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G = C2×C7⋊C24order 336 = 24·3·7

Direct product of C2 and C7⋊C24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C7⋊C24
 Chief series C1 — C7 — C14 — C28 — C4×C7⋊C3 — C7⋊C24 — C2×C7⋊C24
 Lower central C7 — C2×C7⋊C24
 Upper central C1 — C2×C4

Generators and relations for C2×C7⋊C24
G = < a,b,c | a2=b7=c24=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C2×C7⋊C24
On 112 points
Generators in S112
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(65 112)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)
(1 96 112 29 104 37 21)(2 30 22 89 38 97 105)(3 90 106 23 98 31 39)(4 24 40 107 32 91 99)(5 108 100 17 92 25 33)(6 18 34 101 26 109 93)(7 102 94 35 110 19 27)(8 36 28 95 20 103 111)(9 79 71 45 87 53 61)(10 46 62 72 54 80 88)(11 73 65 63 81 47 55)(12 64 56 66 48 74 82)(13 67 83 57 75 41 49)(14 58 50 84 42 68 76)(15 85 77 51 69 59 43)(16 52 44 78 60 86 70)
(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)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 F7 C7⋊C12 C2×F7 C7⋊C12 C7⋊C24 kernel C2×C7⋊C24 C7⋊C24 C2×C4×C7⋊C3 C2×C7⋊C8 C4×C7⋊C3 C22×C7⋊C3 C7⋊C8 C2×C28 C2×C7⋊C3 C28 C2×C14 C14 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 1 1 1 4

Matrix representation of C2×C7⋊C24 in GL7(𝔽337)

 336 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 336 336 336 336 336 336 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 129 0 0 0 0 0 0 0 152 152 229 0 0 152 0 0 152 0 152 152 229 0 0 77 185 185 0 185 0 152 0 152 152 229 0 0 77 185 185 0 185 0 0 185 0 0 77 185 185

G:=sub<GL(7,GF(337))| [336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,336,1,0,0,0,0,0,336,0,1,0,0,0,0,336,0,0,1,0,0,0,336,0,0,0,1,0,0,336,0,0,0,0,1,0,336,0,0,0,0,0],[129,0,0,0,0,0,0,0,152,0,0,152,77,185,0,152,152,77,0,185,0,0,229,0,185,152,185,0,0,0,152,185,152,0,77,0,0,152,0,229,185,185,0,152,229,185,0,0,185] >;

C2×C7⋊C24 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes C_{24}
% in TeX

G:=Group("C2xC7:C24");
// GroupNames label

G:=SmallGroup(336,12);
// by ID

G=gap.SmallGroup(336,12);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,72,69,10373,1745]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^24=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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