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G = A4⋊Dic7order 336 = 24·3·7

The semidirect product of A4 and Dic7 acting via Dic7/C14=C2

non-abelian, soluble, monomial

Aliases: A4⋊Dic7, C14.3S4, C23.D21, C22⋊Dic21, C7⋊(A4⋊C4), (C2×A4).D7, (C7×A4)⋊1C4, C2.1(C7⋊S4), (A4×C14).1C2, (C2×C14)⋊2Dic3, (C22×C14).2S3, SmallGroup(336,120)

Series: Derived Chief Lower central Upper central

C1C22C7×A4 — A4⋊Dic7
C1C22C2×C14C7×A4A4×C14 — A4⋊Dic7
C7×A4 — A4⋊Dic7
C1C2

Generators and relations for A4⋊Dic7
 G = < a,b,c,d,e | a2=b2=c3=d14=1, e2=d7, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

3C2
3C2
4C3
3C22
3C22
42C4
42C4
4C6
3C14
3C14
4C21
21C2×C4
21C2×C4
28Dic3
3C2×C14
3C2×C14
6Dic7
6Dic7
4C42
21C22⋊C4
3C2×Dic7
3C2×Dic7
4Dic21
7A4⋊C4
3C23.D7

Smallest permutation representation of A4⋊Dic7
On 84 points
Generators in S84
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 29)(13 30)(14 31)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(43 83)(44 84)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 66)(16 67)(17 68)(18 69)(19 70)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 71)(53 72)(54 73)(55 74)(56 75)
(1 73 20)(2 74 21)(3 75 22)(4 76 23)(5 77 24)(6 78 25)(7 79 26)(8 80 27)(9 81 28)(10 82 15)(11 83 16)(12 84 17)(13 71 18)(14 72 19)(29 51 61)(30 52 62)(31 53 63)(32 54 64)(33 55 65)(34 56 66)(35 43 67)(36 44 68)(37 45 69)(38 46 70)(39 47 57)(40 48 58)(41 49 59)(42 50 60)
(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 39 8 32)(2 38 9 31)(3 37 10 30)(4 36 11 29)(5 35 12 42)(6 34 13 41)(7 33 14 40)(15 52 22 45)(16 51 23 44)(17 50 24 43)(18 49 25 56)(19 48 26 55)(20 47 27 54)(21 46 28 53)(57 80 64 73)(58 79 65 72)(59 78 66 71)(60 77 67 84)(61 76 68 83)(62 75 69 82)(63 74 70 81)

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

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

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

34 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 7A7B7C14A14B14C14D···14I21A···21F42A···42F
order122234444677714141414···1421···2142···42
size113384242424282222226···68···88···8

34 irreducible representations

dim1112222223366
type+++-+-+-++-
imageC1C2C4S3Dic3D7Dic7D21Dic21S4A4⋊C4C7⋊S4A4⋊Dic7
kernelA4⋊Dic7A4×C14C7×A4C22×C14C2×C14C2×A4A4C23C22C14C7C2C1
# reps1121133662233

Matrix representation of A4⋊Dic7 in GL5(𝔽337)

10000
01000
0033600
00010
000336336
,
10000
01000
00100
0003360
003360336
,
10000
01000
00010
00336336335
00001
,
336335000
98195000
00100
00010
00001
,
1480000
324189000
0033600
00112
0000336

G:=sub<GL(5,GF(337))| [1,0,0,0,0,0,1,0,0,0,0,0,336,0,0,0,0,0,1,336,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,336,0,0,0,336,0,0,0,0,0,336],[1,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,1,336,0,0,0,0,335,1],[336,98,0,0,0,335,195,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[148,324,0,0,0,0,189,0,0,0,0,0,336,1,0,0,0,0,1,0,0,0,0,2,336] >;

A4⋊Dic7 in GAP, Magma, Sage, TeX

A_4\rtimes {\rm Dic}_7
% in TeX

G:=Group("A4:Dic7");
// GroupNames label

G:=SmallGroup(336,120);
// by ID

G=gap.SmallGroup(336,120);
# by ID

G:=PCGroup([6,-2,-2,-3,-7,-2,2,12,146,1731,5044,1276,3029,2285]);
// Polycyclic

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

Export

Subgroup lattice of A4⋊Dic7 in TeX

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