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G = D83D7order 224 = 25·7

The semidirect product of D8 and D7 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83D7, C8.8D14, D14.1D4, D4.1D14, Dic284C2, C56.6C22, C28.3C23, Dic7.12D4, Dic14.1C22, (C8×D7)⋊2C2, (C7×D8)⋊3C2, C72(C4○D8), D4.D72C2, C2.17(D4×D7), C7⋊C8.5C22, D42D72C2, C14.29(C2×D4), C4.3(C22×D7), (C4×D7).8C22, (C7×D4).1C22, SmallGroup(224,107)

Series: Derived Chief Lower central Upper central

C1C28 — D83D7
C1C7C14C28C4×D7D42D7 — D83D7
C7C14C28 — D83D7
C1C2C4D8

Generators and relations for D83D7
 G = < a,b,c,d | a8=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 254 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, D4, Q8, D7, C14, C14, C2×C8, D8, SD16, Q16, C4○D4, Dic7, Dic7, C28, D14, C2×C14, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C8×D7, Dic28, D4.D7, C7×D8, D42D7, D83D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4○D8, C22×D7, D4×D7, D83D7

Smallest permutation representation of D83D7
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 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 50)(10 49)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 42)(18 41)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 112)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 81)(64 88)(65 93)(66 92)(67 91)(68 90)(69 89)(70 96)(71 95)(72 94)(73 98)(74 97)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)
(1 102 14 89 107 47 57)(2 103 15 90 108 48 58)(3 104 16 91 109 41 59)(4 97 9 92 110 42 60)(5 98 10 93 111 43 61)(6 99 11 94 112 44 62)(7 100 12 95 105 45 63)(8 101 13 96 106 46 64)(17 84 29 74 50 66 34)(18 85 30 75 51 67 35)(19 86 31 76 52 68 36)(20 87 32 77 53 69 37)(21 88 25 78 54 70 38)(22 81 26 79 55 71 39)(23 82 27 80 56 72 40)(24 83 28 73 49 65 33)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 110)(10 111)(11 112)(12 105)(13 106)(14 107)(15 108)(16 109)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(41 104)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(65 69)(66 70)(67 71)(68 72)

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

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

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

D83D7 is a maximal subgroup of
D8⋊D14  D163D7  SD32⋊D7  SD323D7  D813D14  D7×C4○D8  D8.10D14  SD16⋊D14  D86D14
D83D7 is a maximal quotient of
Dic76SD16  D4.2Dic14  Dic14.D4  (C8×Dic7)⋊C2  D42D7⋊C4  D14⋊SD16  D4.D28  C561C4⋊C2  Dic286C4  Dic14.2Q8  C56.4Q8  C8.27(C4×D7)  D142Q16  C2.D87D7  D8×Dic7  (C2×D8).D7  C56.22D4  C566D4  Dic14⋊D4

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D14A14B14C14D···14I28A28B28C56A···56F
order1222244444777888814141414···1428282856···56
size11441427728282222214142228···84444···4

35 irreducible representations

dim11111122222244
type++++++++++++-
imageC1C2C2C2C2C2D4D4D7D14D14C4○D8D4×D7D83D7
kernelD83D7C8×D7Dic28D4.D7C7×D8D42D7Dic7D14D8C8D4C7C2C1
# reps11121211336436

Matrix representation of D83D7 in GL4(𝔽113) generated by

18000
04400
001120
000112
,
04400
18000
0010
0001
,
1000
0100
001121
003280
,
1000
011200
001120
00321
G:=sub<GL(4,GF(113))| [18,0,0,0,0,44,0,0,0,0,112,0,0,0,0,112],[0,18,0,0,44,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,32,0,0,1,80],[1,0,0,0,0,112,0,0,0,0,112,32,0,0,0,1] >;

D83D7 in GAP, Magma, Sage, TeX

D_8\rtimes_3D_7
% in TeX

G:=Group("D8:3D7");
// GroupNames label

G:=SmallGroup(224,107);
// by ID

G=gap.SmallGroup(224,107);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,362,116,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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