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## G = A4×C23order 276 = 22·3·23

### Direct product of C23 and A4

Aliases: A4×C23, C22⋊C69, (C2×C46)⋊C3, SmallGroup(276,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C23
 Chief series C1 — C22 — C2×C46 — A4×C23
 Lower central C22 — A4×C23
 Upper central C1 — C23

Generators and relations for A4×C23
G = < a,b,c,d | a23=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C23
On 92 points
Generators in S92
(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)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 24)(19 25)(20 26)(21 27)(22 28)(23 29)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 70)(66 71)(67 72)(68 73)(69 74)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 65)(25 66)(26 67)(27 68)(28 69)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)
(24 70 65)(25 71 66)(26 72 67)(27 73 68)(28 74 69)(29 75 47)(30 76 48)(31 77 49)(32 78 50)(33 79 51)(34 80 52)(35 81 53)(36 82 54)(37 83 55)(38 84 56)(39 85 57)(40 86 58)(41 87 59)(42 88 60)(43 89 61)(44 90 62)(45 91 63)(46 92 64)

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

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

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

92 conjugacy classes

 class 1 2 3A 3B 23A ··· 23V 46A ··· 46V 69A ··· 69AR order 1 2 3 3 23 ··· 23 46 ··· 46 69 ··· 69 size 1 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4

92 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C3 C23 C69 A4 A4×C23 kernel A4×C23 C2×C46 A4 C22 C23 C1 # reps 1 2 22 44 1 22

Matrix representation of A4×C23 in GL3(𝔽139) generated by

 91 0 0 0 91 0 0 0 91
,
 0 1 138 1 0 138 0 0 138
,
 138 0 0 138 0 1 138 1 0
,
 1 138 0 0 138 1 0 138 0
G:=sub<GL(3,GF(139))| [91,0,0,0,91,0,0,0,91],[0,1,0,1,0,0,138,138,138],[138,138,138,0,0,1,0,1,0],[1,0,0,138,138,138,0,1,0] >;

A4×C23 in GAP, Magma, Sage, TeX

A_4\times C_{23}
% in TeX

G:=Group("A4xC23");
// GroupNames label

G:=SmallGroup(276,6);
// by ID

G=gap.SmallGroup(276,6);
# by ID

G:=PCGroup([4,-3,-23,-2,2,1658,3315]);
// Polycyclic

G:=Group<a,b,c,d|a^23=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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