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## G = A4×C2×C14order 336 = 24·3·7

### Direct product of C2×C14 and A4

Aliases: A4×C2×C14, C242C21, C232C42, C22⋊(C2×C42), (C23×C14)⋊1C3, (C22×C14)⋊6C6, (C2×C14)⋊8(C2×C6), SmallGroup(336,221)

Series: Derived Chief Lower central Upper central

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

Generators and relations for A4×C2×C14
G = < a,b,c,d,e | a2=b14=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 184 in 78 conjugacy classes, 30 normal (12 characteristic)
C1, C2, C2, C3, C22, C22, C6, C7, C23, C23, A4, C2×C6, C14, C14, C24, C21, C2×A4, C2×C14, C2×C14, C42, C22×A4, C22×C14, C22×C14, C7×A4, C2×C42, C23×C14, A4×C14, A4×C2×C14
Quotients: C1, C2, C3, C22, C6, C7, A4, C2×C6, C14, C21, C2×A4, C2×C14, C42, C22×A4, C7×A4, C2×C42, A4×C14, A4×C2×C14

Smallest permutation representation of A4×C2×C14
On 84 points
Generators in S84
(1 53)(2 54)(3 55)(4 56)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 29)(26 30)(27 31)(28 32)(57 80)(58 81)(59 82)(60 83)(61 84)(62 71)(63 72)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)
(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)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 43)(13 44)(14 45)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 29)(26 30)(27 31)(28 32)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(1 53)(2 54)(3 55)(4 56)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 71)(70 72)
(1 69 18)(2 70 19)(3 57 20)(4 58 21)(5 59 22)(6 60 23)(7 61 24)(8 62 25)(9 63 26)(10 64 27)(11 65 28)(12 66 15)(13 67 16)(14 68 17)(29 46 71)(30 47 72)(31 48 73)(32 49 74)(33 50 75)(34 51 76)(35 52 77)(36 53 78)(37 54 79)(38 55 80)(39 56 81)(40 43 82)(41 44 83)(42 45 84)

G:=sub<Sym(84)| (1,53)(2,54)(3,55)(4,56)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,29)(26,30)(27,31)(28,32)(57,80)(58,81)(59,82)(60,83)(61,84)(62,71)(63,72)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79), (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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,43)(13,44)(14,45)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,29)(26,30)(27,31)(28,32)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,53)(2,54)(3,55)(4,56)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,71)(70,72), (1,69,18)(2,70,19)(3,57,20)(4,58,21)(5,59,22)(6,60,23)(7,61,24)(8,62,25)(9,63,26)(10,64,27)(11,65,28)(12,66,15)(13,67,16)(14,68,17)(29,46,71)(30,47,72)(31,48,73)(32,49,74)(33,50,75)(34,51,76)(35,52,77)(36,53,78)(37,54,79)(38,55,80)(39,56,81)(40,43,82)(41,44,83)(42,45,84)>;

G:=Group( (1,53)(2,54)(3,55)(4,56)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,29)(26,30)(27,31)(28,32)(57,80)(58,81)(59,82)(60,83)(61,84)(62,71)(63,72)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79), (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), (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,43)(13,44)(14,45)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,29)(26,30)(27,31)(28,32)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,53)(2,54)(3,55)(4,56)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,71)(70,72), (1,69,18)(2,70,19)(3,57,20)(4,58,21)(5,59,22)(6,60,23)(7,61,24)(8,62,25)(9,63,26)(10,64,27)(11,65,28)(12,66,15)(13,67,16)(14,68,17)(29,46,71)(30,47,72)(31,48,73)(32,49,74)(33,50,75)(34,51,76)(35,52,77)(36,53,78)(37,54,79)(38,55,80)(39,56,81)(40,43,82)(41,44,83)(42,45,84) );

G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,29),(26,30),(27,31),(28,32),(57,80),(58,81),(59,82),(60,83),(61,84),(62,71),(63,72),(64,73),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79)], [(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)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,43),(13,44),(14,45),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,29),(26,30),(27,31),(28,32),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(1,53),(2,54),(3,55),(4,56),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,71),(70,72)], [(1,69,18),(2,70,19),(3,57,20),(4,58,21),(5,59,22),(6,60,23),(7,61,24),(8,62,25),(9,63,26),(10,64,27),(11,65,28),(12,66,15),(13,67,16),(14,68,17),(29,46,71),(30,47,72),(31,48,73),(32,49,74),(33,50,75),(34,51,76),(35,52,77),(36,53,78),(37,54,79),(38,55,80),(39,56,81),(40,43,82),(41,44,83),(42,45,84)]])

112 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A ··· 6F 7A ··· 7F 14A ··· 14R 14S ··· 14AP 21A ··· 21L 42A ··· 42AJ order 1 2 2 2 2 2 2 2 3 3 6 ··· 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 1 1 3 3 3 3 4 4 4 ··· 4 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

112 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 C7 C14 C21 C42 A4 C2×A4 C7×A4 A4×C14 kernel A4×C2×C14 A4×C14 C23×C14 C22×C14 C22×A4 C2×A4 C24 C23 C2×C14 C14 C22 C2 # reps 1 3 2 6 6 18 12 36 1 3 6 18

Matrix representation of A4×C2×C14 in GL4(𝔽43) generated by

 42 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 16 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42
,
 1 0 0 0 0 42 0 0 0 0 1 0 0 0 0 42
,
 1 0 0 0 0 1 0 0 0 0 42 0 0 0 0 42
,
 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(43))| [42,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,42,0,0,0,0,1,0,0,0,0,42],[1,0,0,0,0,1,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;

A4×C2×C14 in GAP, Magma, Sage, TeX

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

G:=Group("A4xC2xC14");
// GroupNames label

G:=SmallGroup(336,221);
// by ID

G=gap.SmallGroup(336,221);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,2,1276,2285]);
// Polycyclic

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

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