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## G = C32×D21order 378 = 2·33·7

### Direct product of C32 and D21

Aliases: C32×D21, C331D7, C3⋊(C32×D7), C216(C3×S3), (C3×C21)⋊7S3, C215(C3×C6), C73(S3×C32), (C3×C21)⋊16C6, (C32×C21)⋊2C2, C323(C3×D7), SmallGroup(378,55)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C32×D21
 Chief series C1 — C7 — C21 — C3×C21 — C32×C21 — C32×D21
 Lower central C21 — C32×D21
 Upper central C1 — C32

Generators and relations for C32×D21
G = < a,b,c,d | a3=b3=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 248 in 64 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C7, C32, C32, C32, D7, C3×S3, C3×C6, C21, C21, C21, C33, C3×D7, D21, S3×C32, C3×C21, C3×C21, C3×C21, C32×D7, C3×D21, C32×C21, C32×D21
Quotients: C1, C2, C3, S3, C6, C32, D7, C3×S3, C3×C6, C3×D7, D21, S3×C32, C32×D7, C3×D21, C32×D21

Smallest permutation representation of C32×D21
On 126 points
Generators in S126
(1 28 52)(2 29 53)(3 30 54)(4 31 55)(5 32 56)(6 33 57)(7 34 58)(8 35 59)(9 36 60)(10 37 61)(11 38 62)(12 39 63)(13 40 43)(14 41 44)(15 42 45)(16 22 46)(17 23 47)(18 24 48)(19 25 49)(20 26 50)(21 27 51)(64 98 118)(65 99 119)(66 100 120)(67 101 121)(68 102 122)(69 103 123)(70 104 124)(71 105 125)(72 85 126)(73 86 106)(74 87 107)(75 88 108)(76 89 109)(77 90 110)(78 91 111)(79 92 112)(80 93 113)(81 94 114)(82 95 115)(83 96 116)(84 97 117)
(1 45 35)(2 46 36)(3 47 37)(4 48 38)(5 49 39)(6 50 40)(7 51 41)(8 52 42)(9 53 22)(10 54 23)(11 55 24)(12 56 25)(13 57 26)(14 58 27)(15 59 28)(16 60 29)(17 61 30)(18 62 31)(19 63 32)(20 43 33)(21 44 34)(64 125 91)(65 126 92)(66 106 93)(67 107 94)(68 108 95)(69 109 96)(70 110 97)(71 111 98)(72 112 99)(73 113 100)(74 114 101)(75 115 102)(76 116 103)(77 117 104)(78 118 105)(79 119 85)(80 120 86)(81 121 87)(82 122 88)(83 123 89)(84 124 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 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 105)(32 104)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 126)(55 125)(56 124)(57 123)(58 122)(59 121)(60 120)(61 119)(62 118)(63 117)

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

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

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

108 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 7A 7B 7C 21A ··· 21BZ order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 7 7 7 21 ··· 21 size 1 21 1 ··· 1 2 ··· 2 21 ··· 21 2 2 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 D7 C3×S3 C3×D7 D21 C3×D21 kernel C32×D21 C32×C21 C3×D21 C3×C21 C3×C21 C33 C21 C32 C32 C3 # reps 1 1 8 8 1 3 8 24 6 48

Matrix representation of C32×D21 in GL3(𝔽43) generated by

 1 0 0 0 6 0 0 0 6
,
 6 0 0 0 6 0 0 0 6
,
 1 0 0 0 31 9 0 0 25
,
 1 0 0 0 5 33 0 11 38
G:=sub<GL(3,GF(43))| [1,0,0,0,6,0,0,0,6],[6,0,0,0,6,0,0,0,6],[1,0,0,0,31,0,0,9,25],[1,0,0,0,5,11,0,33,38] >;

C32×D21 in GAP, Magma, Sage, TeX

C_3^2\times D_{21}
% in TeX

G:=Group("C3^2xD21");
// GroupNames label

G:=SmallGroup(378,55);
// by ID

G=gap.SmallGroup(378,55);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,723,8104]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^21=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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