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G = C4×Dic7order 112 = 24·7

Direct product of C4 and Dic7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×Dic7, C7⋊C42, C282C4, C22.3D14, (C2×C4).6D7, C2.2(C4×D7), C14.3(C2×C4), (C2×C28).7C2, C2.2(C2×Dic7), (C2×C14).3C22, (C2×Dic7).4C2, SmallGroup(112,10)

Series: Derived Chief Lower central Upper central

C1C7 — C4×Dic7
C1C7C14C2×C14C2×Dic7 — C4×Dic7
C7 — C4×Dic7
C1C2×C4

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

7C4
7C4
7C4
7C4
7C2×C4
7C2×C4
7C42

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

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

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

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

C4×Dic7 is a maximal subgroup of
Dic7⋊C8  C56⋊C4  D284C4  D42Dic7  D7×C42  C42⋊D7  C23.11D14  C23.D14  Dic74D4  Dic7.D4  Dic73Q8  C28⋊Q8  Dic7.Q8  C28.3Q8  C4⋊C47D7  D28⋊C4  C4⋊C4⋊D7  C23.21D14  C28.17D4  C28⋊D4  Dic7⋊Q8  C28.23D4
C4×Dic7 is a maximal quotient of
C42.D7  C56⋊C4  C14.C42

40 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4L7A7B7C14A···14I28A···28L
order122244444···477714···1428···28
size111111117···72222···22···2

40 irreducible representations

dim111112222
type++++-+
imageC1C2C2C4C4D7Dic7D14C4×D7
kernelC4×Dic7C2×Dic7C2×C28Dic7C28C2×C4C4C22C2
# reps1218436312

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

100
0120
0012
,
2800
001
0283
,
1200
052
01724
G:=sub<GL(3,GF(29))| [1,0,0,0,12,0,0,0,12],[28,0,0,0,0,28,0,1,3],[12,0,0,0,5,17,0,2,24] >;

C4×Dic7 in GAP, Magma, Sage, TeX

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

G:=Group("C4xDic7");
// GroupNames label

G:=SmallGroup(112,10);
// by ID

G=gap.SmallGroup(112,10);
# by ID

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

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

Export

Subgroup lattice of C4×Dic7 in TeX

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