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## G = C22×D23order 184 = 23·23

### Direct product of C22 and D23

Aliases: C22×D23, C23⋊C23, C46⋊C22, (C2×C46)⋊3C2, SmallGroup(184,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C22×D23
 Chief series C1 — C23 — D23 — D46 — C22×D23
 Lower central C23 — C22×D23
 Upper central C1 — C22

Generators and relations for C22×D23
G = < a,b,c,d | a2=b2=c23=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C22×D23
On 92 points
Generators in S92
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)
(1 44)(2 45)(3 46)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 70)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)
(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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 63)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)

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

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

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

C22×D23 is a maximal subgroup of   D46⋊C4
C22×D23 is a maximal quotient of   D925C2  D42D23  D92⋊C2

52 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 23A ··· 23K 46A ··· 46AG order 1 2 2 2 2 2 2 2 23 ··· 23 46 ··· 46 size 1 1 1 1 23 23 23 23 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D23 D46 kernel C22×D23 D46 C2×C46 C22 C2 # reps 1 6 1 11 33

Matrix representation of C22×D23 in GL3(𝔽47) generated by

 1 0 0 0 46 0 0 0 46
,
 46 0 0 0 46 0 0 0 46
,
 1 0 0 0 0 1 0 46 6
,
 46 0 0 0 0 46 0 46 0
G:=sub<GL(3,GF(47))| [1,0,0,0,46,0,0,0,46],[46,0,0,0,46,0,0,0,46],[1,0,0,0,0,46,0,1,6],[46,0,0,0,0,46,0,46,0] >;

C22×D23 in GAP, Magma, Sage, TeX

C_2^2\times D_{23}
% in TeX

G:=Group("C2^2xD23");
// GroupNames label

G:=SmallGroup(184,11);
// by ID

G=gap.SmallGroup(184,11);
# by ID

G:=PCGroup([4,-2,-2,-2,-23,2819]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^23=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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