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G = SD16⋊D14order 448 = 26·7

2nd semidirect product of SD16 and D14 acting via D14/D7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D84D14, SD162D14, C56.1C23, M4(2)⋊8D14, C28.20C24, Dic281C22, D28.13C23, Dic14.13C23, C8⋊C226D7, D83D71C2, D8⋊D72C2, (C2×D4)⋊30D14, D4⋊D76C22, (C4×D7).43D4, C4.190(D4×D7), (D4×D7)⋊9C22, (C7×D8)⋊2C22, (C8×D7)⋊2C22, C7⋊C8.10C23, C8.1(C22×D7), C4○D4.28D14, D14.54(C2×D4), SD16⋊D71C2, C8.D141C2, C28.241(C2×D4), C8⋊D72C22, C56⋊C22C22, (D7×M4(2))⋊2C2, D4.D75C22, (Q8×D7)⋊10C22, C7⋊Q163C22, C22.47(D4×D7), C4.20(C23×D7), SD163D71C2, D4.9D149C2, (D4×C14)⋊22C22, C73(D8⋊C22), Dic7.60(C2×D4), (C7×SD16)⋊2C22, (C7×D4).13C23, D4.13(C22×D7), (C22×D7).43D4, (C4×D7).30C23, D4.D1410C2, Q8.13(C22×D7), (C7×Q8).13C23, D42D710C22, (C2×C28).111C23, (C2×Dic7).195D4, Q82D710C22, C4○D28.28C22, C14.121(C22×D4), (C7×M4(2))⋊2C22, C4.Dic713C22, (C2×Dic14)⋊39C22, C2.94(C2×D4×D7), (D7×C4○D4)⋊4C2, (C7×C8⋊C22)⋊2C2, (C2×C14).66(C2×D4), (C2×D42D7)⋊26C2, (C2×C4×D7).161C22, (C2×C4).95(C22×D7), (C7×C4○D4).24C22, SmallGroup(448,1226)

Series: Derived Chief Lower central Upper central

C1C28 — SD16⋊D14
C1C7C14C28C4×D7C2×C4×D7D7×C4○D4 — SD16⋊D14
C7C14C28 — SD16⋊D14
C1C2C2×C4C8⋊C22

Generators and relations for SD16⋊D14
 G = < a,b,c,d | a8=b2=c14=d2=1, bab=dad=a3, cac-1=a-1, cbc-1=a6b, dbd=a2b, dcd=c-1 >

Subgroups: 1260 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, D8⋊C22, C8×D7, C8⋊D7, C56⋊C2, Dic28, C4.Dic7, D4⋊D7, D4.D7, C7⋊Q16, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, D42D7, Q8×D7, Q82D7, C22×Dic7, C2×C7⋊D4, D4×C14, C7×C4○D4, D7×M4(2), C8.D14, D8⋊D7, D83D7, SD16⋊D7, SD163D7, D4.D14, D4.9D14, C7×C8⋊C22, C2×D42D7, D7×C4○D4, SD16⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D8⋊C22, D4×D7, C23×D7, C2×D4×D7, SD16⋊D14

Smallest permutation representation of SD16⋊D14
On 112 points
Generators in S112
(1 43 36 80 99 86 16 65)(2 66 17 87 100 81 37 44)(3 45 38 82 101 88 18 67)(4 68 19 89 102 83 39 46)(5 47 40 84 103 90 20 69)(6 70 21 91 104 71 41 48)(7 49 42 72 105 92 22 57)(8 58 23 93 106 73 29 50)(9 51 30 74 107 94 24 59)(10 60 25 95 108 75 31 52)(11 53 32 76 109 96 26 61)(12 62 27 97 110 77 33 54)(13 55 34 78 111 98 28 63)(14 64 15 85 112 79 35 56)
(1 58)(2 94)(3 60)(4 96)(5 62)(6 98)(7 64)(8 86)(9 66)(10 88)(11 68)(12 90)(13 70)(14 92)(15 72)(16 93)(17 74)(18 95)(19 76)(20 97)(21 78)(22 85)(23 80)(24 87)(25 82)(26 89)(27 84)(28 91)(29 65)(30 44)(31 67)(32 46)(33 69)(34 48)(35 57)(36 50)(37 59)(38 52)(39 61)(40 54)(41 63)(42 56)(43 106)(45 108)(47 110)(49 112)(51 100)(53 102)(55 104)(71 111)(73 99)(75 101)(77 103)(79 105)(81 107)(83 109)
(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)
(1 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 16)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 85)(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 73)(58 72)(59 71)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)

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

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

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

55 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I7A7B7C8A8B8C8D14A14B14C14D14E14F14G···14O28A···28F28G28H28I56A···56F
order122222222444444444777888814141414141414···1428···2828282856···56
size11244414142822477142828282224428282224448···84···48888···8

55 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D7D14D14D14D14D14D8⋊C22D4×D7D4×D7SD16⋊D14
kernelSD16⋊D14D7×M4(2)C8.D14D8⋊D7D83D7SD16⋊D7SD163D7D4.D14D4.9D14C7×C8⋊C22C2×D42D7D7×C4○D4C4×D7C2×Dic7C22×D7C8⋊C22M4(2)D8SD16C2×D4C4○D4C7C4C22C1
# reps1112222111112113366332333

Matrix representation of SD16⋊D14 in GL6(𝔽113)

11200000
01120000
002438244
0045941591
008012741
008610337101
,
100000
010000
0011211100
000100
00706710550
001032818
,
1031030000
10890000
001120810
0045941591
000010
0031782519
,
10100000
241030000
001120810
0069191722
000010
0036351394

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,24,45,80,86,0,0,38,94,12,103,0,0,2,15,7,37,0,0,44,91,41,101],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,70,103,0,0,111,1,67,28,0,0,0,0,105,1,0,0,0,0,50,8],[103,10,0,0,0,0,103,89,0,0,0,0,0,0,112,45,0,31,0,0,0,94,0,78,0,0,81,15,1,25,0,0,0,91,0,19],[10,24,0,0,0,0,10,103,0,0,0,0,0,0,112,69,0,36,0,0,0,19,0,35,0,0,81,17,1,13,0,0,0,22,0,94] >;

SD16⋊D14 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_{14}
% in TeX

G:=Group("SD16:D14");
// GroupNames label

G:=SmallGroup(448,1226);
// by ID

G=gap.SmallGroup(448,1226);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,185,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=d*a*d=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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