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## G = C2×D6⋊C8order 192 = 26·3

### Direct product of C2 and D6⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×D6⋊C8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C22×C4 — C2×D6⋊C8
 Lower central C3 — C6 — C2×D6⋊C8
 Upper central C1 — C22×C4 — C22×C8

Generators and relations for C2×D6⋊C8
G = < a,b,c,d | a2=b6=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 504 in 202 conjugacy classes, 87 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22⋊C8, C22×C8, C22×C8, C23×C4, C2×C3⋊C8, C2×C3⋊C8, C2×C24, C2×C24, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C2×C22⋊C8, D6⋊C8, C22×C3⋊C8, C22×C24, S3×C22×C4, C2×D6⋊C8
Quotients:

Smallest permutation representation of C2×D6⋊C8
On 96 points
Generators in S96
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(25 96)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(65 78)(66 79)(67 80)(68 73)(69 74)(70 75)(71 76)(72 77)
(1 87 77 51 31 15)(2 88 78 52 32 16)(3 81 79 53 25 9)(4 82 80 54 26 10)(5 83 73 55 27 11)(6 84 74 56 28 12)(7 85 75 49 29 13)(8 86 76 50 30 14)(17 89 63 35 42 67)(18 90 64 36 43 68)(19 91 57 37 44 69)(20 92 58 38 45 70)(21 93 59 39 46 71)(22 94 60 40 47 72)(23 95 61 33 48 65)(24 96 62 34 41 66)
(1 68)(2 57)(3 70)(4 59)(5 72)(6 61)(7 66)(8 63)(9 20)(10 39)(11 22)(12 33)(13 24)(14 35)(15 18)(16 37)(17 76)(19 78)(21 80)(23 74)(25 92)(26 46)(27 94)(28 48)(29 96)(30 42)(31 90)(32 44)(34 75)(36 77)(38 79)(40 73)(41 85)(43 87)(45 81)(47 83)(49 62)(50 67)(51 64)(52 69)(53 58)(54 71)(55 60)(56 65)(82 93)(84 95)(86 89)(88 91)
(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)| (1,40)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(25,96)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(65,78)(66,79)(67,80)(68,73)(69,74)(70,75)(71,76)(72,77), (1,87,77,51,31,15)(2,88,78,52,32,16)(3,81,79,53,25,9)(4,82,80,54,26,10)(5,83,73,55,27,11)(6,84,74,56,28,12)(7,85,75,49,29,13)(8,86,76,50,30,14)(17,89,63,35,42,67)(18,90,64,36,43,68)(19,91,57,37,44,69)(20,92,58,38,45,70)(21,93,59,39,46,71)(22,94,60,40,47,72)(23,95,61,33,48,65)(24,96,62,34,41,66), (1,68)(2,57)(3,70)(4,59)(5,72)(6,61)(7,66)(8,63)(9,20)(10,39)(11,22)(12,33)(13,24)(14,35)(15,18)(16,37)(17,76)(19,78)(21,80)(23,74)(25,92)(26,46)(27,94)(28,48)(29,96)(30,42)(31,90)(32,44)(34,75)(36,77)(38,79)(40,73)(41,85)(43,87)(45,81)(47,83)(49,62)(50,67)(51,64)(52,69)(53,58)(54,71)(55,60)(56,65)(82,93)(84,95)(86,89)(88,91), (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( (1,40)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,54)(18,55)(19,56)(20,49)(21,50)(22,51)(23,52)(24,53)(25,96)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(65,78)(66,79)(67,80)(68,73)(69,74)(70,75)(71,76)(72,77), (1,87,77,51,31,15)(2,88,78,52,32,16)(3,81,79,53,25,9)(4,82,80,54,26,10)(5,83,73,55,27,11)(6,84,74,56,28,12)(7,85,75,49,29,13)(8,86,76,50,30,14)(17,89,63,35,42,67)(18,90,64,36,43,68)(19,91,57,37,44,69)(20,92,58,38,45,70)(21,93,59,39,46,71)(22,94,60,40,47,72)(23,95,61,33,48,65)(24,96,62,34,41,66), (1,68)(2,57)(3,70)(4,59)(5,72)(6,61)(7,66)(8,63)(9,20)(10,39)(11,22)(12,33)(13,24)(14,35)(15,18)(16,37)(17,76)(19,78)(21,80)(23,74)(25,92)(26,46)(27,94)(28,48)(29,96)(30,42)(31,90)(32,44)(34,75)(36,77)(38,79)(40,73)(41,85)(43,87)(45,81)(47,83)(49,62)(50,67)(51,64)(52,69)(53,58)(54,71)(55,60)(56,65)(82,93)(84,95)(86,89)(88,91), (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([[(1,40),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,54),(18,55),(19,56),(20,49),(21,50),(22,51),(23,52),(24,53),(25,96),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(65,78),(66,79),(67,80),(68,73),(69,74),(70,75),(71,76),(72,77)], [(1,87,77,51,31,15),(2,88,78,52,32,16),(3,81,79,53,25,9),(4,82,80,54,26,10),(5,83,73,55,27,11),(6,84,74,56,28,12),(7,85,75,49,29,13),(8,86,76,50,30,14),(17,89,63,35,42,67),(18,90,64,36,43,68),(19,91,57,37,44,69),(20,92,58,38,45,70),(21,93,59,39,46,71),(22,94,60,40,47,72),(23,95,61,33,48,65),(24,96,62,34,41,66)], [(1,68),(2,57),(3,70),(4,59),(5,72),(6,61),(7,66),(8,63),(9,20),(10,39),(11,22),(12,33),(13,24),(14,35),(15,18),(16,37),(17,76),(19,78),(21,80),(23,74),(25,92),(26,46),(27,94),(28,48),(29,96),(30,42),(31,90),(32,44),(34,75),(36,77),(38,79),(40,73),(41,85),(43,87),(45,81),(47,83),(49,62),(50,67),(51,64),(52,69),(53,58),(54,71),(55,60),(56,65),(82,93),(84,95),(86,89),(88,91)], [(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 ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I 4J 4K 4L 6A ··· 6G 8A ··· 8H 8I ··· 8P 12A ··· 12H 24A ··· 24P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 ··· 1 6 6 6 6 2 1 ··· 1 6 6 6 6 2 ··· 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 S3 D4 D6 D6 M4(2) C4×S3 D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel C2×D6⋊C8 D6⋊C8 C22×C3⋊C8 C22×C24 S3×C22×C4 S3×C2×C4 C22×Dic3 S3×C23 C22×S3 C22×C8 C2×C12 C2×C8 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 4 2 2 16 1 4 2 1 4 2 4 4 2 8 8

Matrix representation of C2×D6⋊C8 in GL4(𝔽73) generated by

 1 0 0 0 0 72 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 1
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 72 0
,
 51 0 0 0 0 46 0 0 0 0 3 6 0 0 67 70
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[51,0,0,0,0,46,0,0,0,0,3,67,0,0,6,70] >;

C2×D6⋊C8 in GAP, Magma, Sage, TeX

C_2\times D_6\rtimes C_8
% in TeX

G:=Group("C2xD6:C8");
// GroupNames label

G:=SmallGroup(192,667);
// by ID

G=gap.SmallGroup(192,667);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,58,136,6278]);
// Polycyclic

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

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