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G = Dic7⋊C4order 112 = 24·7

The semidirect product of Dic7 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic7⋊C4, C14.5D4, C14.1Q8, C2.1Dic14, C22.4D14, C71(C4⋊C4), C2.4(C4×D7), (C2×C4).1D7, C14.4(C2×C4), (C2×C28).1C2, C2.1(C7⋊D4), (C2×C14).4C22, (C2×Dic7).1C2, SmallGroup(112,11)

Series: Derived Chief Lower central Upper central

C1C14 — Dic7⋊C4
C1C7C14C2×C14C2×Dic7 — Dic7⋊C4
C7C14 — Dic7⋊C4
C1C22C2×C4

Generators and relations for Dic7⋊C4
 G = < a,b,c | a14=c4=1, b2=a7, bab-1=a-1, ac=ca, cbc-1=a7b >

2C4
7C4
7C4
14C4
7C2×C4
7C2×C4
2C28
2Dic7
7C4⋊C4

Smallest permutation representation of Dic7⋊C4
Regular action 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 71 8 78)(2 84 9 77)(3 83 10 76)(4 82 11 75)(5 81 12 74)(6 80 13 73)(7 79 14 72)(15 70 22 63)(16 69 23 62)(17 68 24 61)(18 67 25 60)(19 66 26 59)(20 65 27 58)(21 64 28 57)(29 109 36 102)(30 108 37 101)(31 107 38 100)(32 106 39 99)(33 105 40 112)(34 104 41 111)(35 103 42 110)(43 95 50 88)(44 94 51 87)(45 93 52 86)(46 92 53 85)(47 91 54 98)(48 90 55 97)(49 89 56 96)
(1 46 28 32)(2 47 15 33)(3 48 16 34)(4 49 17 35)(5 50 18 36)(6 51 19 37)(7 52 20 38)(8 53 21 39)(9 54 22 40)(10 55 23 41)(11 56 24 42)(12 43 25 29)(13 44 26 30)(14 45 27 31)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)

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

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

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

Dic7⋊C4 is a maximal subgroup of
C4×Dic14  C28.6Q8  C42⋊D7  C422D7  C23.11D14  C22⋊Dic14  C23.D14  Dic74D4  D14.D4  D14⋊D4  Dic73Q8  C28⋊Q8  Dic7.Q8  C28.3Q8  D7×C4⋊C4  D14.5D4  D14⋊Q8  C4⋊C4⋊D7  C28.48D4  C4×C7⋊D4  C23.23D14  C23.18D14  Dic7⋊D4  Dic7⋊Q8  D143Q8  Dic7⋊C12  C42.Q8  Dic21⋊C4  C42.4Q8
Dic7⋊C4 is a maximal quotient of
C28.Q8  C4.Dic14  Dic7⋊C8  C28.53D4  C14.C42  C42.Q8  Dic21⋊C4  C42.4Q8

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F7A7B7C14A···14I28A···28L
order122244444477714···1428···28
size111122141414142222···22···2

34 irreducible representations

dim11112222222
type++++-++-
imageC1C2C2C4D4Q8D7D14Dic14C4×D7C7⋊D4
kernelDic7⋊C4C2×Dic7C2×C28Dic7C14C14C2×C4C22C2C2C2
# reps12141133666

Matrix representation of Dic7⋊C4 in GL4(𝔽29) generated by

0100
28700
00828
00218
,
71000
12200
001617
001913
,
17000
01700
002611
0073
G:=sub<GL(4,GF(29))| [0,28,0,0,1,7,0,0,0,0,8,2,0,0,28,18],[7,1,0,0,10,22,0,0,0,0,16,19,0,0,17,13],[17,0,0,0,0,17,0,0,0,0,26,7,0,0,11,3] >;

Dic7⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes C_4
% in TeX

G:=Group("Dic7:C4");
// GroupNames label

G:=SmallGroup(112,11);
// by ID

G=gap.SmallGroup(112,11);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,40,101,26,2404]);
// Polycyclic

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

Export

Subgroup lattice of Dic7⋊C4 in TeX

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