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G = Dic74D4order 224 = 25·7

1st semidirect product of Dic7 and D4 acting through Inn(Dic7)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic74D4, C23.14D14, C7⋊D4⋊C4, C72(C4×D4), C2.2(D4×D7), D142(C2×C4), C22⋊C47D7, C221(C4×D7), D14⋊C410C2, Dic7⋊C49C2, Dic71(C2×C4), (C2×C4).28D14, C14.18(C2×D4), (C4×Dic7)⋊11C2, C14.7(C22×C4), C14.22(C4○D4), C2.2(D42D7), (C2×C28).51C22, (C2×C14).22C23, (C22×Dic7)⋊1C2, C22.14(C22×D7), (C22×C14).11C22, (C2×Dic7).47C22, (C22×D7).16C22, (C2×C4×D7)⋊9C2, C2.9(C2×C4×D7), (C2×C14)⋊2(C2×C4), (C7×C22⋊C4)⋊9C2, (C2×C7⋊D4).2C2, SmallGroup(224,76)

Series: Derived Chief Lower central Upper central

C1C14 — Dic74D4
C1C7C14C2×C14C22×D7C2×C7⋊D4 — Dic74D4
C7C14 — Dic74D4
C1C22C22⋊C4

Generators and relations for Dic74D4
 G = < a,b,c,d | a14=c4=d2=1, b2=a7, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 358 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×C14, C4×Dic7, Dic7⋊C4, D14⋊C4, C7×C22⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, Dic74D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, C22×D7, C2×C4×D7, D4×D7, D42D7, Dic74D4

Smallest permutation representation of Dic74D4
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 40 8 33)(2 39 9 32)(3 38 10 31)(4 37 11 30)(5 36 12 29)(6 35 13 42)(7 34 14 41)(15 68 22 61)(16 67 23 60)(17 66 24 59)(18 65 25 58)(19 64 26 57)(20 63 27 70)(21 62 28 69)(43 99 50 106)(44 112 51 105)(45 111 52 104)(46 110 53 103)(47 109 54 102)(48 108 55 101)(49 107 56 100)(71 98 78 91)(72 97 79 90)(73 96 80 89)(74 95 81 88)(75 94 82 87)(76 93 83 86)(77 92 84 85)
(1 77 69 108)(2 76 70 107)(3 75 57 106)(4 74 58 105)(5 73 59 104)(6 72 60 103)(7 71 61 102)(8 84 62 101)(9 83 63 100)(10 82 64 99)(11 81 65 112)(12 80 66 111)(13 79 67 110)(14 78 68 109)(15 47 34 98)(16 46 35 97)(17 45 36 96)(18 44 37 95)(19 43 38 94)(20 56 39 93)(21 55 40 92)(22 54 41 91)(23 53 42 90)(24 52 29 89)(25 51 30 88)(26 50 31 87)(27 49 32 86)(28 48 33 85)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 99)(14 100)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 50)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 71)

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

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

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

Dic74D4 is a maximal subgroup of
C24.24D14  C24.27D14  C24.31D14  C42.188D14  C42.91D14  C4210D14  C42.96D14  C4×D42D7  C42.104D14  C4×D4×D7  C4211D14  C42.108D14  Dic1423D4  C42.119D14  C24.56D14  C243D14  C24.33D14  C24.35D14  C28⋊(C4○D4)  Dic1420D4  C14.342+ 1+4  C4⋊C421D14  C14.402+ 1+4  C14.422+ 1+4  C14.432+ 1+4  C14.442+ 1+4  C22⋊Q825D7  C4⋊C426D14  Dic1421D4  C14.1182+ 1+4  C14.522+ 1+4  C14.562+ 1+4  C14.572+ 1+4  C4⋊C4.197D14  C14.1212+ 1+4  C14.822- 1+4  C4⋊C428D14  C14.1222+ 1+4  C14.832- 1+4  C14.642+ 1+4  C14.662+ 1+4  C14.672+ 1+4  C14.852- 1+4  C42.233D14  C42.137D14  C42.138D14  Dic1410D4  C4220D14  C4221D14  C42.234D14  C42.143D14  C42.160D14  C42.189D14  C42.161D14  C42.162D14  C42.163D14  C42.164D14
Dic74D4 is a maximal quotient of
C14.(C4×Q8)  Dic7⋊C42  Dic7⋊C4⋊C4  C14.(C4×D4)  D14⋊C42  D14⋊C4⋊C4  D14⋊C45C4  C2.(C4×D28)  C7⋊D4⋊C8  D142M4(2)  Dic7⋊M4(2)  C7⋊C826D4  Dic74D8  D4.D7⋊C4  Dic76SD16  D4⋊D7⋊C4  Dic77SD16  C7⋊Q16⋊C4  Dic74Q16  Q8⋊D7⋊C4  M4(2).22D14  C42.196D14  C22⋊C4×Dic7  C24.3D14  C24.4D14  C24.46D14  C24.12D14  C24.13D14  C23.45D28

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L7A7B7C14A···14I14J···14O28A···28L
order1222222244444444444477714···1414···1428···28
size111122141422227777141414142222···24···44···4

50 irreducible representations

dim11111111122222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D7C4○D4D14D14C4×D7D4×D7D42D7
kernelDic74D4C4×Dic7Dic7⋊C4D14⋊C4C7×C22⋊C4C2×C4×D7C22×Dic7C2×C7⋊D4C7⋊D4Dic7C22⋊C4C14C2×C4C23C22C2C2
# reps111111118232631233

Matrix representation of Dic74D4 in GL4(𝔽29) generated by

12000
12100
00280
00028
,
0700
4000
00170
00017
,
02600
19000
00128
00228
,
28000
02800
0010
00228
G:=sub<GL(4,GF(29))| [1,1,0,0,20,21,0,0,0,0,28,0,0,0,0,28],[0,4,0,0,7,0,0,0,0,0,17,0,0,0,0,17],[0,19,0,0,26,0,0,0,0,0,1,2,0,0,28,28],[28,0,0,0,0,28,0,0,0,0,1,2,0,0,0,28] >;

Dic74D4 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_4D_4
% in TeX

G:=Group("Dic7:4D4");
// GroupNames label

G:=SmallGroup(224,76);
// by ID

G=gap.SmallGroup(224,76);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,188,50,6917]);
// Polycyclic

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

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