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## G = Dic7⋊C4order 112 = 24·7

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

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

 Derived series C1 — C14 — Dic7⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — Dic7⋊C4
 Lower central C7 — C14 — Dic7⋊C4
 Upper central C1 — C22 — C2×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 >

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A ··· 14I 28A ··· 28L order 1 2 2 2 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 14 14 14 14 2 2 2 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + + - image C1 C2 C2 C4 D4 Q8 D7 D14 Dic14 C4×D7 C7⋊D4 kernel Dic7⋊C4 C2×Dic7 C2×C28 Dic7 C14 C14 C2×C4 C22 C2 C2 C2 # reps 1 2 1 4 1 1 3 3 6 6 6

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

 0 1 0 0 28 7 0 0 0 0 8 28 0 0 2 18
,
 7 10 0 0 1 22 0 0 0 0 16 17 0 0 19 13
,
 17 0 0 0 0 17 0 0 0 0 26 11 0 0 7 3
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

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