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## G = A4×C28order 336 = 24·3·7

### Direct product of C28 and A4

Aliases: A4×C28, C22⋊C84, C23.C42, (C22×C4)⋊C21, (C2×C14)⋊6C12, C2.1(A4×C14), (C22×C28)⋊1C3, (C2×A4).2C14, (A4×C14).4C2, C14.10(C2×A4), (C22×C14).6C6, SmallGroup(336,168)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C28
 Chief series C1 — C22 — C23 — C22×C14 — A4×C14 — A4×C28
 Lower central C22 — A4×C28
 Upper central C1 — C28

Generators and relations for A4×C28
G = < a,b,c,d | a28=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C28
On 84 points
Generators in S84
(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 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
(1 43 61)(2 44 62)(3 45 63)(4 46 64)(5 47 65)(6 48 66)(7 49 67)(8 50 68)(9 51 69)(10 52 70)(11 53 71)(12 54 72)(13 55 73)(14 56 74)(15 29 75)(16 30 76)(17 31 77)(18 32 78)(19 33 79)(20 34 80)(21 35 81)(22 36 82)(23 37 83)(24 38 84)(25 39 57)(26 40 58)(27 41 59)(28 42 60)

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

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

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

112 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 7A ··· 7F 12A 12B 12C 12D 14A ··· 14F 14G ··· 14R 21A ··· 21L 28A ··· 28L 28M ··· 28X 42A ··· 42L 84A ··· 84X order 1 2 2 2 3 3 4 4 4 4 6 6 7 ··· 7 12 12 12 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 3 3 4 4 1 1 3 3 4 4 1 ··· 1 4 4 4 4 1 ··· 1 3 ··· 3 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

112 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 C3 C4 C6 C7 C12 C14 C21 C28 C42 C84 A4 C2×A4 C4×A4 C7×A4 A4×C14 A4×C28 kernel A4×C28 A4×C14 C22×C28 C7×A4 C22×C14 C4×A4 C2×C14 C2×A4 C22×C4 A4 C23 C22 C28 C14 C7 C4 C2 C1 # reps 1 1 2 2 2 6 4 6 12 12 12 24 1 1 2 6 6 12

Matrix representation of A4×C28 in GL3(𝔽337) generated by

 49 0 0 0 49 0 0 0 49
,
 336 0 129 0 336 0 0 0 1
,
 336 209 0 0 1 0 0 0 336
,
 208 0 0 0 0 1 335 209 129
G:=sub<GL(3,GF(337))| [49,0,0,0,49,0,0,0,49],[336,0,0,0,336,0,129,0,1],[336,0,0,209,1,0,0,0,336],[208,0,335,0,0,209,0,1,129] >;

A4×C28 in GAP, Magma, Sage, TeX

A_4\times C_{28}
% in TeX

G:=Group("A4xC28");
// GroupNames label

G:=SmallGroup(336,168);
// by ID

G=gap.SmallGroup(336,168);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,-2,2,252,2530,4547]);
// Polycyclic

G:=Group<a,b,c,d|a^28=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations

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