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G = Dic7⋊D4order 224 = 25·7

2nd semidirect product of Dic7 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic73D4, C23.9D14, (C2×D4)⋊5D7, (C2×C14)⋊3D4, (D4×C14)⋊9C2, C75(C4⋊D4), C2.27(D4×D7), D14⋊C415C2, (C2×C4).19D14, C14.51(C2×D4), Dic7⋊C415C2, C221(C7⋊D4), C23.D712C2, C14.32(C4○D4), (C2×C14).54C23, (C2×C28).62C22, (C22×Dic7)⋊6C2, C2.18(D42D7), C22.61(C22×D7), (C22×C14).21C22, (C2×Dic7).38C22, (C22×D7).11C22, (C2×C7⋊D4)⋊6C2, C2.15(C2×C7⋊D4), SmallGroup(224,134)

Series: Derived Chief Lower central Upper central

C1C2×C14 — Dic7⋊D4
C1C7C14C2×C14C22×D7C2×C7⋊D4 — Dic7⋊D4
C7C2×C14 — Dic7⋊D4
C1C22C2×D4

Generators and relations for Dic7⋊D4
 G = < a,b,c,d | a14=c4=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >

Subgroups: 382 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C4⋊D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, Dic7⋊C4, D14⋊C4, C23.D7, C22×Dic7, C2×C7⋊D4, D4×C14, Dic7⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4×D7, D42D7, C2×C7⋊D4, Dic7⋊D4

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

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

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

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

Dic7⋊D4 is a maximal subgroup of
C42.102D14  C42.104D14  C4212D14  Dic1423D4  C4217D14  C42.118D14  C42.119D14  C24.56D14  C243D14  C24.33D14  C24.34D14  C24.35D14  C24.36D14  C28⋊(C4○D4)  C14.682- 1+4  Dic1419D4  Dic1420D4  C14.342+ 1+4  D7×C4⋊D4  C14.372+ 1+4  C14.722- 1+4  C14.402+ 1+4  C14.422+ 1+4  C14.442+ 1+4  C14.452+ 1+4  C14.462+ 1+4  C14.1152+ 1+4  C14.472+ 1+4  C14.482+ 1+4  C14.492+ 1+4  C4⋊C4.197D14  C14.1212+ 1+4  C14.822- 1+4  C14.642+ 1+4  C14.842- 1+4  C14.662+ 1+4  C14.852- 1+4  C14.862- 1+4  C42.137D14  C42.138D14  C4220D14  C42.145D14  C4226D14  Dic1411D4  C42.168D14  C4228D14  D4×C7⋊D4  C247D14  C24.42D14  C14.1042- 1+4  C14.1452+ 1+4  (C2×C28)⋊17D4  C14.1082- 1+4
Dic7⋊D4 is a maximal quotient of
C24.44D14  C24.4D14  C24.6D14  C24.9D14  C24.13D14  C23.45D28  C24.14D14  C23.16D28  Dic7⋊(C4⋊C4)  (C2×C28).54D4  D14⋊C47C4  (C2×C4).45D28  C7⋊C822D4  C4⋊D4⋊D7  C7⋊C823D4  C7⋊C85D4  C7⋊C824D4  C7⋊C86D4  C7⋊C8.29D4  C7⋊C8.6D4  Dic7⋊D8  (C2×D8).D7  Dic73SD16  Dic75SD16  (C7×D4).D4  (C7×Q8).D4  Dic73Q16  (C2×Q16)⋊D7  M4(2).D14  M4(2).13D14  M4(2).15D14  M4(2).16D14  C24.18D14  C24.20D14  C24.21D14

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F7A7B7C14A···14I14J···14U28A···28F
order1222222244444477714···1414···1428···28
size111122428414141414282222···24···44···4

44 irreducible representations

dim1111111222222244
type+++++++++++++-
imageC1C2C2C2C2C2C2D4D4D7C4○D4D14D14C7⋊D4D4×D7D42D7
kernelDic7⋊D4Dic7⋊C4D14⋊C4C23.D7C22×Dic7C2×C7⋊D4D4×C14Dic7C2×C14C2×D4C14C2×C4C23C22C2C2
# reps11111212232361233

Matrix representation of Dic7⋊D4 in GL4(𝔽29) generated by

02800
12200
0010
0001
,
91400
192000
0010
0001
,
91400
152000
00115
002218
,
28000
02800
001824
002411
G:=sub<GL(4,GF(29))| [0,1,0,0,28,22,0,0,0,0,1,0,0,0,0,1],[9,19,0,0,14,20,0,0,0,0,1,0,0,0,0,1],[9,15,0,0,14,20,0,0,0,0,11,22,0,0,5,18],[28,0,0,0,0,28,0,0,0,0,18,24,0,0,24,11] >;

Dic7⋊D4 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes D_4
% in TeX

G:=Group("Dic7:D4");
// GroupNames label

G:=SmallGroup(224,134);
// by ID

G=gap.SmallGroup(224,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,218,188,6917]);
// Polycyclic

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

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