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## G = C2×C19⋊A4order 456 = 23·3·19

### Direct product of C2 and C19⋊A4

Aliases: C2×C19⋊A4, C38⋊A4, C192(C2×A4), C23⋊(C19⋊C3), (C2×C38)⋊7C6, (C22×C38)⋊3C3, C22⋊(C2×C19⋊C3), SmallGroup(456,50)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C19⋊A4
G = < a,b,c,d,e | a2=b19=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b11, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C2×C19⋊A4
On 114 points
Generators in S114
(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 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 58)(53 59)(54 60)(55 61)(56 62)(57 63)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 96)(93 97)(94 98)(95 99)
(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 85 86 87 88 89 90 91 92 93 94 95)(96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114)
(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 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 58)(53 59)(54 60)(55 61)(56 62)(57 63)
(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)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 96)(93 97)(94 98)(95 99)
(1 92 43)(2 80 54)(3 87 46)(4 94 57)(5 82 49)(6 89 41)(7 77 52)(8 84 44)(9 91 55)(10 79 47)(11 86 39)(12 93 50)(13 81 42)(14 88 53)(15 95 45)(16 83 56)(17 90 48)(18 78 40)(19 85 51)(20 96 68)(21 103 60)(22 110 71)(23 98 63)(24 105 74)(25 112 66)(26 100 58)(27 107 69)(28 114 61)(29 102 72)(30 109 64)(31 97 75)(32 104 67)(33 111 59)(34 99 70)(35 106 62)(36 113 73)(37 101 65)(38 108 76)

G:=sub<Sym(114)| (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)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (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)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63), (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)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (1,92,43)(2,80,54)(3,87,46)(4,94,57)(5,82,49)(6,89,41)(7,77,52)(8,84,44)(9,91,55)(10,79,47)(11,86,39)(12,93,50)(13,81,42)(14,88,53)(15,95,45)(16,83,56)(17,90,48)(18,78,40)(19,85,51)(20,96,68)(21,103,60)(22,110,71)(23,98,63)(24,105,74)(25,112,66)(26,100,58)(27,107,69)(28,114,61)(29,102,72)(30,109,64)(31,97,75)(32,104,67)(33,111,59)(34,99,70)(35,106,62)(36,113,73)(37,101,65)(38,108,76)>;

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)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (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)(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95)(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114), (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)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,58)(53,59)(54,60)(55,61)(56,62)(57,63), (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)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,96)(93,97)(94,98)(95,99), (1,92,43)(2,80,54)(3,87,46)(4,94,57)(5,82,49)(6,89,41)(7,77,52)(8,84,44)(9,91,55)(10,79,47)(11,86,39)(12,93,50)(13,81,42)(14,88,53)(15,95,45)(16,83,56)(17,90,48)(18,78,40)(19,85,51)(20,96,68)(21,103,60)(22,110,71)(23,98,63)(24,105,74)(25,112,66)(26,100,58)(27,107,69)(28,114,61)(29,102,72)(30,109,64)(31,97,75)(32,104,67)(33,111,59)(34,99,70)(35,106,62)(36,113,73)(37,101,65)(38,108,76) );

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),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,58),(53,59),(54,60),(55,61),(56,62),(57,63),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,96),(93,97),(94,98),(95,99)], [(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),(77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95),(96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114)], [(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),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,58),(53,59),(54,60),(55,61),(56,62),(57,63)], [(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),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,96),(93,97),(94,98),(95,99)], [(1,92,43),(2,80,54),(3,87,46),(4,94,57),(5,82,49),(6,89,41),(7,77,52),(8,84,44),(9,91,55),(10,79,47),(11,86,39),(12,93,50),(13,81,42),(14,88,53),(15,95,45),(16,83,56),(17,90,48),(18,78,40),(19,85,51),(20,96,68),(21,103,60),(22,110,71),(23,98,63),(24,105,74),(25,112,66),(26,100,58),(27,107,69),(28,114,61),(29,102,72),(30,109,64),(31,97,75),(32,104,67),(33,111,59),(34,99,70),(35,106,62),(36,113,73),(37,101,65),(38,108,76)]])

56 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 19A ··· 19F 38A ··· 38AP order 1 2 2 2 3 3 6 6 19 ··· 19 38 ··· 38 size 1 1 3 3 76 76 76 76 3 ··· 3 3 ··· 3

56 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C6 A4 C2×A4 C19⋊C3 C2×C19⋊C3 C19⋊A4 C2×C19⋊A4 kernel C2×C19⋊A4 C19⋊A4 C22×C38 C2×C38 C38 C19 C23 C22 C2 C1 # reps 1 1 2 2 1 1 6 6 18 18

Matrix representation of C2×C19⋊A4 in GL3(𝔽229) generated by

 228 0 0 0 228 0 0 0 228
,
 44 115 6 0 17 0 0 0 214
,
 228 0 124 0 228 0 0 0 1
,
 228 212 0 0 1 0 0 0 228
,
 107 115 113 0 0 1 131 83 122
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[44,0,0,115,17,0,6,0,214],[228,0,0,0,228,0,124,0,1],[228,0,0,212,1,0,0,0,228],[107,0,131,115,0,83,113,1,122] >;

C2×C19⋊A4 in GAP, Magma, Sage, TeX

C_2\times C_{19}\rtimes A_4
% in TeX

G:=Group("C2xC19:A4");
// GroupNames label

G:=SmallGroup(456,50);
// by ID

G=gap.SmallGroup(456,50);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-19,97,188,2109]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^19=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,e*b*e^-1=b^11,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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