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G = C2×C3⋊D21order 252 = 22·32·7

Direct product of C2 and C3⋊D21

Aliases: C2×C3⋊D21, C6⋊D21, C421S3, C32D42, C216D6, C326D14, C14⋊(C3⋊S3), (C3×C6)⋊2D7, (C3×C42)⋊1C2, (C3×C21)⋊6C22, C72(C2×C3⋊S3), SmallGroup(252,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — C2×C3⋊D21
 Chief series C1 — C7 — C21 — C3×C21 — C3⋊D21 — C2×C3⋊D21
 Lower central C3×C21 — C2×C3⋊D21
 Upper central C1 — C2

Generators and relations for C2×C3⋊D21
G = < a,b,c,d | a2=b3=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 552 in 60 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C3, C22, S3, C6, C7, C32, D6, D7, C14, C3⋊S3, C3×C6, C21, D14, C2×C3⋊S3, D21, C42, C3×C21, D42, C3⋊D21, C3×C42, C2×C3⋊D21
Quotients: C1, C2, C22, S3, D6, D7, C3⋊S3, D14, C2×C3⋊S3, D21, D42, C3⋊D21, C2×C3⋊D21

Smallest permutation representation of C2×C3⋊D21
On 126 points
Generators in S126
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 123)(44 124)(45 125)(46 126)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 113)(55 114)(56 115)(57 116)(58 117)(59 118)(60 119)(61 120)(62 121)(63 122)
(1 62 33)(2 63 34)(3 43 35)(4 44 36)(5 45 37)(6 46 38)(7 47 39)(8 48 40)(9 49 41)(10 50 42)(11 51 22)(12 52 23)(13 53 24)(14 54 25)(15 55 26)(16 56 27)(17 57 28)(18 58 29)(19 59 30)(20 60 31)(21 61 32)(64 110 91)(65 111 92)(66 112 93)(67 113 94)(68 114 95)(69 115 96)(70 116 97)(71 117 98)(72 118 99)(73 119 100)(74 120 101)(75 121 102)(76 122 103)(77 123 104)(78 124 105)(79 125 85)(80 126 86)(81 106 87)(82 107 88)(83 108 89)(84 109 90)
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
(1 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 84)(13 83)(14 82)(15 81)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 110)(23 109)(24 108)(25 107)(26 106)(27 126)(28 125)(29 124)(30 123)(31 122)(32 121)(33 120)(34 119)(35 118)(36 117)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 99)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)

G:=sub<Sym(126)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,123)(44,124)(45,125)(46,126)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122), (1,62,33)(2,63,34)(3,43,35)(4,44,36)(5,45,37)(6,46,38)(7,47,39)(8,48,40)(9,49,41)(10,50,42)(11,51,22)(12,52,23)(13,53,24)(14,54,25)(15,55,26)(16,56,27)(17,57,28)(18,58,29)(19,59,30)(20,60,31)(21,61,32)(64,110,91)(65,111,92)(66,112,93)(67,113,94)(68,114,95)(69,115,96)(70,116,97)(71,117,98)(72,118,99)(73,119,100)(74,120,101)(75,121,102)(76,122,103)(77,123,104)(78,124,105)(79,125,85)(80,126,86)(81,106,87)(82,107,88)(83,108,89)(84,109,90), (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,110)(23,109)(24,108)(25,107)(26,106)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)>;

G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,123)(44,124)(45,125)(46,126)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,113)(55,114)(56,115)(57,116)(58,117)(59,118)(60,119)(61,120)(62,121)(63,122), (1,62,33)(2,63,34)(3,43,35)(4,44,36)(5,45,37)(6,46,38)(7,47,39)(8,48,40)(9,49,41)(10,50,42)(11,51,22)(12,52,23)(13,53,24)(14,54,25)(15,55,26)(16,56,27)(17,57,28)(18,58,29)(19,59,30)(20,60,31)(21,61,32)(64,110,91)(65,111,92)(66,112,93)(67,113,94)(68,114,95)(69,115,96)(70,116,97)(71,117,98)(72,118,99)(73,119,100)(74,120,101)(75,121,102)(76,122,103)(77,123,104)(78,124,105)(79,125,85)(80,126,86)(81,106,87)(82,107,88)(83,108,89)(84,109,90), (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126), (1,74)(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,84)(13,83)(14,82)(15,81)(16,80)(17,79)(18,78)(19,77)(20,76)(21,75)(22,110)(23,109)(24,108)(25,107)(26,106)(27,126)(28,125)(29,124)(30,123)(31,122)(32,121)(33,120)(34,119)(35,118)(36,117)(37,116)(38,115)(39,114)(40,113)(41,112)(42,111)(43,99)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,85)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100) );

G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,123),(44,124),(45,125),(46,126),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,113),(55,114),(56,115),(57,116),(58,117),(59,118),(60,119),(61,120),(62,121),(63,122)], [(1,62,33),(2,63,34),(3,43,35),(4,44,36),(5,45,37),(6,46,38),(7,47,39),(8,48,40),(9,49,41),(10,50,42),(11,51,22),(12,52,23),(13,53,24),(14,54,25),(15,55,26),(16,56,27),(17,57,28),(18,58,29),(19,59,30),(20,60,31),(21,61,32),(64,110,91),(65,111,92),(66,112,93),(67,113,94),(68,114,95),(69,115,96),(70,116,97),(71,117,98),(72,118,99),(73,119,100),(74,120,101),(75,121,102),(76,122,103),(77,123,104),(78,124,105),(79,125,85),(80,126,86),(81,106,87),(82,107,88),(83,108,89),(84,109,90)], [(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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)], [(1,74),(2,73),(3,72),(4,71),(5,70),(6,69),(7,68),(8,67),(9,66),(10,65),(11,64),(12,84),(13,83),(14,82),(15,81),(16,80),(17,79),(18,78),(19,77),(20,76),(21,75),(22,110),(23,109),(24,108),(25,107),(26,106),(27,126),(28,125),(29,124),(30,123),(31,122),(32,121),(33,120),(34,119),(35,118),(36,117),(37,116),(38,115),(39,114),(40,113),(41,112),(42,111),(43,99),(44,98),(45,97),(46,96),(47,95),(48,94),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(57,85),(58,105),(59,104),(60,103),(61,102),(62,101),(63,100)]])

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 7A 7B 7C 14A 14B 14C 21A ··· 21X 42A ··· 42X order 1 2 2 2 3 3 3 3 6 6 6 6 7 7 7 14 14 14 21 ··· 21 42 ··· 42 size 1 1 63 63 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

66 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D7 D14 D21 D42 kernel C2×C3⋊D21 C3⋊D21 C3×C42 C42 C21 C3×C6 C32 C6 C3 # reps 1 2 1 4 4 3 3 24 24

Matrix representation of C2×C3⋊D21 in GL4(𝔽43) generated by

 1 0 0 0 0 1 0 0 0 0 42 0 0 0 0 42
,
 3 20 0 0 23 39 0 0 0 0 3 35 0 0 7 39
,
 4 33 0 0 10 29 0 0 0 0 2 3 0 0 35 10
,
 4 33 0 0 23 39 0 0 0 0 1 42 0 0 0 42
G:=sub<GL(4,GF(43))| [1,0,0,0,0,1,0,0,0,0,42,0,0,0,0,42],[3,23,0,0,20,39,0,0,0,0,3,7,0,0,35,39],[4,10,0,0,33,29,0,0,0,0,2,35,0,0,3,10],[4,23,0,0,33,39,0,0,0,0,1,0,0,0,42,42] >;

C2×C3⋊D21 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes D_{21}
% in TeX

G:=Group("C2xC3:D21");
// GroupNames label

G:=SmallGroup(252,45);
// by ID

G=gap.SmallGroup(252,45);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,122,483,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^21=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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