1 86 75)(2 76 87)(3 88 77)(4 78 81)(5 82 79)(6 80 83)(7 84 73)(8 74 85)(9 47 50)(10 51 48)(11 41 52)(12 53 42)(13 43 54)(14 55 44)(15 45 56)(16 49 46)(17 72 31)(18 32 65)(19 66 25)(20 26 67)(21 68 27)(22 28 69)(23 70 29)(24 30 71)(33 62 92)(34 93 63)(35 64 94)(36 95 57)(37 58 96)(38 89 59)(39 60 90)(40 91 61)
(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)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,89)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76), (1,75,86)(2,76,87)(3,77,88)(4,78,81)(5,79,82)(6,80,83)(7,73,84)(8,74,85)(9,47,50)(10,48,51)(11,41,52)(12,42,53)(13,43,54)(14,44,55)(15,45,56)(16,46,49)(17,72,31)(18,65,32)(19,66,25)(20,67,26)(21,68,27)(22,69,28)(23,70,29)(24,71,30)(33,62,92)(34,63,93)(35,64,94)(36,57,95)(37,58,96)(38,59,89)(39,60,90)(40,61,91), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,86)(42,87)(43,88)(44,81)(45,82)(46,83)(47,84)(48,85)(49,80)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79), (1,86,75)(2,76,87)(3,88,77)(4,78,81)(5,82,79)(6,80,83)(7,84,73)(8,74,85)(9,47,50)(10,51,48)(11,41,52)(12,53,42)(13,43,54)(14,55,44)(15,45,56)(16,49,46)(17,72,31)(18,32,65)(19,66,25)(20,26,67)(21,68,27)(22,28,69)(23,70,29)(24,30,71)(33,62,92)(34,93,63)(35,64,94)(36,95,57)(37,58,96)(38,89,59)(39,60,90)(40,91,61), (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( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,57)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,89)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76), (1,75,86)(2,76,87)(3,77,88)(4,78,81)(5,79,82)(6,80,83)(7,73,84)(8,74,85)(9,47,50)(10,48,51)(11,41,52)(12,42,53)(13,43,54)(14,44,55)(15,45,56)(16,46,49)(17,72,31)(18,65,32)(19,66,25)(20,67,26)(21,68,27)(22,69,28)(23,70,29)(24,71,30)(33,62,92)(34,63,93)(35,64,94)(36,57,95)(37,58,96)(38,59,89)(39,60,90)(40,61,91), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,86)(42,87)(43,88)(44,81)(45,82)(46,83)(47,84)(48,85)(49,80)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79), (1,86,75)(2,76,87)(3,88,77)(4,78,81)(5,82,79)(6,80,83)(7,84,73)(8,74,85)(9,47,50)(10,51,48)(11,41,52)(12,53,42)(13,43,54)(14,55,44)(15,45,56)(16,49,46)(17,72,31)(18,32,65)(19,66,25)(20,26,67)(21,68,27)(22,28,69)(23,70,29)(24,30,71)(33,62,92)(34,93,63)(35,64,94)(36,95,57)(37,58,96)(38,89,59)(39,60,90)(40,91,61), (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([[(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,57),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,54),(34,55),(35,56),(36,49),(37,50),(38,51),(39,52),(40,53),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,89),(65,77),(66,78),(67,79),(68,80),(69,73),(70,74),(71,75),(72,76)], [(1,75,86),(2,76,87),(3,77,88),(4,78,81),(5,79,82),(6,80,83),(7,73,84),(8,74,85),(9,47,50),(10,48,51),(11,41,52),(12,42,53),(13,43,54),(14,44,55),(15,45,56),(16,46,49),(17,72,31),(18,65,32),(19,66,25),(20,67,26),(21,68,27),(22,69,28),(23,70,29),(24,71,30),(33,62,92),(34,63,93),(35,64,94),(36,57,95),(37,58,96),(38,59,89),(39,60,90),(40,61,91)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(25,93),(26,94),(27,95),(28,96),(29,89),(30,90),(31,91),(32,92),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,86),(42,87),(43,88),(44,81),(45,82),(46,83),(47,84),(48,85),(49,80),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79)], [(1,86,75),(2,76,87),(3,88,77),(4,78,81),(5,82,79),(6,80,83),(7,84,73),(8,74,85),(9,47,50),(10,51,48),(11,41,52),(12,53,42),(13,43,54),(14,55,44),(15,45,56),(16,49,46),(17,72,31),(18,32,65),(19,66,25),(20,26,67),(21,68,27),(22,28,69),(23,70,29),(24,30,71),(33,62,92),(34,93,63),(35,64,94),(36,95,57),(37,58,96),(38,89,59),(39,60,90),(40,91,61)], [(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)]])

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 8A ··· 8H 8I ··· 8P 12A ··· 12H 12I 12J 12K 12L 12M 12N 12O 12P 24A ··· 24H order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 8 ··· 8 8 ··· 8 12 ··· 12 12 12 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 3 3 3 3 2 2 4 1 1 1 1 3 3 3 3 2 ··· 2 4 4 4 6 6 6 6 3 ··· 3 9 ··· 9 2 ··· 2 4 4 4 4 6 6 6 6 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + - + - + - + - image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 S3 D6 Dic3 D6 Dic3 D6 Dic3 C4×S3 C3⋊C8 C4×S3 S3×C8 S32 S3×Dic3 C2×S32 S3×Dic3 S3×C3⋊C8 kernel C2×S3×C3⋊C8 S3×C3⋊C8 C6×C3⋊C8 C2×C32⋊4C8 S3×C2×C12 S3×C12 C6×Dic3 S3×C2×C6 S3×C6 C2×C3⋊C8 S3×C2×C4 C3⋊C8 C4×S3 C4×S3 C2×Dic3 C2×C12 C22×S3 C12 D6 C2×C6 C6 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 1 1 2 2 2 1 2 1 2 8 2 8 1 1 1 1 4

Matrix representation of C2×S3×C3⋊C8 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 0 1 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 72 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 72
,
 27 0 0 0 0 27 0 0 0 0 0 51 0 0 51 0
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,72],[27,0,0,0,0,27,0,0,0,0,0,51,0,0,51,0] >;

C2×S3×C3⋊C8 in GAP, Magma, Sage, TeX

C_2\times S_3\times C_3\rtimes C_8
% in TeX

G:=Group("C2xS3xC3:C8");
// GroupNames label

G:=SmallGroup(288,460);
// by ID

G=gap.SmallGroup(288,460);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,80,1356,9414]);
// Polycyclic

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

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