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G = C7×C4⋊C4order 112 = 24·7

Direct product of C7 and C4⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C4⋊C4, C4⋊C28, C283C4, C14.3Q8, C14.13D4, C2.(C7×Q8), C2.2(C7×D4), (C2×C4).1C14, C2.2(C2×C28), (C2×C28).2C2, C14.11(C2×C4), C22.3(C2×C14), (C2×C14).14C22, SmallGroup(112,21)

Series: Derived Chief Lower central Upper central

C1C2 — C7×C4⋊C4
C1C2C22C2×C14C2×C28 — C7×C4⋊C4
C1C2 — C7×C4⋊C4
C1C2×C14 — C7×C4⋊C4

Generators and relations for C7×C4⋊C4
 G = < a,b,c | a7=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C28
2C28

Smallest permutation representation of C7×C4⋊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 59 38 67)(2 60 39 68)(3 61 40 69)(4 62 41 70)(5 63 42 64)(6 57 36 65)(7 58 37 66)(8 81 27 105)(9 82 28 99)(10 83 22 100)(11 84 23 101)(12 78 24 102)(13 79 25 103)(14 80 26 104)(15 90 109 98)(16 91 110 92)(17 85 111 93)(18 86 112 94)(19 87 106 95)(20 88 107 96)(21 89 108 97)(29 72 44 55)(30 73 45 56)(31 74 46 50)(32 75 47 51)(33 76 48 52)(34 77 49 53)(35 71 43 54)
(1 95 46 102)(2 96 47 103)(3 97 48 104)(4 98 49 105)(5 92 43 99)(6 93 44 100)(7 94 45 101)(8 70 15 77)(9 64 16 71)(10 65 17 72)(11 66 18 73)(12 67 19 74)(13 68 20 75)(14 69 21 76)(22 57 111 55)(23 58 112 56)(24 59 106 50)(25 60 107 51)(26 61 108 52)(27 62 109 53)(28 63 110 54)(29 83 36 85)(30 84 37 86)(31 78 38 87)(32 79 39 88)(33 80 40 89)(34 81 41 90)(35 82 42 91)

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

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

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

C7×C4⋊C4 is a maximal subgroup of
C28.Q8  C4.Dic14  C14.D8  C14.Q16  Dic73Q8  C28⋊Q8  Dic7.Q8  C28.3Q8  C4⋊C47D7  D28⋊C4  D14.5D4  C4⋊D28  D14⋊Q8  D142Q8  C4⋊C4⋊D7  D4×C28  Q8×C28

70 conjugacy classes

class 1 2A2B2C4A···4F7A···7F14A···14R28A···28AJ
order12224···47···714···1428···28
size11112···21···11···12···2

70 irreducible representations

dim1111112222
type+++-
imageC1C2C4C7C14C28D4Q8C7×D4C7×Q8
kernelC7×C4⋊C4C2×C28C28C4⋊C4C2×C4C4C14C14C2C2
# reps134618241166

Matrix representation of C7×C4⋊C4 in GL3(𝔽29) generated by

100
0240
0024
,
2800
0028
010
,
1700
02716
0162
G:=sub<GL(3,GF(29))| [1,0,0,0,24,0,0,0,24],[28,0,0,0,0,1,0,28,0],[17,0,0,0,27,16,0,16,2] >;

C7×C4⋊C4 in GAP, Magma, Sage, TeX

C_7\times C_4\rtimes C_4
% in TeX

G:=Group("C7xC4:C4");
// GroupNames label

G:=SmallGroup(112,21);
// by ID

G=gap.SmallGroup(112,21);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,280,301,146]);
// Polycyclic

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

Export

Subgroup lattice of C7×C4⋊C4 in TeX

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