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G = SD16⋊D7order 224 = 25·7

2nd semidirect product of SD16 and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.2D14, SD162D7, D14.8D4, D4.4D14, Q8.1D14, Dic286C2, C28.6C23, C56.9C22, Dic7.10D4, Dic14.2C22, (Q8×D7)⋊2C2, C8⋊D72C2, D4.D74C2, D42D7.C2, C7⋊Q161C2, C2.20(D4×D7), C7⋊C8.1C22, (C7×SD16)⋊2C2, C14.32(C2×D4), C72(C8.C22), C4.6(C22×D7), (C4×D7).3C22, (C7×D4).4C22, (C7×Q8).1C22, SmallGroup(224,110)

Series: Derived Chief Lower central Upper central

C1C28 — SD16⋊D7
C1C7C14C28C4×D7Q8×D7 — SD16⋊D7
C7C14C28 — SD16⋊D7
C1C2C4SD16

Generators and relations for SD16⋊D7
 G = < a,b,c,d | a8=b2=c7=d2=1, bab=a3, ac=ca, dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 246 in 60 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×4], C22 [×2], C7, C8, C8, C2×C4 [×3], D4, D4, Q8, Q8 [×3], D7, C14, C14, M4(2), SD16, SD16, Q16 [×2], C2×Q8, C4○D4, Dic7, Dic7 [×2], C28, C28, D14, C2×C14, C8.C22, C7⋊C8, C56, Dic14 [×2], Dic14, C4×D7, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C7×Q8, C8⋊D7, Dic28, D4.D7, C7⋊Q16, C7×SD16, D42D7, Q8×D7, SD16⋊D7
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8.C22, C22×D7, D4×D7, SD16⋊D7

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

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

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

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

SD16⋊D7 is a maximal subgroup of
D28.29D4  D810D14  D8.10D14  SD16⋊D14  D86D14  D7×C8.C22  D28.44D4
SD16⋊D7 is a maximal quotient of
D4.D7⋊C4  Dic7.D8  C28⋊Q8⋊C2  Dic14.D4  D4⋊(C4×D7)  D14.D8  C7⋊C81D4  D4.D28  C7⋊Q16⋊C4  Dic7⋊Q16  Q8.Dic14  C56⋊C4.C2  (Q8×D7)⋊C4  D144Q16  D14⋊C8.C2  C7⋊C8.D4  Dic289C4  Dic14⋊Q8  C563Q8  Dic14.Q8  C8⋊(C4×D7)  D14.2SD16  C28.(C4○D4)  C8.2D28  Dic73SD16  SD16⋊Dic7  (C7×Q8).D4  C56.31D4  Dic147D4  Dic14.16D4  C568D4

32 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A7B7C8A8B14A14B14C14D14E14F28A28B28C28D28E28F56A···56F
order1222444447778814141414141428282828282856···56
size11414241428282224282228884448884···4

32 irreducible representations

dim11111111222222444
type++++++++++++++-+-
imageC1C2C2C2C2C2C2C2D4D4D7D14D14D14C8.C22D4×D7SD16⋊D7
kernelSD16⋊D7C8⋊D7Dic28D4.D7C7⋊Q16C7×SD16D42D7Q8×D7Dic7D14SD16C8D4Q8C7C2C1
# reps11111111113333136

Matrix representation of SD16⋊D7 in GL4(𝔽113) generated by

00791
00530
0322096
1711793
,
11594232
7610210
0299654
68825917
,
88100
533400
0001
001129
,
34900
607900
0001
0010
G:=sub<GL(4,GF(113))| [0,0,0,1,0,0,32,71,79,53,20,17,1,0,96,93],[11,76,0,68,59,102,29,82,42,1,96,59,32,0,54,17],[88,53,0,0,1,34,0,0,0,0,0,112,0,0,1,9],[34,60,0,0,9,79,0,0,0,0,0,1,0,0,1,0] >;

SD16⋊D7 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes D_7
% in TeX

G:=Group("SD16:D7");
// GroupNames label

G:=SmallGroup(224,110);
// by ID

G=gap.SmallGroup(224,110);
# by ID

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

G:=Group<a,b,c,d|a^8=b^2=c^7=d^2=1,b*a*b=a^3,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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