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G = C2×C3⋊D21order 252 = 22·32·7

Direct product of C2 and C3⋊D21

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

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

C1C3×C21 — C2×C3⋊D21
C1C7C21C3×C21C3⋊D21 — C2×C3⋊D21
C3×C21 — C2×C3⋊D21
C1C2

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 2A2B2C3A3B3C3D6A6B6C6D7A7B7C14A14B14C21A···21X42A···42X
order12223333666677714141421···2142···42
size116363222222222222222···22···2

66 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D7D14D21D42
kernelC2×C3⋊D21C3⋊D21C3×C42C42C21C3×C6C32C6C3
# reps12144332424

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

1000
0100
00420
00042
,
32000
233900
00335
00739
,
43300
102900
0023
003510
,
43300
233900
00142
00042
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

׿
×
𝔽