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G = M5(2)⋊D7order 448 = 26·7

3rd semidirect product of M5(2) and D7 acting via D7/C7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C56.46D4, C8.25D28, M5(2)⋊3D7, C28.6M4(2), (C2×Dic7).C8, (C22×D7).C8, C71(C23.C8), C22.5(C8×D7), (C2×C8).152D14, C8.46(C7⋊D4), (C7×M5(2))⋊7C2, C28.C810C2, C2.10(D14⋊C8), C4.42(D14⋊C4), C4.10(C8⋊D7), C14.9(C22⋊C8), C28.57(C22⋊C4), (C2×C56).220C22, (C2×C7⋊C8).2C4, (C2×C4×D7).1C4, (C2×C14).3(C2×C8), (C2×C28).52(C2×C4), (C2×C4).137(C4×D7), (C2×C8⋊D7).10C2, SmallGroup(448,71)

Series: Derived Chief Lower central Upper central

C1C2×C14 — M5(2)⋊D7
C1C7C14C28C56C2×C56C2×C8⋊D7 — M5(2)⋊D7
C7C14C2×C14 — M5(2)⋊D7
C1C4C2×C8M5(2)

Generators and relations for M5(2)⋊D7
 G = < a,b,c,d | a16=b2=c7=d2=1, bab=a9, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 276 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C16, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, D14, C2×C14, M5(2), M5(2), C2×M4(2), C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C23.C8, C7⋊C16, C112, C8⋊D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C28.C8, C7×M5(2), C2×C8⋊D7, M5(2)⋊D7
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D14, C22⋊C8, C4×D7, D28, C7⋊D4, C23.C8, C8×D7, C8⋊D7, D14⋊C4, D14⋊C8, M5(2)⋊D7

Smallest permutation representation of M5(2)⋊D7
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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)
(1 111 47 74 23 96 58)(2 112 48 75 24 81 59)(3 97 33 76 25 82 60)(4 98 34 77 26 83 61)(5 99 35 78 27 84 62)(6 100 36 79 28 85 63)(7 101 37 80 29 86 64)(8 102 38 65 30 87 49)(9 103 39 66 31 88 50)(10 104 40 67 32 89 51)(11 105 41 68 17 90 52)(12 106 42 69 18 91 53)(13 107 43 70 19 92 54)(14 108 44 71 20 93 55)(15 109 45 72 21 94 56)(16 110 46 73 22 95 57)
(1 58)(2 51)(3 52)(4 61)(5 62)(6 55)(7 56)(8 49)(9 50)(10 59)(11 60)(12 53)(13 54)(14 63)(15 64)(16 57)(17 33)(18 42)(19 43)(20 36)(21 37)(22 46)(23 47)(24 40)(25 41)(26 34)(27 35)(28 44)(29 45)(30 38)(31 39)(32 48)(67 75)(68 76)(71 79)(72 80)(81 104)(82 105)(83 98)(84 99)(85 108)(86 109)(87 102)(88 103)(89 112)(90 97)(91 106)(92 107)(93 100)(94 101)(95 110)(96 111)

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

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

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

82 conjugacy classes

class 1 2A2B2C4A4B4C4D7A7B7C8A8B8C8D8E8F14A14B14C14D14E14F16A16B16C16D16E16F16G16H28A···28F28G28H28I56A···56L56M···56R112A···112X
order12224444777888888141414141414161616161616161628···2828282856···5656···56112···112
size1122811228222222228282224444444282828282···24442···24···44···4

82 irreducible representations

dim1111111122222222244
type++++++++
imageC1C2C2C2C4C4C8C8D4D7M4(2)D14D28C7⋊D4C4×D7C8⋊D7C8×D7C23.C8M5(2)⋊D7
kernelM5(2)⋊D7C28.C8C7×M5(2)C2×C8⋊D7C2×C7⋊C8C2×C4×D7C2×Dic7C22×D7C56M5(2)C28C2×C8C8C8C2×C4C4C22C7C1
# reps1111224423236661212212

Matrix representation of M5(2)⋊D7 in GL4(𝔽113) generated by

9808071
098420
97107150
9821015
,
1000
0100
0331120
80790112
,
0100
112900
00341
005388
,
0100
1000
0088112
005925
G:=sub<GL(4,GF(113))| [98,0,97,98,0,98,107,21,80,42,15,0,71,0,0,15],[1,0,0,80,0,1,33,79,0,0,112,0,0,0,0,112],[0,112,0,0,1,9,0,0,0,0,34,53,0,0,1,88],[0,1,0,0,1,0,0,0,0,0,88,59,0,0,112,25] >;

M5(2)⋊D7 in GAP, Magma, Sage, TeX

M_5(2)\rtimes D_7
% in TeX

G:=Group("M5(2):D7");
// GroupNames label

G:=SmallGroup(448,71);
// by ID

G=gap.SmallGroup(448,71);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,100,570,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^16=b^2=c^7=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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