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## G = C3×C7⋊F5order 420 = 22·3·5·7

### Direct product of C3 and C7⋊F5

Aliases: C3×C7⋊F5, C212F5, C1052C4, C356C12, C152Dic7, C5⋊(C3×Dic7), C73(C3×F5), D5.(C3×D7), (C3×D5).2D7, (C7×D5).3C6, (D5×C21).2C2, SmallGroup(420,21)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C3×C7⋊F5
 Chief series C1 — C7 — C35 — C7×D5 — D5×C21 — C3×C7⋊F5
 Lower central C35 — C3×C7⋊F5
 Upper central C1 — C3

Generators and relations for C3×C7⋊F5
G = < a,b,c,d | a3=b7=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Smallest permutation representation of C3×C7⋊F5
On 105 points
Generators in S105
(1 71 36)(2 72 37)(3 73 38)(4 74 39)(5 75 40)(6 76 41)(7 77 42)(8 78 43)(9 79 44)(10 80 45)(11 81 46)(12 82 47)(13 83 48)(14 84 49)(15 85 50)(16 86 51)(17 87 52)(18 88 53)(19 89 54)(20 90 55)(21 91 56)(22 92 57)(23 93 58)(24 94 59)(25 95 60)(26 96 61)(27 97 62)(28 98 63)(29 99 64)(30 100 65)(31 101 66)(32 102 67)(33 103 68)(34 104 69)(35 105 70)
(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)
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)(71 99 92 85 78)(72 100 93 86 79)(73 101 94 87 80)(74 102 95 88 81)(75 103 96 89 82)(76 104 97 90 83)(77 105 98 91 84)
(2 7)(3 6)(4 5)(8 15 29 22)(9 21 30 28)(10 20 31 27)(11 19 32 26)(12 18 33 25)(13 17 34 24)(14 16 35 23)(37 42)(38 41)(39 40)(43 50 64 57)(44 56 65 63)(45 55 66 62)(46 54 67 61)(47 53 68 60)(48 52 69 59)(49 51 70 58)(72 77)(73 76)(74 75)(78 85 99 92)(79 91 100 98)(80 90 101 97)(81 89 102 96)(82 88 103 95)(83 87 104 94)(84 86 105 93)

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

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

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

51 conjugacy classes

 class 1 2 3A 3B 4A 4B 5 6A 6B 7A 7B 7C 12A 12B 12C 12D 14A 14B 14C 15A 15B 21A ··· 21F 35A ··· 35F 42A ··· 42F 105A ··· 105L order 1 2 3 3 4 4 5 6 6 7 7 7 12 12 12 12 14 14 14 15 15 21 ··· 21 35 ··· 35 42 ··· 42 105 ··· 105 size 1 5 1 1 35 35 4 5 5 2 2 2 35 35 35 35 10 10 10 4 4 2 ··· 2 4 ··· 4 10 ··· 10 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + image C1 C2 C3 C4 C6 C12 D7 Dic7 C3×D7 C3×Dic7 F5 C3×F5 C7⋊F5 C3×C7⋊F5 kernel C3×C7⋊F5 D5×C21 C7⋊F5 C105 C7×D5 C35 C3×D5 C15 D5 C5 C21 C7 C3 C1 # reps 1 1 2 2 2 4 3 3 6 6 1 2 6 12

Matrix representation of C3×C7⋊F5 in GL6(𝔽421)

 20 0 0 0 0 0 0 20 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 116 420 0 0 0 0 1 0 0 0 0 0 0 0 0 420 0 0 0 0 1 403 0 0 0 0 0 0 0 420 0 0 0 0 1 403
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 369 263 1 0 0 0 158 51 0 1 0 0 420 0 0 0 0 0 0 420 0 0
,
 392 0 0 0 0 0 4 29 0 0 0 0 0 0 1 403 420 18 0 0 0 420 0 1 0 0 0 0 369 252 0 0 0 0 158 52

G:=sub<GL(6,GF(421))| [20,0,0,0,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[116,1,0,0,0,0,420,0,0,0,0,0,0,0,0,1,0,0,0,0,420,403,0,0,0,0,0,0,0,1,0,0,0,0,420,403],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,369,158,420,0,0,0,263,51,0,420,0,0,1,0,0,0,0,0,0,1,0,0],[392,4,0,0,0,0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,403,420,0,0,0,0,420,0,369,158,0,0,18,1,252,52] >;

C3×C7⋊F5 in GAP, Magma, Sage, TeX

C_3\times C_7\rtimes F_5
% in TeX

G:=Group("C3xC7:F5");
// GroupNames label

G:=SmallGroup(420,21);
// by ID

G=gap.SmallGroup(420,21);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-7,30,723,173,9004]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^7=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations

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