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G = M4(2).D14order 448 = 26·7

12nd non-split extension by M4(2) of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).12D14, C7⋊C8.30D4, C8⋊C222D7, (C7×D4).11D4, C4.178(D4×D7), (C7×Q8).11D4, D4⋊D144C2, C4○D4.24D14, (C2×D4).79D14, C28.195(C2×D4), C28.D49C2, C75(D4.4D4), D4.4(C7⋊D4), Q8.Dic75C2, Q8.4(C7⋊D4), C28.53D49C2, (C2×C28).14C23, C28.46D410C2, C14.124(C4⋊D4), (D4×C14).104C22, (C2×D28).129C22, C4.Dic7.24C22, C2.30(Dic7⋊D4), C22.13(D42D7), (C7×M4(2)).22C22, (C2×D4⋊D7)⋊22C2, (C7×C8⋊C22)⋊6C2, C4.51(C2×C7⋊D4), (C2×C7⋊C8).170C22, (C2×C4).14(C22×D7), (C2×C14).36(C4○D4), (C7×C4○D4).12C22, SmallGroup(448,733)

Series: Derived Chief Lower central Upper central

C1C2×C28 — M4(2).D14
C1C7C14C28C2×C28C2×D28D4⋊D14 — M4(2).D14
C7C14C2×C28 — M4(2).D14
C1C2C2×C4C8⋊C22

Generators and relations for M4(2).D14
 G = < a,b,c,d | a8=b2=c14=1, d2=a6, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 556 in 108 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, C28, C28, D14, C2×C14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C8⋊C22, C7⋊C8, C7⋊C8, C56, D28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, D4.4D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, D4⋊D7, Q8⋊D7, C7×M4(2), C7×D8, C7×SD16, C2×D28, D4×C14, C7×C4○D4, C28.53D4, C28.46D4, C28.D4, C2×D4⋊D7, Q8.Dic7, D4⋊D14, C7×C8⋊C22, M4(2).D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.4D4, D4×D7, D42D7, C2×C7⋊D4, Dic7⋊D4, M4(2).D14

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

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

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

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

49 conjugacy classes

class 1 2A2B2C2D2E4A4B4C7A7B7C8A8B8C8D8E8F8G14A14B14C14D14E14F14G···14O28A···28F28G28H28I56A···56F
order122222444777888888814141414141414···1428···2828282856···56
size112485622422281414282828562224448···84···48888···8

49 irreducible representations

dim1111111122222222224448
type+++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D7C4○D4D14D14D14C7⋊D4C7⋊D4D4.4D4D4×D7D42D7M4(2).D14
kernelM4(2).D14C28.53D4C28.46D4C28.D4C2×D4⋊D7Q8.Dic7D4⋊D14C7×C8⋊C22C7⋊C8C7×D4C7×Q8C8⋊C22C2×C14M4(2)C2×D4C4○D4D4Q8C7C4C22C1
# reps1111111121132333662333

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

236644960000
3911100840000
522357840000
383217220000
000083835162
00001515051
0000820150
000000980
,
10000000
01000000
00100000
00010000
0000112000
0000011200
0000262610
0000878701
,
285112990000
6188102120000
93842720000
877996680000
000087871110
00000011
00005511213100
000058210013
,
2091410000
5435101110000
29841710000
1025445170000
000026260111
00000011
000057110013
00005511113100

G:=sub<GL(8,GF(113))| [23,39,52,38,0,0,0,0,66,11,23,32,0,0,0,0,44,100,57,17,0,0,0,0,96,84,84,22,0,0,0,0,0,0,0,0,83,15,82,0,0,0,0,0,83,15,0,0,0,0,0,0,51,0,15,98,0,0,0,0,62,51,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,26,87,0,0,0,0,0,112,26,87,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,61,9,87,0,0,0,0,5,88,38,79,0,0,0,0,112,102,42,96,0,0,0,0,99,12,72,68,0,0,0,0,0,0,0,0,87,0,55,58,0,0,0,0,87,0,112,2,0,0,0,0,111,1,13,100,0,0,0,0,0,1,100,13],[20,54,2,102,0,0,0,0,9,35,98,54,0,0,0,0,14,101,41,45,0,0,0,0,1,11,71,17,0,0,0,0,0,0,0,0,26,0,57,55,0,0,0,0,26,0,1,111,0,0,0,0,0,1,100,13,0,0,0,0,111,1,13,100] >;

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

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

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

G:=SmallGroup(448,733);
// by ID

G=gap.SmallGroup(448,733);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,219,1123,297,136,1684,851,438,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^14=1,d^2=a^6,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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