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## G = A4×D19order 456 = 23·3·19

### Direct product of A4 and D19

Aliases: A4×D19, C193(C2×A4), (C2×C38)⋊1C6, C22⋊(C3×D19), (A4×C19)⋊2C2, (C22×D19)⋊1C3, SmallGroup(456,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C38 — A4×D19
 Chief series C1 — C19 — C2×C38 — A4×C19 — A4×D19
 Lower central C2×C38 — A4×D19
 Upper central C1

Generators and relations for A4×D19
G = < a,b,c,d,e | a2=b2=c3=d19=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

3C2
19C2
57C2
4C3
57C22
57C22
76C6
3C38
3D19
4C57
19C23
3D38
3D38
19C2×A4

Smallest permutation representation of A4×D19
On 76 points
Generators in S76
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(57 65)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 73)(21 74)(22 75)(23 76)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)
(20 46 73)(21 47 74)(22 48 75)(23 49 76)(24 50 58)(25 51 59)(26 52 60)(27 53 61)(28 54 62)(29 55 63)(30 56 64)(31 57 65)(32 39 66)(33 40 67)(34 41 68)(35 42 69)(36 43 70)(37 44 71)(38 45 72)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 57)(54 56)(58 68)(59 67)(60 66)(61 65)(62 64)(69 76)(70 75)(71 74)(72 73)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 19A ··· 19I 38A ··· 38I 57A ··· 57R order 1 2 2 2 3 3 6 6 19 ··· 19 38 ··· 38 57 ··· 57 size 1 3 19 57 4 4 76 76 2 ··· 2 6 ··· 6 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 type + + + + + + image C1 C2 C3 C6 D19 C3×D19 A4 C2×A4 A4×D19 kernel A4×D19 A4×C19 C22×D19 C2×C38 A4 C22 D19 C19 C1 # reps 1 1 2 2 9 18 1 1 9

Matrix representation of A4×D19 in GL5(𝔽229)

 1 0 0 0 0 0 1 0 0 0 0 0 228 0 0 0 0 227 228 104 0 0 44 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 228 227 104 0 0 0 228 0 0 0 0 44 1
,
 94 0 0 0 0 0 94 0 0 0 0 0 228 228 104 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 228 102 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(229))| [1,0,0,0,0,0,1,0,0,0,0,0,228,227,44,0,0,0,228,0,0,0,0,104,1],[1,0,0,0,0,0,1,0,0,0,0,0,228,0,0,0,0,227,228,44,0,0,104,0,1],[94,0,0,0,0,0,94,0,0,0,0,0,228,1,0,0,0,228,0,0,0,0,104,0,1],[0,228,0,0,0,1,102,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×D19 in GAP, Magma, Sage, TeX

A_4\times D_{19}
% in TeX

G:=Group("A4xD19");
// GroupNames label

G:=SmallGroup(456,45);
// by ID

G=gap.SmallGroup(456,45);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-19,142,68,10804]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^19=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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