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G = D7×C2×C8order 224 = 25·7

Direct product of C2×C8 and D7

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×C2×C8, C5610C22, C28.35C23, C141(C2×C8), (C2×C56)⋊8C2, C71(C22×C8), C7⋊C813C22, (C4×D7).5C4, C4.23(C4×D7), C28.26(C2×C4), D14.9(C2×C4), (C2×C4).97D14, (C2×Dic7).8C4, (C22×D7).5C4, C4.35(C22×D7), C22.13(C4×D7), C14.12(C22×C4), Dic7.10(C2×C4), (C4×D7).17C22, (C2×C28).110C22, (C2×C7⋊C8)⋊13C2, C2.2(C2×C4×D7), (C2×C4×D7).12C2, (C2×C14).14(C2×C4), SmallGroup(224,94)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C2×C8
C1C7C14C28C4×D7C2×C4×D7 — D7×C2×C8
C7 — D7×C2×C8
C1C2×C8

Generators and relations for D7×C2×C8
 G = < a,b,c,d | a2=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 238 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, C2×C14, C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, D7×C2×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, D14, C22×C8, C4×D7, C22×D7, C8×D7, C2×C4×D7, D7×C2×C8

Smallest permutation representation of D7×C2×C8
On 112 points
Generators in S112
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 101)(18 102)(19 103)(20 104)(21 97)(22 98)(23 99)(24 100)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)
(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 83 94 103 64 68 79)(2 84 95 104 57 69 80)(3 85 96 97 58 70 73)(4 86 89 98 59 71 74)(5 87 90 99 60 72 75)(6 88 91 100 61 65 76)(7 81 92 101 62 66 77)(8 82 93 102 63 67 78)(9 18 29 33 44 107 56)(10 19 30 34 45 108 49)(11 20 31 35 46 109 50)(12 21 32 36 47 110 51)(13 22 25 37 48 111 52)(14 23 26 38 41 112 53)(15 24 27 39 42 105 54)(16 17 28 40 43 106 55)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 73)(8 74)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 21)(18 22)(19 23)(20 24)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(41 108)(42 109)(43 110)(44 111)(45 112)(46 105)(47 106)(48 107)(57 91)(58 92)(59 93)(60 94)(61 95)(62 96)(63 89)(64 90)(65 84)(66 85)(67 86)(68 87)(69 88)(70 81)(71 82)(72 83)(97 101)(98 102)(99 103)(100 104)

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

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

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

D7×C2×C8 is a maximal subgroup of
D14⋊C16  D14.C42  C89D28  D14.4C42  C7⋊D4⋊C8  D14⋊C8⋊C2  D142M4(2)  D42D7⋊C4  D14⋊D8  D14⋊SD16  Q82D7⋊C4  D142SD16  D14⋊Q16  D28⋊C8  D143M4(2)  C42.30D14  (C8×D7)⋊C4  C88D28  C8.27(C4×D7)  C87D28  D142Q16  C56⋊D4  C566D4  C5614D4  D143Q16
D7×C2×C8 is a maximal quotient of
C42.282D14  Dic7.5M4(2)  C7⋊D4⋊C8  Dic14⋊C8  C42.200D14  D28⋊C8  D28.4C8  C16.12D14

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H7A7B7C8A···8H8I···8P14A···14I28A···28L56A···56X
order12222222444444447778···88···814···1428···2856···56
size11117777111177772221···17···72···22···22···2

80 irreducible representations

dim111111111222222
type++++++++
imageC1C2C2C2C2C4C4C4C8D7D14D14C4×D7C4×D7C8×D7
kernelD7×C2×C8C8×D7C2×C7⋊C8C2×C56C2×C4×D7C4×D7C2×Dic7C22×D7D14C2×C8C8C2×C4C4C22C2
# reps14111422163636624

Matrix representation of D7×C2×C8 in GL3(𝔽113) generated by

11200
010
001
,
11200
0690
0069
,
100
01121
03280
,
11200
010
081112
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[112,0,0,0,69,0,0,0,69],[1,0,0,0,112,32,0,1,80],[112,0,0,0,1,81,0,0,112] >;

D7×C2×C8 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_8
% in TeX

G:=Group("D7xC2xC8");
// GroupNames label

G:=SmallGroup(224,94);
// by ID

G=gap.SmallGroup(224,94);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,50,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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