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## G = A4×C29order 348 = 22·3·29

### Direct product of C29 and A4

Aliases: A4×C29, C22⋊C87, (C2×C58)⋊C3, SmallGroup(348,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C29
 Chief series C1 — C22 — C2×C58 — A4×C29
 Lower central C22 — A4×C29
 Upper central C1 — C29

Generators and relations for A4×C29
G = < a,b,c,d | a29=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C29
On 116 points
Generators in S116
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 88)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 59)(56 60)(57 61)(58 62)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 87)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 116)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)
(30 116 63)(31 88 64)(32 89 65)(33 90 66)(34 91 67)(35 92 68)(36 93 69)(37 94 70)(38 95 71)(39 96 72)(40 97 73)(41 98 74)(42 99 75)(43 100 76)(44 101 77)(45 102 78)(46 103 79)(47 104 80)(48 105 81)(49 106 82)(50 107 83)(51 108 84)(52 109 85)(53 110 86)(54 111 87)(55 112 59)(56 113 60)(57 114 61)(58 115 62)

G:=sub<Sym(116)| (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)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,88)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,59)(56,60)(57,61)(58,62), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,116)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115), (30,116,63)(31,88,64)(32,89,65)(33,90,66)(34,91,67)(35,92,68)(36,93,69)(37,94,70)(38,95,71)(39,96,72)(40,97,73)(41,98,74)(42,99,75)(43,100,76)(44,101,77)(45,102,78)(46,103,79)(47,104,80)(48,105,81)(49,106,82)(50,107,83)(51,108,84)(52,109,85)(53,110,86)(54,111,87)(55,112,59)(56,113,60)(57,114,61)(58,115,62)>;

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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,88)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,59)(56,60)(57,61)(58,62), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,82)(19,83)(20,84)(21,85)(22,86)(23,87)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,116)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115), (30,116,63)(31,88,64)(32,89,65)(33,90,66)(34,91,67)(35,92,68)(36,93,69)(37,94,70)(38,95,71)(39,96,72)(40,97,73)(41,98,74)(42,99,75)(43,100,76)(44,101,77)(45,102,78)(46,103,79)(47,104,80)(48,105,81)(49,106,82)(50,107,83)(51,108,84)(52,109,85)(53,110,86)(54,111,87)(55,112,59)(56,113,60)(57,114,61)(58,115,62) );

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),(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87),(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,88),(30,63),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,59),(56,60),(57,61),(58,62)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,81),(18,82),(19,83),(20,84),(21,85),(22,86),(23,87),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,116),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115)], [(30,116,63),(31,88,64),(32,89,65),(33,90,66),(34,91,67),(35,92,68),(36,93,69),(37,94,70),(38,95,71),(39,96,72),(40,97,73),(41,98,74),(42,99,75),(43,100,76),(44,101,77),(45,102,78),(46,103,79),(47,104,80),(48,105,81),(49,106,82),(50,107,83),(51,108,84),(52,109,85),(53,110,86),(54,111,87),(55,112,59),(56,113,60),(57,114,61),(58,115,62)]])

116 conjugacy classes

 class 1 2 3A 3B 29A ··· 29AB 58A ··· 58AB 87A ··· 87BD order 1 2 3 3 29 ··· 29 58 ··· 58 87 ··· 87 size 1 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4

116 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C3 C29 C87 A4 A4×C29 kernel A4×C29 C2×C58 A4 C22 C29 C1 # reps 1 2 28 56 1 28

Matrix representation of A4×C29 in GL3(𝔽349) generated by

 228 0 0 0 228 0 0 0 228
,
 0 1 348 1 0 348 0 0 348
,
 348 0 0 348 0 1 348 1 0
,
 1 348 0 0 348 1 0 348 0
G:=sub<GL(3,GF(349))| [228,0,0,0,228,0,0,0,228],[0,1,0,1,0,0,348,348,348],[348,348,348,0,0,1,0,1,0],[1,0,0,348,348,348,0,1,0] >;

A4×C29 in GAP, Magma, Sage, TeX

A_4\times C_{29}
% in TeX

G:=Group("A4xC29");
// GroupNames label

G:=SmallGroup(348,8);
// by ID

G=gap.SmallGroup(348,8);
# by ID

G:=PCGroup([4,-3,-29,-2,2,2090,4179]);
// Polycyclic

G:=Group<a,b,c,d|a^29=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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