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## G = C3×C3⋊D21order 378 = 2·33·7

### Direct product of C3 and C3⋊D21

Aliases: C3×C3⋊D21, C332D7, C323D21, C3⋊(C3×D21), C215(C3×S3), (C3×C21)⋊6S3, C212(C3⋊S3), (C3×C21)⋊15C6, (C32×C21)⋊3C2, C324(C3×D7), C73(C3×C3⋊S3), SmallGroup(378,57)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C3⋊D21
G = < a,b,c,d | a3=b3=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 408 in 64 conjugacy classes, 26 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C7, C32, C32, C32, D7, C3×S3, C3⋊S3, C21, C21, C21, C33, C3×D7, D21, C3×C3⋊S3, C3×C21, C3×C21, C3×C21, C3×D21, C3⋊D21, C32×C21, C3×C3⋊D21
Quotients: C1, C2, C3, S3, C6, D7, C3×S3, C3⋊S3, C3×D7, D21, C3×C3⋊S3, C3×D21, C3⋊D21, C3×C3⋊D21

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

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

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

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

99 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 7A 7B 7C 21A ··· 21BZ order 1 2 3 3 3 ··· 3 6 6 7 7 7 21 ··· 21 size 1 63 1 1 2 ··· 2 63 63 2 2 2 2 ··· 2

99 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 C3×C3⋊D21 C32×C21 C3⋊D21 C3×C21 C3×C21 C33 C21 C32 C32 C3 # reps 1 1 2 2 4 3 8 6 24 48

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

 36 0 0 0 0 36 0 0 0 0 6 0 0 0 0 6
,
 6 0 0 0 2 36 0 0 0 0 1 0 0 0 0 1
,
 21 0 0 0 30 41 0 0 0 0 14 0 0 0 26 40
,
 16 25 0 0 7 27 0 0 0 0 23 23 0 0 35 20
G:=sub<GL(4,GF(43))| [36,0,0,0,0,36,0,0,0,0,6,0,0,0,0,6],[6,2,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[21,30,0,0,0,41,0,0,0,0,14,26,0,0,0,40],[16,7,0,0,25,27,0,0,0,0,23,35,0,0,23,20] >;

C3×C3⋊D21 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC3:D21");
// GroupNames label

G:=SmallGroup(378,57);
// by ID

G=gap.SmallGroup(378,57);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,182,723,8104]);
// 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,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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