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## G = SD16⋊D7order 224 = 25·7

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — SD16⋊D7
 Chief series C1 — C7 — C14 — C28 — C4×D7 — Q8×D7 — SD16⋊D7
 Lower central C7 — C14 — C28 — SD16⋊D7
 Upper central C1 — C2 — C4 — SD16

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)])

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 C8.C22 D4×D7 SD16⋊D7 kernel SD16⋊D7 C8⋊D7 Dic28 D4.D7 C7⋊Q16 C7×SD16 D4⋊2D7 Q8×D7 Dic7 D14 SD16 C8 D4 Q8 C7 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 3 3 3 3 1 3 6

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

 0 0 79 1 0 0 53 0 0 32 20 96 1 71 17 93
,
 11 59 42 32 76 102 1 0 0 29 96 54 68 82 59 17
,
 88 1 0 0 53 34 0 0 0 0 0 1 0 0 112 9
,
 34 9 0 0 60 79 0 0 0 0 0 1 0 0 1 0
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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