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G = D56:7C4order 448 = 26·7

7th semidirect product of D56 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D56:7C4, Dic28:7C4, M4(2).27D14, C7:3(C8oD8), C7:C8.38D4, C56:C2:6C4, C8.16(C4xD7), C56.35(C2xC4), (C8xDic7):1C2, C4.213(D4xD7), C14.56(C4xD4), C8.C4:8D7, D28:4C4:8C2, D28.C4:11C2, D28.11(C2xC4), (C2xC8).252D14, C28.372(C2xD4), D56:7C2.5C2, (C2xC56).42C22, C28.55(C22xC4), (C2xC28).311C23, Dic14.11(C2xC4), C4oD28.18C22, C2.16(D28:C4), C22.2(Q8:2D7), (C4xDic7).235C22, (C7xM4(2)).21C22, C4.47(C2xC4xD7), (C7xC8.C4):5C2, (C2xC14).2(C4oD4), (C2xC7:C8).238C22, (C2xC4).414(C22xD7), SmallGroup(448,429)

Series: Derived Chief Lower central Upper central

C1C28 — D56:7C4
C1C7C14C28C2xC28C4oD28D56:7C2 — D56:7C4
C7C14C28 — D56:7C4
C1C4C2xC4C8.C4

Generators and relations for D56:7C4
 G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a41, cbc-1=a54b >

Subgroups: 508 in 106 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, Q8, D7, C14, C14, C42, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C4oD4, Dic7, C28, D14, C2xC14, C4xC8, C4wrC2, C8.C4, C8oD4, C4oD8, C7:C8, C56, C56, Dic14, C4xD7, D28, C2xDic7, C7:D4, C2xC28, C8oD8, C8xD7, C8:D7, C56:C2, D56, Dic28, C2xC7:C8, C4xDic7, C2xC56, C7xM4(2), C4oD28, C8xDic7, D28:4C4, C7xC8.C4, D56:7C2, D28.C4, D56:7C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D7, C22xC4, C2xD4, C4oD4, D14, C4xD4, C4xD7, C22xD7, C8oD8, C2xC4xD7, D4xD7, Q8:2D7, D28:C4, D56:7C4

Smallest permutation representation of D56:7C4
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 61)(2 60)(3 59)(4 58)(5 57)(6 112)(7 111)(8 110)(9 109)(10 108)(11 107)(12 106)(13 105)(14 104)(15 103)(16 102)(17 101)(18 100)(19 99)(20 98)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 90)(29 89)(30 88)(31 87)(32 86)(33 85)(34 84)(35 83)(36 82)(37 81)(38 80)(39 79)(40 78)(41 77)(42 76)(43 75)(44 74)(45 73)(46 72)(47 71)(48 70)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)
(1 29)(2 14)(3 55)(4 40)(5 25)(6 10)(7 51)(8 36)(9 21)(11 47)(12 32)(13 17)(15 43)(16 28)(18 54)(19 39)(20 24)(22 50)(23 35)(26 46)(27 31)(30 42)(33 53)(34 38)(37 49)(41 45)(44 56)(48 52)(57 67 85 95)(58 108 86 80)(59 93 87 65)(60 78 88 106)(61 63 89 91)(62 104 90 76)(64 74 92 102)(66 100 94 72)(68 70 96 98)(69 111 97 83)(71 81 99 109)(73 107 101 79)(75 77 103 105)(82 84 110 112)

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

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

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

70 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I7A7B7C8A8B8C8D8E8F8G8H8I8J8K8L8M8N14A14B14C14D14E14F28A···28F28G28H28I56A···56L56M···56X
order122224444444447778888888888888814141414141428···2828282856···5656···56
size112282811214141414282822222224444777714142224442···24444···48···8

70 irreducible representations

dim1111111112222222444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4D4D7C4oD4D14D14C4xD7C8oD8D4xD7Q8:2D7D56:7C4
kernelD56:7C4C8xDic7D28:4C4C7xC8.C4D56:7C2D28.C4C56:C2D56Dic28C7:C8C8.C4C2xC14C2xC8M4(2)C8C7C4C22C1
# reps112112422232361283312

Matrix representation of D56:7C4 in GL4(F113) generated by

7911200
1000
00180
00044
,
7911200
253400
00098
00150
,
1000
7911200
001120
00015
G:=sub<GL(4,GF(113))| [79,1,0,0,112,0,0,0,0,0,18,0,0,0,0,44],[79,25,0,0,112,34,0,0,0,0,0,15,0,0,98,0],[1,79,0,0,0,112,0,0,0,0,112,0,0,0,0,15] >;

D56:7C4 in GAP, Magma, Sage, TeX

D_{56}\rtimes_7C_4
% in TeX

G:=Group("D56:7C4");
// GroupNames label

G:=SmallGroup(448,429);
// by ID

G=gap.SmallGroup(448,429);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,555,58,136,1684,438,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^54*b>;
// generators/relations

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