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## G = A4×C34order 408 = 23·3·17

### Direct product of C34 and A4

Aliases: A4×C34, C23⋊C51, C22⋊C102, (C22×C34)⋊C3, (C2×C34)⋊2C6, SmallGroup(408,42)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C34
 Chief series C1 — C22 — C2×C34 — A4×C17 — A4×C34
 Lower central C22 — A4×C34
 Upper central C1 — C34

Generators and relations for A4×C34
G = < a,b,c,d | a34=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C34
On 102 points
Generators in S102
(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)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)
(1 93 39)(2 94 40)(3 95 41)(4 96 42)(5 97 43)(6 98 44)(7 99 45)(8 100 46)(9 101 47)(10 102 48)(11 69 49)(12 70 50)(13 71 51)(14 72 52)(15 73 53)(16 74 54)(17 75 55)(18 76 56)(19 77 57)(20 78 58)(21 79 59)(22 80 60)(23 81 61)(24 82 62)(25 83 63)(26 84 64)(27 85 65)(28 86 66)(29 87 67)(30 88 68)(31 89 35)(32 90 36)(33 91 37)(34 92 38)

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 17A ··· 17P 34A ··· 34P 34Q ··· 34AV 51A ··· 51AF 102A ··· 102AF order 1 2 2 2 3 3 6 6 17 ··· 17 34 ··· 34 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 3 3 4 4 4 4 1 ··· 1 1 ··· 1 3 ··· 3 4 ··· 4 4 ··· 4

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C6 C17 C34 C51 C102 A4 C2×A4 A4×C17 A4×C34 kernel A4×C34 A4×C17 C22×C34 C2×C34 C2×A4 A4 C23 C22 C34 C17 C2 C1 # reps 1 1 2 2 16 16 32 32 1 1 16 16

Matrix representation of A4×C34 in GL3(𝔽103) generated by

 94 0 0 0 94 0 0 0 94
,
 102 0 0 0 102 0 20 0 1
,
 102 0 0 3 1 0 0 0 102
,
 100 101 0 66 3 1 81 20 0
G:=sub<GL(3,GF(103))| [94,0,0,0,94,0,0,0,94],[102,0,20,0,102,0,0,0,1],[102,3,0,0,1,0,0,0,102],[100,66,81,101,3,20,0,1,0] >;

A4×C34 in GAP, Magma, Sage, TeX

A_4\times C_{34}
% in TeX

G:=Group("A4xC34");
// GroupNames label

G:=SmallGroup(408,42);
// by ID

G=gap.SmallGroup(408,42);
# by ID

G:=PCGroup([5,-2,-3,-17,-2,2,2048,3834]);
// Polycyclic

G:=Group<a,b,c,d|a^34=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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