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G = C357D4order 280 = 23·5·7

1st semidirect product of C35 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C357D4, C22⋊D35, D702C2, C2.5D70, Dic351C2, C10.12D14, C14.12D10, C70.12C22, (C2×C70)⋊2C2, (C2×C14)⋊2D5, (C2×C10)⋊2D7, C53(C7⋊D4), C73(C5⋊D4), SmallGroup(280,28)

Series: Derived Chief Lower central Upper central

C1C70 — C357D4
C1C7C35C70D70 — C357D4
C35C70 — C357D4
C1C2C22

Generators and relations for C357D4
 G = < a,b,c | a35=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
70C2
35C22
35C4
2C10
14D5
2C14
10D7
35D4
7D10
7Dic5
5D14
5Dic7
2C70
2D35
7C5⋊D4
5C7⋊D4

Smallest permutation representation of C357D4
On 140 points
Generators in S140
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)
(1 77 63 106)(2 76 64 140)(3 75 65 139)(4 74 66 138)(5 73 67 137)(6 72 68 136)(7 71 69 135)(8 105 70 134)(9 104 36 133)(10 103 37 132)(11 102 38 131)(12 101 39 130)(13 100 40 129)(14 99 41 128)(15 98 42 127)(16 97 43 126)(17 96 44 125)(18 95 45 124)(19 94 46 123)(20 93 47 122)(21 92 48 121)(22 91 49 120)(23 90 50 119)(24 89 51 118)(25 88 52 117)(26 87 53 116)(27 86 54 115)(28 85 55 114)(29 84 56 113)(30 83 57 112)(31 82 58 111)(32 81 59 110)(33 80 60 109)(34 79 61 108)(35 78 62 107)
(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 140)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)

G:=sub<Sym(140)| (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,77,63,106)(2,76,64,140)(3,75,65,139)(4,74,66,138)(5,73,67,137)(6,72,68,136)(7,71,69,135)(8,105,70,134)(9,104,36,133)(10,103,37,132)(11,102,38,131)(12,101,39,130)(13,100,40,129)(14,99,41,128)(15,98,42,127)(16,97,43,126)(17,96,44,125)(18,95,45,124)(19,94,46,123)(20,93,47,122)(21,92,48,121)(22,91,49,120)(23,90,50,119)(24,89,51,118)(25,88,52,117)(26,87,53,116)(27,86,54,115)(28,85,55,114)(29,84,56,113)(30,83,57,112)(31,82,58,111)(32,81,59,110)(33,80,60,109)(34,79,61,108)(35,78,62,107), (2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)>;

G:=Group( (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,77,63,106)(2,76,64,140)(3,75,65,139)(4,74,66,138)(5,73,67,137)(6,72,68,136)(7,71,69,135)(8,105,70,134)(9,104,36,133)(10,103,37,132)(11,102,38,131)(12,101,39,130)(13,100,40,129)(14,99,41,128)(15,98,42,127)(16,97,43,126)(17,96,44,125)(18,95,45,124)(19,94,46,123)(20,93,47,122)(21,92,48,121)(22,91,49,120)(23,90,50,119)(24,89,51,118)(25,88,52,117)(26,87,53,116)(27,86,54,115)(28,85,55,114)(29,84,56,113)(30,83,57,112)(31,82,58,111)(32,81,59,110)(33,80,60,109)(34,79,61,108)(35,78,62,107), (2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113) );

G=PermutationGroup([(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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)], [(1,77,63,106),(2,76,64,140),(3,75,65,139),(4,74,66,138),(5,73,67,137),(6,72,68,136),(7,71,69,135),(8,105,70,134),(9,104,36,133),(10,103,37,132),(11,102,38,131),(12,101,39,130),(13,100,40,129),(14,99,41,128),(15,98,42,127),(16,97,43,126),(17,96,44,125),(18,95,45,124),(19,94,46,123),(20,93,47,122),(21,92,48,121),(22,91,49,120),(23,90,50,119),(24,89,51,118),(25,88,52,117),(26,87,53,116),(27,86,54,115),(28,85,55,114),(29,84,56,113),(30,83,57,112),(31,82,58,111),(32,81,59,110),(33,80,60,109),(34,79,61,108),(35,78,62,107)], [(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,140),(79,139),(80,138),(81,137),(82,136),(83,135),(84,134),(85,133),(86,132),(87,131),(88,130),(89,129),(90,128),(91,127),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113)])

73 conjugacy classes

class 1 2A2B2C 4 5A5B7A7B7C10A···10F14A···14I35A···35L70A···70AJ
order122245577710···1014···1435···3570···70
size1127070222222···22···22···22···2

73 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2D4D5D7D10D14C5⋊D4C7⋊D4D35D70C357D4
kernelC357D4Dic35D70C2×C70C35C2×C14C2×C10C14C10C7C5C22C2C1
# reps11111232346121224

Matrix representation of C357D4 in GL2(𝔽281) generated by

111215
66100
,
127162
230154
,
10
47280
G:=sub<GL(2,GF(281))| [111,66,215,100],[127,230,162,154],[1,47,0,280] >;

C357D4 in GAP, Magma, Sage, TeX

C_{35}\rtimes_7D_4
% in TeX

G:=Group("C35:7D4");
// GroupNames label

G:=SmallGroup(280,28);
// by ID

G=gap.SmallGroup(280,28);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,61,643,6004]);
// Polycyclic

G:=Group<a,b,c|a^35=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C357D4 in TeX

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