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G = D815D14order 448 = 26·7

4th semidirect product of D8 and D14 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D815D14, Q1613D14, D28.45D4, SD1611D14, D5619C22, C28.16C24, C56.38C23, Dic14.45D4, D28.11C23, C4○D84D7, (D7×D8)⋊7C2, C73(D4○D8), C4○D41D14, (C2×C8)⋊13D14, C7⋊D4.1D4, C7⋊C8.7C23, (C2×D56)⋊23C2, D56⋊C26C2, D4⋊D73C22, C4.143(D4×D7), (D4×D7)⋊2C22, (C8×D7)⋊8C22, Q8⋊D72C22, D48D145C2, Q8.D147C2, D4⋊D147C2, C22.8(D4×D7), (C2×C56)⋊12C22, D14.29(C2×D4), C28.349(C2×D4), (C7×D8)⋊13C22, (C4×D7).9C23, C8.16(C22×D7), C4.16(C23×D7), D28.2C47C2, (C2×D28)⋊34C22, C8⋊D711C22, Dic7.34(C2×D4), Q82D72C22, (C7×Q16)⋊11C22, (C7×D4).10C23, D4.10(C22×D7), (C7×Q8).10C23, Q8.10(C22×D7), (C2×C28).533C23, (C7×SD16)⋊11C22, C4○D28.54C22, C14.117(C22×D4), C4.Dic730C22, C2.90(C2×D4×D7), (C7×C4○D8)⋊5C2, (C2×C14).13(C2×D4), (C7×C4○D4)⋊3C22, (C2×C4).232(C22×D7), SmallGroup(448,1222)

Series: Derived Chief Lower central Upper central

C1C28 — D815D14
C1C7C14C28C4×D7C4○D28D48D14 — D815D14
C7C14C28 — D815D14
C1C2C2×C4C4○D8

Generators and relations for D815D14
 G = < a,b,c,d | a8=b2=c14=d2=1, bab=dad=a-1, ac=ca, cbc-1=a4b, dbd=a2b, dcd=c-1 >

Subgroups: 1620 in 268 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, SD16, Q16, C2×D4, C4○D4, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×D8, C4○D8, C4○D8, C8⋊C22, 2+ 1+4, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, D28, D28, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4○D8, C8×D7, C8⋊D7, D56, C4.Dic7, D4⋊D7, Q8⋊D7, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×D28, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, Q82D7, C7×C4○D4, D28.2C4, C2×D56, D7×D8, D56⋊C2, Q8.D14, D4⋊D14, C7×C4○D8, D48D14, D815D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○D8, D4×D7, C23×D7, C2×D4×D7, D815D14

Smallest permutation representation of D815D14
On 112 points
Generators in S112
(1 16 40 54 65 109 71 92)(2 17 41 55 66 110 72 93)(3 18 42 56 67 111 73 94)(4 19 29 43 68 112 74 95)(5 20 30 44 69 99 75 96)(6 21 31 45 70 100 76 97)(7 22 32 46 57 101 77 98)(8 23 33 47 58 102 78 85)(9 24 34 48 59 103 79 86)(10 25 35 49 60 104 80 87)(11 26 36 50 61 105 81 88)(12 27 37 51 62 106 82 89)(13 28 38 52 63 107 83 90)(14 15 39 53 64 108 84 91)
(2 66)(4 68)(6 70)(8 58)(10 60)(12 62)(14 64)(15 53)(16 92)(17 55)(18 94)(19 43)(20 96)(21 45)(22 98)(23 47)(24 86)(25 49)(26 88)(27 51)(28 90)(30 75)(32 77)(34 79)(36 81)(38 83)(40 71)(42 73)(44 99)(46 101)(48 103)(50 105)(52 107)(54 109)(56 111)(85 102)(87 104)(89 106)(91 108)(93 110)(95 112)(97 100)
(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 46)(2 45)(3 44)(4 43)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 33)(16 32)(17 31)(18 30)(19 29)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(57 92)(58 91)(59 90)(60 89)(61 88)(62 87)(63 86)(64 85)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 101)(72 100)(73 99)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)

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

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

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

64 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F7A7B7C8A8B8C8D8E14A14B14C14D14E14F14G···14L28A···28F28G28H28I28J···28O56A···56L
order122222222224444447778888814141414141414···1428···2828282828···2856···56
size112441414282828282244141422222428282224448···82···24448···84···4

64 irreducible representations

dim1111111112222222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D7D14D14D14D14D14D4○D8D4×D7D4×D7D815D14
kernelD815D14D28.2C4C2×D56D7×D8D56⋊C2Q8.D14D4⋊D14C7×C4○D8D48D14Dic14D28C7⋊D4C4○D8C2×C8D8SD16Q16C4○D4C7C4C22C1
# reps11124221211233363623312

Matrix representation of D815D14 in GL4(𝔽113) generated by

820310
082031
820820
082082
,
1000
0100
001120
000112
,
009419
009413
199400
1910000
,
16169797
37977616
97979797
76167616
G:=sub<GL(4,GF(113))| [82,0,82,0,0,82,0,82,31,0,82,0,0,31,0,82],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[0,0,19,19,0,0,94,100,94,94,0,0,19,13,0,0],[16,37,97,76,16,97,97,16,97,76,97,76,97,16,97,16] >;

D815D14 in GAP, Magma, Sage, TeX

D_8\rtimes_{15}D_{14}
% in TeX

G:=Group("D8:15D14");
// GroupNames label

G:=SmallGroup(448,1222);
// by ID

G=gap.SmallGroup(448,1222);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,570,185,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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