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G = Q16⋊D7order 224 = 25·7

2nd semidirect product of Q16 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q162D7, C8.3D14, D14.9D4, Q8.4D14, C28.9C23, C56.10C22, Dic7.11D4, D28.4C22, Dic14.5C22, Q8⋊D73C2, (Q8×D7)⋊3C2, C8⋊D74C2, C56⋊C24C2, (C7×Q16)⋊4C2, C7⋊Q164C2, C2.23(D4×D7), C7⋊C8.2C22, Q82D7.C2, C14.35(C2×D4), C73(C8.C22), C4.9(C22×D7), (C4×D7).4C22, (C7×Q8).4C22, SmallGroup(224,113)

Series: Derived Chief Lower central Upper central

C1C28 — Q16⋊D7
C1C7C14C28C4×D7Q8×D7 — Q16⋊D7
C7C14C28 — Q16⋊D7
C1C2C4Q16

Generators and relations for Q16⋊D7
 G = < a,b,c,d | a8=c7=d2=1, b2=a4, bab-1=a-1, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 270 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22 [×2], C7, C8, C8, C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×2], D7 [×2], C14, M4(2), SD16 [×2], Q16, Q16, C2×Q8, C4○D4, Dic7, Dic7, C28, C28 [×2], D14, D14, C8.C22, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7 [×2], D28, D28, C7×Q8 [×2], C8⋊D7, C56⋊C2, Q8⋊D7, C7⋊Q16, C7×Q16, Q8×D7, Q82D7, Q16⋊D7
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8.C22, C22×D7, D4×D7, Q16⋊D7

Smallest permutation representation of Q16⋊D7
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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 69 13 65)(10 68 14 72)(11 67 15 71)(12 66 16 70)(17 61 21 57)(18 60 22 64)(19 59 23 63)(20 58 24 62)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(41 89 45 93)(42 96 46 92)(43 95 47 91)(44 94 48 90)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)
(1 102 91 12 23 54 36)(2 103 92 13 24 55 37)(3 104 93 14 17 56 38)(4 97 94 15 18 49 39)(5 98 95 16 19 50 40)(6 99 96 9 20 51 33)(7 100 89 10 21 52 34)(8 101 90 11 22 53 35)(25 74 43 66 63 105 88)(26 75 44 67 64 106 81)(27 76 45 68 57 107 82)(28 77 46 69 58 108 83)(29 78 47 70 59 109 84)(30 79 48 71 60 110 85)(31 80 41 72 61 111 86)(32 73 42 65 62 112 87)
(1 84)(2 81)(3 86)(4 83)(5 88)(6 85)(7 82)(8 87)(9 71)(10 68)(11 65)(12 70)(13 67)(14 72)(15 69)(16 66)(17 41)(18 46)(19 43)(20 48)(21 45)(22 42)(23 47)(24 44)(25 40)(26 37)(27 34)(28 39)(29 36)(30 33)(31 38)(32 35)(49 77)(50 74)(51 79)(52 76)(53 73)(54 78)(55 75)(56 80)(57 89)(58 94)(59 91)(60 96)(61 93)(62 90)(63 95)(64 92)(97 108)(98 105)(99 110)(100 107)(101 112)(102 109)(103 106)(104 111)

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

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

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

Q16⋊D7 is a maximal subgroup of
D28.30D4  D810D14  D811D14  D7×C8.C22  D56⋊C22  C56.C23  D28.44D4
Q16⋊D7 is a maximal quotient of
C7⋊Q16⋊C4  Q8⋊Dic14  Q8⋊C4⋊D7  C56⋊C4.C2  Dic14.11D4  Q8.2Dic14  (Q8×D7)⋊C4  Q8⋊(C4×D7)  D14.1SD16  Q82D28  Q8.D28  C7⋊(C8⋊D4)  (C2×C8).D14  C7⋊C8.D4  Q8⋊D7⋊C4  Dic7⋊SD16  C564Q8  Dic14.2Q8  C56⋊(C2×C4)  D14.5D8  C83D28  C2.D87D7  C56⋊C2⋊C4  D282Q8  Dic73Q16  Q16⋊Dic7  (C2×Q16)⋊D7  D145Q16  D28.17D4  C56.36D4  C56.37D4

32 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A7B7C8A8B14A14B14C28A28B28C28D···28I56A···56F
order1222444447778814141428282828···2856···56
size11142824414282224282224448···84···4

32 irreducible representations

dim1111111122222444
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D7D14D14C8.C22D4×D7Q16⋊D7
kernelQ16⋊D7C8⋊D7C56⋊C2Q8⋊D7C7⋊Q16C7×Q16Q8×D7Q82D7Dic7D14Q16C8Q8C7C2C1
# reps1111111111336136

Matrix representation of Q16⋊D7 in GL6(𝔽113)

11200000
01120000
0082484141
004004
00109377237
0042377272
,
11200000
01120000
001120360
00001121
0069010
006911210
,
11210000
32800000
001000
000100
000010
000001
,
100000
811120000
001120360
00440112112
000010
006911210

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,82,4,109,42,0,0,48,0,37,37,0,0,41,0,72,72,0,0,41,4,37,72],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,69,69,0,0,0,0,0,112,0,0,36,112,1,1,0,0,0,1,0,0],[112,32,0,0,0,0,1,80,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,81,0,0,0,0,0,112,0,0,0,0,0,0,112,44,0,69,0,0,0,0,0,112,0,0,36,112,1,1,0,0,0,112,0,0] >;

Q16⋊D7 in GAP, Magma, Sage, TeX

Q_{16}\rtimes D_7
% in TeX

G:=Group("Q16:D7");
// GroupNames label

G:=SmallGroup(224,113);
// by ID

G=gap.SmallGroup(224,113);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,362,116,86,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^7=d^2=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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