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G = M4(2).19D14order 448 = 26·7

2nd non-split extension by M4(2) of D14 acting via D14/D7=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).19D14, (C4×D7).35D4, C28.90(C2×D4), C4.147(D4×D7), C4.D46D7, C23.7(C4×D7), C28.D44C2, (D7×M4(2))⋊6C2, (C2×C28).2C23, (C2×D4).122D14, C4.12D285C2, D14.2(C22⋊C4), (D4×C14).12C22, (C22×Dic7).4C4, C4.Dic7.1C22, (C7×M4(2)).9C22, Dic7.16(C22⋊C4), (C2×Dic14).44C22, C71(M4(2).8C22), (C2×C7⋊D4).2C4, (C2×C4×D7).2C22, C22.15(C2×C4×D7), (C7×C4.D4)⋊6C2, C2.14(D7×C22⋊C4), (C2×C4).2(C22×D7), (C2×D42D7).2C2, C14.13(C2×C22⋊C4), (C22×C14).7(C2×C4), (C2×C14).9(C22×C4), (C2×Dic7).2(C2×C4), (C22×D7).15(C2×C4), SmallGroup(448,279)

Series: Derived Chief Lower central Upper central

C1C2×C14 — M4(2).19D14
C1C7C14C28C2×C28C2×C4×D7C2×D42D7 — M4(2).19D14
C7C14C2×C14 — M4(2).19D14
C1C2C2×C4C4.D4

Generators and relations for M4(2).19D14
 G = < a,b,c,d | a8=b2=c14=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >

Subgroups: 716 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C4.D4, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, M4(2).8C22, C8×D7, C8⋊D7, C4.Dic7, C7×M4(2), C2×Dic14, C2×C4×D7, D42D7, C22×Dic7, C2×C7⋊D4, D4×C14, C4.12D28, C28.D4, C7×C4.D4, D7×M4(2), C2×D42D7, M4(2).19D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, C22×D7, M4(2).8C22, C2×C4×D7, D4×D7, D7×C22⋊C4, M4(2).19D14

Smallest permutation representation of M4(2).19D14
On 112 points
Generators in S112
(1 98 48 19 70 77 111 37)(2 78 112 20 57 85 49 38)(3 86 50 21 58 79 99 39)(4 80 100 22 59 87 51 40)(5 88 52 23 60 81 101 41)(6 82 102 24 61 89 53 42)(7 90 54 25 62 83 103 29)(8 84 104 26 63 91 55 30)(9 92 56 27 64 71 105 31)(10 72 106 28 65 93 43 32)(11 94 44 15 66 73 107 33)(12 74 108 16 67 95 45 34)(13 96 46 17 68 75 109 35)(14 76 110 18 69 97 47 36)
(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 29)(26 30)(27 31)(28 32)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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 14 70 69)(2 68 57 13)(3 12 58 67)(4 66 59 11)(5 10 60 65)(6 64 61 9)(7 8 62 63)(15 40 33 22)(16 21 34 39)(17 38 35 20)(18 19 36 37)(23 32 41 28)(24 27 42 31)(25 30 29 26)(43 101 106 52)(44 51 107 100)(45 99 108 50)(46 49 109 112)(47 111 110 48)(53 105 102 56)(54 55 103 104)(71 89 92 82)(72 81 93 88)(73 87 94 80)(74 79 95 86)(75 85 96 78)(76 77 97 98)(83 91 90 84)

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

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

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

55 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G7A7B7C8A8B8C8D8E8F8G8H14A14B14C14D14E14F14G···14L28A···28F56A···56L
order122222244444447778888888814141414141414···1428···2856···56
size11244141422771428282224444282828282224448···84···48···8

55 irreducible representations

dim1111111122222448
type+++++++++++-
imageC1C2C2C2C2C2C4C4D4D7D14D14C4×D7M4(2).8C22D4×D7M4(2).19D14
kernelM4(2).19D14C4.12D28C28.D4C7×C4.D4D7×M4(2)C2×D42D7C22×Dic7C2×C7⋊D4C4×D7C4.D4M4(2)C2×D4C23C7C4C1
# reps12112144436312263

Matrix representation of M4(2).19D14 in GL8(𝔽113)

150000000
098000000
001500000
000980000
000096112112111
000000980
00000100
00002481617
,
1120000000
0112000000
0011200000
0001120000
00001000
0000011200
00000010
0000960112112
,
0800800000
8008000000
033090000
330900000
00000010
000096112112111
00001000
0000801051
,
0330330000
3303300000
01040800000
10408000000
0000001120
000017112
00001000
00001041128112

G:=sub<GL(8,GF(113))| [15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,98,0,0,0,0,0,0,0,0,96,0,0,24,0,0,0,0,112,0,1,8,0,0,0,0,112,98,0,16,0,0,0,0,111,0,0,17],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,96,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,112,0,0,0,0,0,0,0,112],[0,80,0,33,0,0,0,0,80,0,33,0,0,0,0,0,0,80,0,9,0,0,0,0,80,0,9,0,0,0,0,0,0,0,0,0,0,96,1,8,0,0,0,0,0,112,0,0,0,0,0,0,1,112,0,105,0,0,0,0,0,111,0,1],[0,33,0,104,0,0,0,0,33,0,104,0,0,0,0,0,0,33,0,80,0,0,0,0,33,0,80,0,0,0,0,0,0,0,0,0,0,17,1,104,0,0,0,0,0,1,0,112,0,0,0,0,112,1,0,8,0,0,0,0,0,2,0,112] >;

M4(2).19D14 in GAP, Magma, Sage, TeX

M_4(2)._{19}D_{14}
% in TeX

G:=Group("M4(2).19D14");
// GroupNames label

G:=SmallGroup(448,279);
// by ID

G=gap.SmallGroup(448,279);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,219,58,570,136,438,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^4,b*a*b=a^5,c*a*c^-1=a*b,d*a*d^-1=a^5*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations

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