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## G = M4(2).Dic7order 448 = 26·7

### 1st non-split extension by M4(2) of Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — M4(2).Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C2×C4.Dic7 — M4(2).Dic7
 Lower central C7 — C14 — C28 — M4(2).Dic7
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2).Dic7
G = < a,b,c,d | a8=b2=1, c14=a4, d2=a4c7, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c13 >

Subgroups: 292 in 102 conjugacy classes, 67 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C8.C4, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×C28, C2×C28, C22×C14, M4(2).C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C22×C28, C56.C4, C2×C4.Dic7, C14×M4(2), M4(2).Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic7, D14, C2×C4⋊C4, Dic14, D28, C2×Dic7, C22×D7, M4(2).C4, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C2×C4⋊Dic7, M4(2).Dic7

Smallest permutation representation of M4(2).Dic7
On 112 points
Generators in S112
(1 35 22 56 15 49 8 42)(2 36 23 29 16 50 9 43)(3 37 24 30 17 51 10 44)(4 38 25 31 18 52 11 45)(5 39 26 32 19 53 12 46)(6 40 27 33 20 54 13 47)(7 41 28 34 21 55 14 48)(57 85 64 92 71 99 78 106)(58 86 65 93 72 100 79 107)(59 87 66 94 73 101 80 108)(60 88 67 95 74 102 81 109)(61 89 68 96 75 103 82 110)(62 90 69 97 76 104 83 111)(63 91 70 98 77 105 84 112)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 57 22 78 15 71 8 64)(2 70 23 63 16 84 9 77)(3 83 24 76 17 69 10 62)(4 68 25 61 18 82 11 75)(5 81 26 74 19 67 12 60)(6 66 27 59 20 80 13 73)(7 79 28 72 21 65 14 58)(29 112 50 105 43 98 36 91)(30 97 51 90 44 111 37 104)(31 110 52 103 45 96 38 89)(32 95 53 88 46 109 39 102)(33 108 54 101 47 94 40 87)(34 93 55 86 48 107 41 100)(35 106 56 99 49 92 42 85)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E ··· 8L 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 1 1 2 2 2 2 2 2 4 4 4 4 28 ··· 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + + - + - + - image C1 C2 C2 C2 C4 D4 Q8 Q8 D7 D14 Dic7 D14 Dic14 D28 Dic14 M4(2).C4 M4(2).Dic7 kernel M4(2).Dic7 C56.C4 C2×C4.Dic7 C14×M4(2) C7×M4(2) C2×C28 C2×C28 C22×C14 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C2×C4 C23 C7 C1 # reps 1 4 2 1 8 2 1 1 3 6 12 3 6 12 6 2 12

Matrix representation of M4(2).Dic7 in GL4(𝔽113) generated by

 0 1 0 0 98 0 0 0 0 0 0 15 0 0 1 0
,
 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 112
,
 60 0 0 0 0 60 0 0 0 0 32 0 0 0 0 32
,
 0 0 1 0 0 0 0 1 98 0 0 0 0 98 0 0
G:=sub<GL(4,GF(113))| [0,98,0,0,1,0,0,0,0,0,0,1,0,0,15,0],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,112],[60,0,0,0,0,60,0,0,0,0,32,0,0,0,0,32],[0,0,98,0,0,0,0,98,1,0,0,0,0,1,0,0] >;

M4(2).Dic7 in GAP, Magma, Sage, TeX

M_4(2).{\rm Dic}_7
% in TeX

G:=Group("M4(2).Dic7");
// GroupNames label

G:=SmallGroup(448,659);
// by ID

G=gap.SmallGroup(448,659);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,100,136,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^14=a^4,d^2=a^4*c^7,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^13>;
// generators/relations

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