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G = SD163D7order 224 = 25·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD163D7, D4.5D14, D14.2D4, C8.11D14, Q8.2D14, C28.7C23, C56.11C22, Dic7.13D4, D28.3C22, Dic14.3C22, D4⋊D74C2, (C8×D7)⋊5C2, C73(C4○D8), C56⋊C26C2, C7⋊Q162C2, C2.21(D4×D7), C7⋊C8.6C22, D42D73C2, Q82D72C2, (C7×SD16)⋊4C2, C14.33(C2×D4), C4.7(C22×D7), (C7×D4).5C22, (C7×Q8).2C22, (C4×D7).10C22, SmallGroup(224,111)

Series: Derived Chief Lower central Upper central

C1C28 — SD163D7
C1C7C14C28C4×D7D42D7 — SD163D7
C7C14C28 — SD163D7
C1C2C4SD16

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)])

SD163D7 is a maximal subgroup of
D28.29D4  D7×C4○D8  D811D14  SD16⋊D14  D85D14  D56⋊C22  D28.44D4
SD163D7 is a maximal quotient of
Dic74D8  D4.Dic14  (C8×Dic7)⋊C2  D42D7⋊C4  D14⋊D8  C8⋊Dic7⋊C2  D43D28  D28.D4  Dic74Q16  Dic14.11D4  Q8.2Dic14  Q8⋊Dic7⋊C2  Q82D7⋊C4  Q8.D28  D14⋊Q16  (C2×C8).D14  Dic78SD16  Dic14.Q8  C56.8Q8  (C8×D7)⋊C4  C88D28  C4.Q8⋊D7  C28.(C4○D4)  D28.Q8  SD16×Dic7  (C7×D4).D4  (C7×Q8).D4  C56.43D4  C5614D4  D287D4  Dic14.16D4

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D14A14B14C14D14E14F28A28B28C28D28E28F56A···56F
order1222244444777888814141414141428282828282856···56
size11414282477282222214142228884448884···4

35 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D7D14D14D14C4○D8D4×D7SD163D7
kernelSD163D7C8×D7C56⋊C2D4⋊D7C7⋊Q16C7×SD16D42D7Q82D7Dic7D14SD16C8D4Q8C7C2C1
# reps11111111113333436

Matrix representation of SD163D7 in GL4(𝔽113) generated by

112000
011200
00081
006087
,
1000
0100
00085
0040
,
103100
1113400
0010
0001
,
248900
1048900
001550
0010498
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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