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G = A4xC28order 336 = 24·3·7

Direct product of C28 and A4

direct product, metabelian, soluble, monomial, A-group

Aliases: A4xC28, C22:C84, C23.C42, (C22xC4):C21, (C2xC14):6C12, C2.1(A4xC14), (C22xC28):1C3, (C2xA4).2C14, (A4xC14).4C2, C14.10(C2xA4), (C22xC14).6C6, SmallGroup(336,168)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC28
C1C22C23C22xC14A4xC14 — A4xC28
C22 — A4xC28
C1C28

Generators and relations for A4xC28
 G = < a,b,c,d | a28=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 84 in 38 conjugacy classes, 18 normal (all characteristic)
Quotients: C1, C2, C3, C4, C6, C7, C12, A4, C14, C21, C2xA4, C28, C42, C4xA4, C84, C7xA4, A4xC14, A4xC28
3C2
3C2
4C3
3C4
3C22
3C22
4C6
3C14
3C14
4C21
3C2xC4
3C2xC4
4C12
3C2xC14
3C28
3C2xC14
4C42
3C2xC28
3C2xC28
4C84

Smallest permutation representation of A4xC28
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 2A2B2C3A3B4A4B4C4D6A6B7A···7F12A12B12C12D14A···14F14G···14R21A···21L28A···28L28M···28X42A···42L84A···84X
order1222334444667···71212121214···1414···1421···2128···2828···2842···4284···84
size1133441133441···144441···13···34···41···13···34···44···4

112 irreducible representations

dim111111111111333333
type++++
imageC1C2C3C4C6C7C12C14C21C28C42C84A4C2xA4C4xA4C7xA4A4xC14A4xC28
kernelA4xC28A4xC14C22xC28C7xA4C22xC14C4xA4C2xC14C2xA4C22xC4A4C23C22C28C14C7C4C2C1
# reps11222646121212241126612

Matrix representation of A4xC28 in GL3(F337) generated by

4900
0490
0049
,
3360129
03360
001
,
3362090
010
00336
,
20800
001
335209129
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] >;

A4xC28 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

Export

Subgroup lattice of A4xC28 in TeX

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