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## G = C4×Dic7order 112 = 24·7

### Direct product of C4 and Dic7

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

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

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

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 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 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 1 1 1 1 7 ··· 7 2 2 2 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 D7 Dic7 D14 C4×D7 kernel C4×Dic7 C2×Dic7 C2×C28 Dic7 C28 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 3 6 3 12

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

 1 0 0 0 12 0 0 0 12
,
 28 0 0 0 0 1 0 28 3
,
 12 0 0 0 5 2 0 17 24
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

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