Copied to
clipboard

## G = Q8.Dic7order 224 = 25·7

### The non-split extension by Q8 of Dic7 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — Q8.Dic7
 Chief series C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8 — Q8.Dic7
 Lower central C7 — C14 — Q8.Dic7
 Upper central C1 — C4 — C4○D4

Generators and relations for Q8.Dic7
G = < a,b,c,d | a4=b2=1, c14=a2, d2=a2c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c13 >

Subgroups: 134 in 62 conjugacy classes, 45 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, D4, Q8, C14, C14, C2×C8, M4(2), C4○D4, C28, C28, C2×C14, C8○D4, C7⋊C8, C7⋊C8, C2×C28, C7×D4, C7×Q8, C2×C7⋊C8, C4.Dic7, C7×C4○D4, Q8.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, Dic7, D14, C8○D4, C2×Dic7, C22×D7, C22×Dic7, Q8.Dic7

Smallest permutation representation of Q8.Dic7
On 112 points
Generators in S112
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)
(1 56)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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 57 22 78 15 71 8 64)(2 70 23 63 16 84 9 77)(3 83 24 76 17 69 10 62)(4 68 25 61 18 82 11 75)(5 81 26 74 19 67 12 60)(6 66 27 59 20 80 13 73)(7 79 28 72 21 65 14 58)(29 105 50 98 43 91 36 112)(30 90 51 111 44 104 37 97)(31 103 52 96 45 89 38 110)(32 88 53 109 46 102 39 95)(33 101 54 94 47 87 40 108)(34 86 55 107 48 100 41 93)(35 99 56 92 49 85 42 106)

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

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

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

Q8.Dic7 is a maximal subgroup of
M4(2).22D14  C42.196D14  D85Dic7  D84Dic7  M4(2).D14  M4(2).13D14  M4(2).15D14  M4(2).16D14  D7×C8○D4  C56.49C23  C28.76C24  D28.32C23  D28.33C23  D28.34C23  D28.35C23
Q8.Dic7 is a maximal quotient of
C28.5C42  C42.43D14  C42.187D14  D4×C7⋊C8  C42.47D14  C283M4(2)  Q8×C7⋊C8  C42.210D14  (D4×C14).11C4

50 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E ··· 8J 14A 14B 14C 14D ··· 14L 28A ··· 28F 28G ··· 28O order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 ··· 8 14 14 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 1 1 2 2 2 2 2 2 7 7 7 7 14 ··· 14 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + - - image C1 C2 C2 C2 C4 C4 D7 D14 Dic7 Dic7 C8○D4 Q8.Dic7 kernel Q8.Dic7 C2×C7⋊C8 C4.Dic7 C7×C4○D4 C7×D4 C7×Q8 C4○D4 C2×C4 D4 Q8 C7 C1 # reps 1 3 3 1 6 2 3 9 9 3 4 6

Matrix representation of Q8.Dic7 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 15 0 0 0 0 98
,
 1 0 0 0 0 1 0 0 0 0 0 98 0 0 15 0
,
 103 1 0 0 13 89 0 0 0 0 15 0 0 0 0 15
,
 33 12 0 0 60 80 0 0 0 0 44 0 0 0 0 44
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,98],[1,0,0,0,0,1,0,0,0,0,0,15,0,0,98,0],[103,13,0,0,1,89,0,0,0,0,15,0,0,0,0,15],[33,60,0,0,12,80,0,0,0,0,44,0,0,0,0,44] >;

Q8.Dic7 in GAP, Magma, Sage, TeX

Q_8.{\rm Dic}_7
% in TeX

G:=Group("Q8.Dic7");
// GroupNames label

G:=SmallGroup(224,143);
// by ID

G=gap.SmallGroup(224,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,188,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^14=a^2,d^2=a^2*c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations

׿
×
𝔽