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G = D4xDic7order 224 = 25·7

Direct product of D4 and Dic7

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

Aliases: D4xDic7, C23.17D14, C7:5(C4xD4), C28:3(C2xC4), (C7xD4):3C4, C2.5(D4xD7), (C2xD4).7D7, C4:1(C2xDic7), C4:Dic7:13C2, (C4xDic7):4C2, (D4xC14).4C2, C14.37(C2xD4), (C2xC4).49D14, C23.D7:7C2, C22:1(C2xDic7), C14.28(C4oD4), C2.5(D4:2D7), C14.25(C22xC4), (C2xC14).49C23, (C2xC28).32C22, (C22xDic7):4C2, C2.6(C22xDic7), C22.25(C22xD7), (C22xC14).17C22, (C2xDic7).37C22, (C2xC14):3(C2xC4), SmallGroup(224,129)

Series: Derived Chief Lower central Upper central

C1C14 — D4xDic7
C1C7C14C2xC14C2xDic7C22xDic7 — D4xDic7
C7C14 — D4xDic7
C1C22C2xD4

Generators and relations for D4xDic7
 G = < a,b,c,d | a4=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 286 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, D4, C23, C14, C14, C42, C22:C4, C4:C4, C22xC4, C2xD4, Dic7, Dic7, C28, C2xC14, C2xC14, C2xC14, C4xD4, C2xDic7, C2xDic7, C2xDic7, C2xC28, C7xD4, C22xC14, C4xDic7, C4:Dic7, C23.D7, C22xDic7, D4xC14, D4xDic7
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22xC4, C2xD4, C4oD4, Dic7, D14, C4xD4, C2xDic7, C22xD7, D4xD7, D4:2D7, C22xDic7, D4xDic7

Smallest permutation representation of D4xDic7
On 112 points
Generators in S112
(1 81 54 26)(2 82 55 27)(3 83 56 28)(4 84 43 15)(5 71 44 16)(6 72 45 17)(7 73 46 18)(8 74 47 19)(9 75 48 20)(10 76 49 21)(11 77 50 22)(12 78 51 23)(13 79 52 24)(14 80 53 25)(29 93 61 100)(30 94 62 101)(31 95 63 102)(32 96 64 103)(33 97 65 104)(34 98 66 105)(35 85 67 106)(36 86 68 107)(37 87 69 108)(38 88 70 109)(39 89 57 110)(40 90 58 111)(41 91 59 112)(42 92 60 99)
(1 26)(2 27)(3 28)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 99)(43 84)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)
(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 60 8 67)(2 59 9 66)(3 58 10 65)(4 57 11 64)(5 70 12 63)(6 69 13 62)(7 68 14 61)(15 89 22 96)(16 88 23 95)(17 87 24 94)(18 86 25 93)(19 85 26 92)(20 98 27 91)(21 97 28 90)(29 46 36 53)(30 45 37 52)(31 44 38 51)(32 43 39 50)(33 56 40 49)(34 55 41 48)(35 54 42 47)(71 109 78 102)(72 108 79 101)(73 107 80 100)(74 106 81 99)(75 105 82 112)(76 104 83 111)(77 103 84 110)

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

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

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

D4xDic7 is a maximal subgroup of
Dic7:4D8  D4.D7:C4  Dic7:6SD16  Dic7.D8  D4:Dic14  D4.Dic14  D4.2Dic14  D4:D7:C4  Dic7:D8  D8:Dic7  (C2xD8).D7  Dic7:3SD16  SD16:Dic7  (C7xD4).D4  C4xD4:2D7  D4:5Dic14  D4:6Dic14  C4xD4xD7  C42:11D14  C42.108D14  C24.56D14  C24.32D14  C24.33D14  C24.35D14  C28:(C4oD4)  Dic14:19D4  C4:C4.178D14  C14.342+ 1+4  C14.352+ 1+4  C14.712- 1+4  C4:C4:21D14  C14.732- 1+4  C14.432+ 1+4  C14.452+ 1+4  C14.462+ 1+4  C14.1152+ 1+4  C14.472+ 1+4  C4:C4.197D14  C14.802- 1+4  C14.1222+ 1+4  C14.852- 1+4  C42.139D14  C42.234D14  C42.143D14  C42.144D14  C42.166D14  C42.238D14  Dic14:11D4  C42.168D14  C24.38D14  C24.42D14  C14.1042- 1+4  C14.1062- 1+4  C14.1452+ 1+4
D4xDic7 is a maximal quotient of
C24.47D14  C24.8D14  C4:C4:5Dic7  C4:(C4:Dic7)  C42.47D14  C28:3M4(2)  D8:Dic7  SD16:Dic7  Q16:Dic7  D8:5Dic7  D8:4Dic7  C24.18D14  C24.19D14

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L7A7B7C14A···14I14J···14U28A···28F
order122222224444444···477714···1414···1428···28
size1111222222777714···142222···24···44···4

50 irreducible representations

dim111111122222244
type+++++++++-++-
imageC1C2C2C2C2C2C4D4D7C4oD4D14Dic7D14D4xD7D4:2D7
kernelD4xDic7C4xDic7C4:Dic7C23.D7C22xDic7D4xC14C7xD4Dic7C2xD4C14C2xC4D4C23C2C2
# reps1112218232312633

Matrix representation of D4xDic7 in GL4(F29) generated by

1000
0100
0001
00280
,
1000
0100
0001
0010
,
262800
1000
00280
00028
,
01200
12000
00120
00012
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[26,1,0,0,28,0,0,0,0,0,28,0,0,0,0,28],[0,12,0,0,12,0,0,0,0,0,12,0,0,0,0,12] >;

D4xDic7 in GAP, Magma, Sage, TeX

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

G:=Group("D4xDic7");
// GroupNames label

G:=SmallGroup(224,129);
// by ID

G=gap.SmallGroup(224,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,188,6917]);
// Polycyclic

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

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