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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

 Derived series C1 — C2 — C7×C4⋊C4
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4
 Lower central C1 — C2 — C7×C4⋊C4
 Upper central C1 — C2×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 >

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

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

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

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 2A 2B 2C 4A ··· 4F 7A ··· 7F 14A ··· 14R 28A ··· 28AJ order 1 2 2 2 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C7 C14 C28 D4 Q8 C7×D4 C7×Q8 kernel C7×C4⋊C4 C2×C28 C28 C4⋊C4 C2×C4 C4 C14 C14 C2 C2 # reps 1 3 4 6 18 24 1 1 6 6

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

 1 0 0 0 24 0 0 0 24
,
 28 0 0 0 0 28 0 1 0
,
 17 0 0 0 27 16 0 16 2
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

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