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G = A4xC29order 348 = 22·3·29

Direct product of C29 and A4

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

Aliases: A4xC29, C22:C87, (C2xC58):C3, SmallGroup(348,8)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC29
C1C22C2xC58 — A4xC29
C22 — A4xC29
C1C29

Generators and relations for A4xC29
 G = < a,b,c,d | a29=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Subgroups: 20 in 10 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C3, A4, C29, C87, A4xC29
3C2
4C3
3C58
4C87

Smallest permutation representation of A4xC29
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 3A3B29A···29AB58A···58AB87A···87BD
order123329···2958···5887···87
size13441···13···34···4

116 irreducible representations

dim111133
type++
imageC1C3C29C87A4A4xC29
kernelA4xC29C2xC58A4C22C29C1
# reps122856128

Matrix representation of A4xC29 in GL3(F349) generated by

22800
02280
00228
,
01348
10348
00348
,
34800
34801
34810
,
13480
03481
03480
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] >;

A4xC29 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

Export

Subgroup lattice of A4xC29 in TeX

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