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G = C4⋊Dic7order 112 = 24·7

The semidirect product of C4 and Dic7 acting via Dic7/C14=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic7, C281C4, C14.4D4, C2.1D28, C14.2Q8, C2.2Dic14, C22.5D14, C72(C4⋊C4), (C2×C4).3D7, C14.8(C2×C4), (C2×C28).3C2, C2.4(C2×Dic7), (C2×C14).5C22, (C2×Dic7).2C2, SmallGroup(112,12)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4⋊Dic7
 G = < a,b,c | a4=b14=1, c2=b7, ab=ba, cac-1=a-1, cbc-1=b-1 >

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

Smallest permutation representation of C4⋊Dic7
Regular action on 112 points
Generators in S112
(1 106 91 48)(2 107 92 49)(3 108 93 50)(4 109 94 51)(5 110 95 52)(6 111 96 53)(7 112 97 54)(8 99 98 55)(9 100 85 56)(10 101 86 43)(11 102 87 44)(12 103 88 45)(13 104 89 46)(14 105 90 47)(15 34 67 72)(16 35 68 73)(17 36 69 74)(18 37 70 75)(19 38 57 76)(20 39 58 77)(21 40 59 78)(22 41 60 79)(23 42 61 80)(24 29 62 81)(25 30 63 82)(26 31 64 83)(27 32 65 84)(28 33 66 71)
(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 27 8 20)(2 26 9 19)(3 25 10 18)(4 24 11 17)(5 23 12 16)(6 22 13 15)(7 21 14 28)(29 44 36 51)(30 43 37 50)(31 56 38 49)(32 55 39 48)(33 54 40 47)(34 53 41 46)(35 52 42 45)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 112 78 105)(72 111 79 104)(73 110 80 103)(74 109 81 102)(75 108 82 101)(76 107 83 100)(77 106 84 99)

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

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

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

C4⋊Dic7 is a maximal subgroup of
C28.Q8  C4.Dic14  C28.44D4  C8⋊Dic7  C561C4  C2.D56  D4⋊Dic7  Q8⋊Dic7  C4×Dic14  C282Q8  C28.6Q8  C4×D28  C22⋊Dic14  C23.D14  D14.D4  C22.D28  C28⋊Q8  Dic7.Q8  C28.3Q8  D7×C4⋊C4  C4⋊C47D7  D142Q8  C4⋊C4⋊D7  C28.48D4  C23.21D14  C287D4  D4×Dic7  C282D4  Q8×Dic7  D143Q8  C28⋊C12  C14.Dic6  C84⋊C4
C4⋊Dic7 is a maximal quotient of
C28⋊C8  C8⋊Dic7  C561C4  C56.C4  C14.C42  C14.Dic6  C84⋊C4

34 conjugacy classes

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

34 irreducible representations

dim11112222222
type++++-+-+-+
imageC1C2C2C4D4Q8D7Dic7D14Dic14D28
kernelC4⋊Dic7C2×Dic7C2×C28C28C14C14C2×C4C4C22C2C2
# reps12141136366

Matrix representation of C4⋊Dic7 in GL3(𝔽29) generated by

2800
026
0427
,
2800
0281
0208
,
1200
02425
065
G:=sub<GL(3,GF(29))| [28,0,0,0,2,4,0,6,27],[28,0,0,0,28,20,0,1,8],[12,0,0,0,24,6,0,25,5] >;

C4⋊Dic7 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(112,12);
// by ID

G=gap.SmallGroup(112,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,20,101,46,2404]);
// Polycyclic

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

Export

Subgroup lattice of C4⋊Dic7 in TeX

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