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## G = C2×C8×C7⋊C3order 336 = 24·3·7

### Direct product of C2×C8 and C7⋊C3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C8×C7⋊C3
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C2×C4×C7⋊C3 — C2×C8×C7⋊C3
 Lower central C7 — C2×C8×C7⋊C3
 Upper central C1 — C2×C8

Generators and relations for C2×C8×C7⋊C3
G = < a,b,c,d | a2=b8=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Smallest permutation representation of C2×C8×C7⋊C3
On 112 points
Generators in S112
(1 112)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 71)(34 72)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 79)(42 80)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 87)(50 88)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)
(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)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)
(1 72 89 60 85 97 80)(2 65 90 61 86 98 73)(3 66 91 62 87 99 74)(4 67 92 63 88 100 75)(5 68 93 64 81 101 76)(6 69 94 57 82 102 77)(7 70 95 58 83 103 78)(8 71 96 59 84 104 79)(9 30 55 17 42 112 34)(10 31 56 18 43 105 35)(11 32 49 19 44 106 36)(12 25 50 20 45 107 37)(13 26 51 21 46 108 38)(14 27 52 22 47 109 39)(15 28 53 23 48 110 40)(16 29 54 24 41 111 33)
(9 55 34)(10 56 35)(11 49 36)(12 50 37)(13 51 38)(14 52 39)(15 53 40)(16 54 33)(17 30 42)(18 31 43)(19 32 44)(20 25 45)(21 26 46)(22 27 47)(23 28 48)(24 29 41)(57 77 102)(58 78 103)(59 79 104)(60 80 97)(61 73 98)(62 74 99)(63 75 100)(64 76 101)(65 90 86)(66 91 87)(67 92 88)(68 93 81)(69 94 82)(70 95 83)(71 96 84)(72 89 85)

G:=sub<Sym(112)| (1,112)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86), (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)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112), (1,72,89,60,85,97,80)(2,65,90,61,86,98,73)(3,66,91,62,87,99,74)(4,67,92,63,88,100,75)(5,68,93,64,81,101,76)(6,69,94,57,82,102,77)(7,70,95,58,83,103,78)(8,71,96,59,84,104,79)(9,30,55,17,42,112,34)(10,31,56,18,43,105,35)(11,32,49,19,44,106,36)(12,25,50,20,45,107,37)(13,26,51,21,46,108,38)(14,27,52,22,47,109,39)(15,28,53,23,48,110,40)(16,29,54,24,41,111,33), (9,55,34)(10,56,35)(11,49,36)(12,50,37)(13,51,38)(14,52,39)(15,53,40)(16,54,33)(17,30,42)(18,31,43)(19,32,44)(20,25,45)(21,26,46)(22,27,47)(23,28,48)(24,29,41)(57,77,102)(58,78,103)(59,79,104)(60,80,97)(61,73,98)(62,74,99)(63,75,100)(64,76,101)(65,90,86)(66,91,87)(67,92,88)(68,93,81)(69,94,82)(70,95,83)(71,96,84)(72,89,85)>;

G:=Group( (1,112)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,71)(34,72)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,79)(42,80)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86), (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)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112), (1,72,89,60,85,97,80)(2,65,90,61,86,98,73)(3,66,91,62,87,99,74)(4,67,92,63,88,100,75)(5,68,93,64,81,101,76)(6,69,94,57,82,102,77)(7,70,95,58,83,103,78)(8,71,96,59,84,104,79)(9,30,55,17,42,112,34)(10,31,56,18,43,105,35)(11,32,49,19,44,106,36)(12,25,50,20,45,107,37)(13,26,51,21,46,108,38)(14,27,52,22,47,109,39)(15,28,53,23,48,110,40)(16,29,54,24,41,111,33), (9,55,34)(10,56,35)(11,49,36)(12,50,37)(13,51,38)(14,52,39)(15,53,40)(16,54,33)(17,30,42)(18,31,43)(19,32,44)(20,25,45)(21,26,46)(22,27,47)(23,28,48)(24,29,41)(57,77,102)(58,78,103)(59,79,104)(60,80,97)(61,73,98)(62,74,99)(63,75,100)(64,76,101)(65,90,86)(66,91,87)(67,92,88)(68,93,81)(69,94,82)(70,95,83)(71,96,84)(72,89,85) );

G=PermutationGroup([[(1,112),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,71),(34,72),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,79),(42,80),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,87),(50,88),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86)], [(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),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112)], [(1,72,89,60,85,97,80),(2,65,90,61,86,98,73),(3,66,91,62,87,99,74),(4,67,92,63,88,100,75),(5,68,93,64,81,101,76),(6,69,94,57,82,102,77),(7,70,95,58,83,103,78),(8,71,96,59,84,104,79),(9,30,55,17,42,112,34),(10,31,56,18,43,105,35),(11,32,49,19,44,106,36),(12,25,50,20,45,107,37),(13,26,51,21,46,108,38),(14,27,52,22,47,109,39),(15,28,53,23,48,110,40),(16,29,54,24,41,111,33)], [(9,55,34),(10,56,35),(11,49,36),(12,50,37),(13,51,38),(14,52,39),(15,53,40),(16,54,33),(17,30,42),(18,31,43),(19,32,44),(20,25,45),(21,26,46),(22,27,47),(23,28,48),(24,29,41),(57,77,102),(58,78,103),(59,79,104),(60,80,97),(61,73,98),(62,74,99),(63,75,100),(64,76,101),(65,90,86),(66,91,87),(67,92,88),(68,93,81),(69,94,82),(70,95,83),(71,96,84),(72,89,85)]])

80 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A ··· 6F 7A 7B 8A ··· 8H 12A ··· 12H 14A ··· 14F 24A ··· 24P 28A ··· 28H 56A ··· 56P order 1 2 2 2 3 3 4 4 4 4 6 ··· 6 7 7 8 ··· 8 12 ··· 12 14 ··· 14 24 ··· 24 28 ··· 28 56 ··· 56 size 1 1 1 1 7 7 1 1 1 1 7 ··· 7 3 3 1 ··· 1 7 ··· 7 3 ··· 3 7 ··· 7 3 ··· 3 3 ··· 3

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 C7⋊C3 C2×C7⋊C3 C2×C7⋊C3 C4×C7⋊C3 C4×C7⋊C3 C8×C7⋊C3 kernel C2×C8×C7⋊C3 C8×C7⋊C3 C2×C4×C7⋊C3 C2×C56 C4×C7⋊C3 C22×C7⋊C3 C56 C2×C28 C2×C7⋊C3 C28 C2×C14 C14 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 4 2 4 4 16

Matrix representation of C2×C8×C7⋊C3 in GL4(𝔽337) generated by

 336 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 336 0 0 0 0 111 0 0 0 0 111 0 0 0 0 111
,
 1 0 0 0 0 336 212 1 0 0 212 1 0 336 213 1
,
 208 0 0 0 0 213 1 125 0 1 0 0 0 1 1 124
G:=sub<GL(4,GF(337))| [336,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[336,0,0,0,0,111,0,0,0,0,111,0,0,0,0,111],[1,0,0,0,0,336,0,336,0,212,212,213,0,1,1,1],[208,0,0,0,0,213,1,1,0,1,0,1,0,125,0,124] >;

C2×C8×C7⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_8\times C_7\rtimes C_3
% in TeX

G:=Group("C2xC8xC7:C3");
// GroupNames label

G:=SmallGroup(336,51);
// by ID

G=gap.SmallGroup(336,51);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,79,69,881]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^7=d^3=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^4>;
// generators/relations

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