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## G = SD16⋊3D7order 224 = 25·7

### The semidirect product of SD16 and D7 acting through Inn(SD16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — SD16⋊3D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — D4⋊2D7 — SD16⋊3D7
 Lower central C7 — C14 — C28 — SD16⋊3D7
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD163D7
G = < a,b,c,d | a8=b2=c7=d2=1, bab=a3, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 278 in 62 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C7, C8, C8, C2×C4 [×3], D4, D4 [×3], Q8, Q8, D7 [×2], C14, C14, C2×C8, D8, SD16, SD16, Q16, C4○D4 [×2], Dic7, Dic7, C28, C28, D14, D14, C2×C14, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7×D4, C7×Q8, C8×D7, C56⋊C2, D4⋊D7, C7⋊Q16, C7×SD16, D42D7, Q82D7, SD163D7
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C4○D8, C22×D7, D4×D7, SD163D7

Smallest permutation representation of SD163D7
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 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 36)(10 39)(11 34)(12 37)(13 40)(14 35)(15 38)(16 33)(17 60)(18 63)(19 58)(20 61)(21 64)(22 59)(23 62)(24 57)(41 81)(42 84)(43 87)(44 82)(45 85)(46 88)(47 83)(48 86)(49 98)(50 101)(51 104)(52 99)(53 102)(54 97)(55 100)(56 103)(65 111)(66 106)(67 109)(68 112)(69 107)(70 110)(71 105)(72 108)(73 95)(74 90)(75 93)(76 96)(77 91)(78 94)(79 89)(80 92)
(1 89 16 105 88 61 50)(2 90 9 106 81 62 51)(3 91 10 107 82 63 52)(4 92 11 108 83 64 53)(5 93 12 109 84 57 54)(6 94 13 110 85 58 55)(7 95 14 111 86 59 56)(8 96 15 112 87 60 49)(17 98 29 76 38 68 43)(18 99 30 77 39 69 44)(19 100 31 78 40 70 45)(20 101 32 79 33 71 46)(21 102 25 80 34 72 47)(22 103 26 73 35 65 48)(23 104 27 74 36 66 41)(24 97 28 75 37 67 42)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 97)(7 98)(8 99)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 95)(18 96)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 73)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 28A 28B 28C 28D 28E 28F 56A ··· 56F order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 28 28 28 28 28 28 56 ··· 56 size 1 1 4 14 28 2 4 7 7 28 2 2 2 2 2 14 14 2 2 2 8 8 8 4 4 4 8 8 8 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 C4○D8 D4×D7 SD16⋊3D7 kernel SD16⋊3D7 C8×D7 C56⋊C2 D4⋊D7 C7⋊Q16 C7×SD16 D4⋊2D7 Q8⋊2D7 Dic7 D14 SD16 C8 D4 Q8 C7 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 3 3 3 4 3 6

Matrix representation of SD163D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 0 81 0 0 60 87
,
 1 0 0 0 0 1 0 0 0 0 0 85 0 0 4 0
,
 103 1 0 0 111 34 0 0 0 0 1 0 0 0 0 1
,
 24 89 0 0 104 89 0 0 0 0 15 50 0 0 104 98
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,0,60,0,0,81,87],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,85,0],[103,111,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[24,104,0,0,89,89,0,0,0,0,15,104,0,0,50,98] >;

SD163D7 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_3D_7
% in TeX

G:=Group("SD16:3D7");
// GroupNames label

G:=SmallGroup(224,111);
// by ID

G=gap.SmallGroup(224,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,362,116,86,297,159,69,6917]);
// 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,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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