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G = D146SD16order 448 = 26·7

2nd semidirect product of D14 and SD16 acting via SD16/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D146SD16, D28.16D4, (C2×C8)⋊17D14, (C2×Q8)⋊3D14, (C7×D4).9D4, C4.62(D4×D7), D14⋊C832C2, C28.47(C2×D4), (C2×C56)⋊33C22, D143Q83C2, (C2×SD16)⋊10D7, C74(C22⋊SD16), D4.8(C7⋊D4), C2.D5635C2, C2.28(D7×SD16), (Q8×C14)⋊3C22, C14.57C22≀C2, (C14×SD16)⋊20C2, (C2×D4).146D14, D4⋊Dic733C2, C4⋊Dic720C22, (C2×Dic7).71D4, C14.45(C2×SD16), (C22×D7).91D4, C22.266(D4×D7), C2.28(D56⋊C2), C14.78(C8⋊C22), (C2×C28).446C23, (D4×C14).95C22, C2.25(C23⋊D14), (C2×D28).120C22, (C2×D4×D7).6C2, (C2×C7⋊C8)⋊8C22, (C2×Q8⋊D7)⋊17C2, C4.42(C2×C7⋊D4), (C2×C4×D7).47C22, (C2×C14).358(C2×D4), (C2×C4).535(C22×D7), SmallGroup(448,703)

Series: Derived Chief Lower central Upper central

C1C2×C28 — D146SD16
C1C7C14C2×C14C2×C28C2×C4×D7C2×D4×D7 — D146SD16
C7C14C2×C28 — D146SD16
C1C22C2×C4C2×SD16

Generators and relations for D146SD16
 G = < a,b,c,d | a14=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c3 >

Subgroups: 1284 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C22⋊SD16, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D14⋊C4, Q8⋊D7, C2×C56, C7×SD16, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, D14⋊C8, C2.D56, D4⋊Dic7, C2×Q8⋊D7, D143Q8, C14×SD16, C2×D4×D7, D146SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, C7⋊D4, C22×D7, C22⋊SD16, D4×D7, C2×C7⋊D4, D7×SD16, D56⋊C2, C23⋊D14, D146SD16

Smallest permutation representation of D146SD16
On 112 points
Generators in S112
(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 21)(16 20)(17 19)(22 28)(23 27)(24 26)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(43 50)(44 49)(45 48)(46 47)(51 56)(52 55)(53 54)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 94)(86 93)(87 92)(88 91)(89 90)(95 98)(96 97)(99 107)(100 106)(101 105)(102 104)(108 112)(109 111)
(1 107 47 15 79 64 90 35)(2 108 48 16 80 65 91 36)(3 109 49 17 81 66 92 37)(4 110 50 18 82 67 93 38)(5 111 51 19 83 68 94 39)(6 112 52 20 84 69 95 40)(7 99 53 21 71 70 96 41)(8 100 54 22 72 57 97 42)(9 101 55 23 73 58 98 29)(10 102 56 24 74 59 85 30)(11 103 43 25 75 60 86 31)(12 104 44 26 76 61 87 32)(13 105 45 27 77 62 88 33)(14 106 46 28 78 63 89 34)
(1 54)(2 55)(3 56)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 85)(82 86)(83 87)(84 88)

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

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

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

61 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E7A7B7C8A8B8C8D14A···14I14J···14O28A···28F28G···28L56A···56L
order122222222244444777888814···1414···1428···2828···2856···56
size1111441414282822828562224428282···28···84···48···84···4

61 irreducible representations

dim11111111222222222244444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D7SD16D14D14D14C7⋊D4C8⋊C22D4×D7D4×D7D7×SD16D56⋊C2
kernelD146SD16D14⋊C8C2.D56D4⋊Dic7C2×Q8⋊D7D143Q8C14×SD16C2×D4×D7D28C2×Dic7C7×D4C22×D7C2×SD16D14C2×C8C2×D4C2×Q8D4C14C4C22C2C2
# reps111111112121343331213366

Matrix representation of D146SD16 in GL4(𝔽113) generated by

1000
0100
002320
0010311
,
1000
0100
0089104
008924
,
10010000
1310000
004199
00772
,
0100
1000
001120
000112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,23,103,0,0,20,11],[1,0,0,0,0,1,0,0,0,0,89,89,0,0,104,24],[100,13,0,0,100,100,0,0,0,0,41,7,0,0,99,72],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112] >;

D146SD16 in GAP, Magma, Sage, TeX

D_{14}\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("D14:6SD16");
// GroupNames label

G:=SmallGroup(448,703);
// by ID

G=gap.SmallGroup(448,703);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,184,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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