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G = D14:2Q8order 224 = 25·7

2nd semidirect product of D14 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14:2Q8, C28.11D4, C4.13D28, C4:C4:5D7, C2.7(Q8xD7), C4:Dic7:6C2, C14.8(C2xD4), C7:3(C22:Q8), D14:C4.3C2, C2.10(C2xD28), (C2xC4).14D14, C14.14(C2xQ8), (C2xDic14):7C2, (C2xC28).6C22, C14.27(C4oD4), (C2xC14).38C23, C2.13(D4:2D7), C22.52(C22xD7), (C2xDic7).13C22, (C22xD7).22C22, (C7xC4:C4):8C2, (C2xC4xD7).3C2, SmallGroup(224,92)

Series: Derived Chief Lower central Upper central

C1C2xC14 — D14:2Q8
C1C7C14C2xC14C22xD7C2xC4xD7 — D14:2Q8
C7C2xC14 — D14:2Q8
C1C22C4:C4

Generators and relations for D14:2Q8
 G = < a,b,c,d | a14=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 310 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2xC4, C2xC4, C2xC4, Q8, C23, D7, C14, C22:C4, C4:C4, C4:C4, C22xC4, C2xQ8, Dic7, C28, C28, D14, D14, C2xC14, C22:Q8, Dic14, C4xD7, C2xDic7, C2xDic7, C2xC28, C2xC28, C22xD7, C4:Dic7, D14:C4, C7xC4:C4, C2xDic14, C2xC4xD7, D14:2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2xD4, C2xQ8, C4oD4, D14, C22:Q8, D28, C22xD7, C2xD28, D4:2D7, Q8xD7, D14:2Q8

Smallest permutation representation of D14:2Q8
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 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 56)(13 55)(14 54)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 84)(70 83)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 112)(95 111)(96 110)(97 109)(98 108)
(1 97 54 103)(2 96 55 102)(3 95 56 101)(4 94 43 100)(5 93 44 99)(6 92 45 112)(7 91 46 111)(8 90 47 110)(9 89 48 109)(10 88 49 108)(11 87 50 107)(12 86 51 106)(13 85 52 105)(14 98 53 104)(15 83 42 58)(16 82 29 57)(17 81 30 70)(18 80 31 69)(19 79 32 68)(20 78 33 67)(21 77 34 66)(22 76 35 65)(23 75 36 64)(24 74 37 63)(25 73 38 62)(26 72 39 61)(27 71 40 60)(28 84 41 59)
(1 19 54 32)(2 20 55 33)(3 21 56 34)(4 22 43 35)(5 23 44 36)(6 24 45 37)(7 25 46 38)(8 26 47 39)(9 27 48 40)(10 28 49 41)(11 15 50 42)(12 16 51 29)(13 17 52 30)(14 18 53 31)(57 106 82 86)(58 107 83 87)(59 108 84 88)(60 109 71 89)(61 110 72 90)(62 111 73 91)(63 112 74 92)(64 99 75 93)(65 100 76 94)(66 101 77 95)(67 102 78 96)(68 103 79 97)(69 104 80 98)(70 105 81 85)

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

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

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

D14:2Q8 is a maximal subgroup of
D4.6D28  D14:SD16  C7:C8:1D4  D4.D28  D14:4Q16  Q8.D28  D14:Q16  C7:C8.D4  D14.2SD16  C8:8D28  C28.(C4oD4)  C8.2D28  D14.2Q16  C8:3D28  D14:2Q16  C2.D8:7D7  C14.2+ 1+4  C14.102+ 1+4  C14.52- 1+4  C42:8D14  C42.92D14  C42.94D14  C42.98D14  D4:5D28  D4:6D28  C42.229D14  C42.115D14  Q8xD28  Q8:5D28  C42.232D14  C42.134D14  Dic14:19D4  C4:C4:21D14  C14.722- 1+4  D28:20D4  D7xC22:Q8  C14.162- 1+4  D28:22D4  Dic14:21D4  C14.512+ 1+4  C14.1182+ 1+4  C14.242- 1+4  C14.262- 1+4  C14.612+ 1+4  C14.1222+ 1+4  C14.852- 1+4  C14.862- 1+4  C42.148D14  D28:7Q8  C42.237D14  C42.150D14  C42.152D14  C42.155D14  C42.157D14  C42:24D14  C42.161D14  C42.164D14  C42.165D14  C42.241D14  D28:9Q8  C42.177D14  C42.178D14
D14:2Q8 is a maximal quotient of
C4:Dic7:7C4  (C2xDic7):Q8  (C2xC28).28D4  D14:C4:C4  (C2xC4).20D28  (C2xC28).33D4  Dic14.3Q8  D28:3Q8  D28:4Q8  D28.3Q8  C28.7Q16  Dic14:4Q8  (C2xDic7):6Q8  (C2xC4).44D28  C4:(C4:Dic7)  C4:(D14:C4)  (C2xC4).45D28

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H7A7B7C14A···14I28A···28R
order1222224444444477714···1428···28
size111114142244141428282222···24···4

44 irreducible representations

dim11111122222244
type+++++++-+++--
imageC1C2C2C2C2C2D4Q8D7C4oD4D14D28D4:2D7Q8xD7
kernelD14:2Q8C4:Dic7D14:C4C7xC4:C4C2xDic14C2xC4xD7C28D14C4:C4C14C2xC4C4C2C2
# reps122111223291233

Matrix representation of D14:2Q8 in GL4(F29) generated by

4400
251800
0010
0001
,
4400
182500
00280
00028
,
13500
71600
001816
002511
,
28000
02800
001312
001016
G:=sub<GL(4,GF(29))| [4,25,0,0,4,18,0,0,0,0,1,0,0,0,0,1],[4,18,0,0,4,25,0,0,0,0,28,0,0,0,0,28],[13,7,0,0,5,16,0,0,0,0,18,25,0,0,16,11],[28,0,0,0,0,28,0,0,0,0,13,10,0,0,12,16] >;

D14:2Q8 in GAP, Magma, Sage, TeX

D_{14}\rtimes_2Q_8
% in TeX

G:=Group("D14:2Q8");
// GroupNames label

G:=SmallGroup(224,92);
// by ID

G=gap.SmallGroup(224,92);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,122,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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