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G = D14:D4order 224 = 25·7

1st semidirect product of D14 and D4 acting via D4/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14:1D4, Dic7:2D4, C23.5D14, C2.9(D4xD7), (C2xD28):3C2, C7:1(C4:D4), C22:C4:4D7, (C2xC4).7D14, D14:C4:11C2, Dic7:C4:5C2, C14.20(C2xD4), C14.9(C4oD4), C2.11(C4oD28), (C2xC28).53C22, (C2xC14).25C23, (C2xDic7).7C22, C22.43(C22xD7), (C22xC14).14C22, (C22xD7).18C22, (C2xC4xD7):11C2, (C2xC7:D4):2C2, (C7xC22:C4):6C2, SmallGroup(224,79)

Series: Derived Chief Lower central Upper central

C1C2xC14 — D14:D4
C1C7C14C2xC14C22xD7C2xC4xD7 — D14:D4
C7C2xC14 — D14:D4
C1C22C22:C4

Generators and relations for D14:D4
 G = < a,b,c,d | a14=b2=c4=d2=1, bab=cac-1=dad=a-1, cbc-1=a12b, dbd=a5b, dcd=c-1 >

Subgroups: 454 in 94 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2xC4, C2xC4, D4, C23, C23, D7, C14, C14, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, Dic7, Dic7, C28, D14, D14, C2xC14, C2xC14, C4:D4, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C22xD7, C22xC14, Dic7:C4, D14:C4, C7xC22:C4, C2xC4xD7, C2xD28, C2xC7:D4, D14:D4
Quotients: C1, C2, C22, D4, C23, D7, C2xD4, C4oD4, D14, C4:D4, C22xD7, C4oD28, D4xD7, D14:D4

Smallest permutation representation of D14:D4
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 91)(2 90)(3 89)(4 88)(5 87)(6 86)(7 85)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 56)(26 55)(27 54)(28 53)(29 80)(30 79)(31 78)(32 77)(33 76)(34 75)(35 74)(36 73)(37 72)(38 71)(39 84)(40 83)(41 82)(42 81)(57 109)(58 108)(59 107)(60 106)(61 105)(62 104)(63 103)(64 102)(65 101)(66 100)(67 99)(68 112)(69 111)(70 110)
(1 58 77 22)(2 57 78 21)(3 70 79 20)(4 69 80 19)(5 68 81 18)(6 67 82 17)(7 66 83 16)(8 65 84 15)(9 64 71 28)(10 63 72 27)(11 62 73 26)(12 61 74 25)(13 60 75 24)(14 59 76 23)(29 50 88 99)(30 49 89 112)(31 48 90 111)(32 47 91 110)(33 46 92 109)(34 45 93 108)(35 44 94 107)(36 43 95 106)(37 56 96 105)(38 55 97 104)(39 54 98 103)(40 53 85 102)(41 52 86 101)(42 51 87 100)
(1 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 22)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 88)(38 87)(39 86)(40 85)(41 98)(42 97)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 66)(58 65)(59 64)(60 63)(61 62)(67 70)(68 69)(99 105)(100 104)(101 103)(106 112)(107 111)(108 110)

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

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

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

D14:D4 is a maximal subgroup of
C24.27D14  C24.30D14  C42.93D14  C42.95D14  C42.97D14  C42.100D14  C42.104D14  C42:12D14  C42.228D14  D28:23D4  Dic14:24D4  C42.113D14  C42:17D14  C42.116D14  C24:2D14  C24:3D14  C24.34D14  C24.36D14  C14.682- 1+4  Dic14:20D4  D7xC4:D4  C14.722- 1+4  D28:19D4  C14.402+ 1+4  C14.442+ 1+4  C14.482+ 1+4  C14.172- 1+4  D28:22D4  Dic14:22D4  C14.202- 1+4  C14.222- 1+4  C14.262- 1+4  C14.1202+ 1+4  C14.1212+ 1+4  C14.822- 1+4  C4:C4:28D14  C14.612+ 1+4  C14.642+ 1+4  C14.662+ 1+4  C14.682+ 1+4  C14.862- 1+4  C42.233D14  C42.138D14  C42:18D14  D28:10D4  Dic14:10D4  C42:20D14  C42.145D14  C42:23D14  C42.189D14  C42.161D14  C42.163D14  C42.164D14  C42:25D14
D14:D4 is a maximal quotient of
C14.(C4xQ8)  C2.(C28:Q8)  (C2xC4):9D28  D14:(C4:C4)  D14:C4:5C4  (C2xDic7):3D4  (C22xD7).9D4  (C22xD7).Q8  D28.2D4  D28.3D4  D28.6D4  D28.7D4  Dic14:2D4  Dic14.D4  D14:D8  D14:SD16  C7:C8:1D4  C7:C8:D4  D28:3D4  D28.D4  Dic7:Q16  Dic14.11D4  D14:2SD16  C7:(C8:D4)  D14:Q16  C7:C8.D4  Dic7:SD16  D28.12D4  C24.6D14  C24.8D14  C24.12D14  C24.13D14  C24.14D14  C23:2D28

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C14A···14I14J···14O28A···28L
order1222222244444477714···1414···1428···28
size111141414282241414282222···24···44···4

44 irreducible representations

dim111111122222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D4D7C4oD4D14D14C4oD28D4xD7
kernelD14:D4Dic7:C4D14:C4C7xC22:C4C2xC4xD7C2xD28C2xC7:D4Dic7D14C22:C4C14C2xC4C23C2C2
# reps1111112223263126

Matrix representation of D14:D4 in GL6(F29)

2800000
0280000
00272100
00162000
000010
000001
,
14270000
25150000
0082800
0052100
0000280
0000028
,
2350000
1060000
002800
00252700
000001
0000280
,
100000
14280000
002800
00252700
0000280
000001

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,27,16,0,0,0,0,21,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,25,0,0,0,0,27,15,0,0,0,0,0,0,8,5,0,0,0,0,28,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[23,10,0,0,0,0,5,6,0,0,0,0,0,0,2,25,0,0,0,0,8,27,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,14,0,0,0,0,0,28,0,0,0,0,0,0,2,25,0,0,0,0,8,27,0,0,0,0,0,0,28,0,0,0,0,0,0,1] >;

D14:D4 in GAP, Magma, Sage, TeX

D_{14}\rtimes D_4
% in TeX

G:=Group("D14:D4");
// GroupNames label

G:=SmallGroup(224,79);
// by ID

G=gap.SmallGroup(224,79);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,55,506,188,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^12*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations

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