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G = C2×C14.A4order 336 = 24·3·7

Direct product of C2 and C14.A4

Aliases: C2×C14.A4, C14⋊SL2(𝔽3), (C7×Q8)⋊8C6, (Q8×C14)⋊2C3, C14.5(C2×A4), (C2×C14).3A4, C22.2(C7⋊A4), C72(C2×SL2(𝔽3)), Q8⋊(C2×C7⋊C3), (C2×Q8)⋊(C7⋊C3), C2.2(C2×C7⋊A4), SmallGroup(336,172)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×Q8 — C2×C14.A4
 Chief series C1 — C2 — C14 — C7×Q8 — C14.A4 — C2×C14.A4
 Lower central C7×Q8 — C2×C14.A4
 Upper central C1 — C22

Generators and relations for C2×C14.A4
G = < a,b,c,d,e | a2=b14=e3=1, c2=d2=b7, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b11, dcd-1=b7c, ece-1=b7cd, ede-1=c >

Smallest permutation representation of C2×C14.A4
On 112 points
Generators in S112
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(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 29 8 36)(2 30 9 37)(3 31 10 38)(4 32 11 39)(5 33 12 40)(6 34 13 41)(7 35 14 42)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 56)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(71 99 78 106)(72 100 79 107)(73 101 80 108)(74 102 81 109)(75 103 82 110)(76 104 83 111)(77 105 84 112)
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 71 64 78)(58 72 65 79)(59 73 66 80)(60 74 67 81)(61 75 68 82)(62 76 69 83)(63 77 70 84)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)
(2 10 12)(3 5 9)(4 14 6)(7 13 11)(15 43 29)(16 52 40)(17 47 37)(18 56 34)(19 51 31)(20 46 42)(21 55 39)(22 50 36)(23 45 33)(24 54 30)(25 49 41)(26 44 38)(27 53 35)(28 48 32)(58 66 68)(59 61 65)(60 70 62)(63 69 67)(71 99 85)(72 108 96)(73 103 93)(74 112 90)(75 107 87)(76 102 98)(77 111 95)(78 106 92)(79 101 89)(80 110 86)(81 105 97)(82 100 94)(83 109 91)(84 104 88)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 6A ··· 6F 7A 7B 14A ··· 14F 28A ··· 28L order 1 2 2 2 3 3 4 4 6 ··· 6 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 28 28 6 6 28 ··· 28 3 3 3 ··· 3 6 ··· 6

34 irreducible representations

 dim 1 1 1 1 2 2 3 3 3 3 3 3 6 type + + - + + image C1 C2 C3 C6 SL2(𝔽3) SL2(𝔽3) A4 C7⋊C3 C2×A4 C2×C7⋊C3 C7⋊A4 C2×C7⋊A4 C14.A4 kernel C2×C14.A4 C14.A4 Q8×C14 C7×Q8 C14 C14 C2×C14 C2×Q8 C14 Q8 C22 C2 C2 # reps 1 1 2 2 2 4 1 2 1 2 6 6 4

Matrix representation of C2×C14.A4 in GL5(𝔽337)

 336 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336 0 0 0 0 0 336
,
 336 0 0 0 0 0 336 0 0 0 0 0 1 213 212 0 0 212 336 336 0 0 336 336 124
,
 80 256 0 0 0 208 257 0 0 0 0 0 196 64 151 0 0 151 7 61 0 0 61 1 133
,
 336 2 0 0 0 336 1 0 0 0 0 0 193 186 250 0 0 250 197 277 0 0 277 276 283
,
 1 0 0 0 0 209 128 0 0 0 0 0 1 0 0 0 0 212 336 336 0 0 0 1 0

G:=sub<GL(5,GF(337))| [336,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336,0,0,0,0,0,336],[336,0,0,0,0,0,336,0,0,0,0,0,1,212,336,0,0,213,336,336,0,0,212,336,124],[80,208,0,0,0,256,257,0,0,0,0,0,196,151,61,0,0,64,7,1,0,0,151,61,133],[336,336,0,0,0,2,1,0,0,0,0,0,193,250,277,0,0,186,197,276,0,0,250,277,283],[1,209,0,0,0,0,128,0,0,0,0,0,1,212,0,0,0,0,336,1,0,0,0,336,0] >;

C2×C14.A4 in GAP, Magma, Sage, TeX

C_2\times C_{14}.A_4
% in TeX

G:=Group("C2xC14.A4");
// GroupNames label

G:=SmallGroup(336,172);
// by ID

G=gap.SmallGroup(336,172);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-7,-2,116,518,225,735,357,730]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^14=e^3=1,c^2=d^2=b^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^11,d*c*d^-1=b^7*c,e*c*e^-1=b^7*c*d,e*d*e^-1=c>;
// generators/relations

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