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## G = C7×D4⋊C4order 224 = 25·7

### Direct product of C7 and D4⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×D4⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — C7×C4⋊C4 — C7×D4⋊C4
 Lower central C1 — C2 — C4 — C7×D4⋊C4
 Upper central C1 — C2×C14 — C2×C28 — C7×D4⋊C4

Generators and relations for C7×D4⋊C4
G = < a,b,c,d | a7=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Smallest permutation representation of C7×D4⋊C4
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 50 78 59)(2 51 79 60)(3 52 80 61)(4 53 81 62)(5 54 82 63)(6 55 83 57)(7 56 84 58)(8 35 110 37)(9 29 111 38)(10 30 112 39)(11 31 106 40)(12 32 107 41)(13 33 108 42)(14 34 109 36)(15 43 24 99)(16 44 25 100)(17 45 26 101)(18 46 27 102)(19 47 28 103)(20 48 22 104)(21 49 23 105)(64 91 73 92)(65 85 74 93)(66 86 75 94)(67 87 76 95)(68 88 77 96)(69 89 71 97)(70 90 72 98)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 85)(7 86)(8 101)(9 102)(10 103)(11 104)(12 105)(13 99)(14 100)(15 42)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 31)(23 32)(24 33)(25 34)(26 35)(27 29)(28 30)(43 108)(44 109)(45 110)(46 111)(47 112)(48 106)(49 107)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 74)(58 75)(59 76)(60 77)(61 71)(62 72)(63 73)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)
(1 22 76 40)(2 23 77 41)(3 24 71 42)(4 25 72 36)(5 26 73 37)(6 27 74 38)(7 28 75 39)(8 63 101 91)(9 57 102 85)(10 58 103 86)(11 59 104 87)(12 60 105 88)(13 61 99 89)(14 62 100 90)(15 69 33 80)(16 70 34 81)(17 64 35 82)(18 65 29 83)(19 66 30 84)(20 67 31 78)(21 68 32 79)(43 97 108 52)(44 98 109 53)(45 92 110 54)(46 93 111 55)(47 94 112 56)(48 95 106 50)(49 96 107 51)

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

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

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

98 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 28A ··· 28L 28M ··· 28X 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 4 4 2 2 4 4 1 ··· 1 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 D4 D4 D8 SD16 C7×D4 C7×D4 C7×D8 C7×SD16 kernel C7×D4⋊C4 C7×C4⋊C4 C2×C56 D4×C14 C7×D4 D4⋊C4 C4⋊C4 C2×C8 C2×D4 D4 C28 C2×C14 C14 C14 C4 C22 C2 C2 # reps 1 1 1 1 4 6 6 6 6 24 1 1 2 2 6 6 12 12

Matrix representation of C7×D4⋊C4 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 112 0 0 0 0 112 0 0 0 0 112 111 0 0 1 1
,
 1 0 0 0 0 112 0 0 0 0 1 2 0 0 0 112
,
 0 1 0 0 112 0 0 0 0 0 26 26 0 0 100 87
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[112,0,0,0,0,112,0,0,0,0,112,1,0,0,111,1],[1,0,0,0,0,112,0,0,0,0,1,0,0,0,2,112],[0,112,0,0,1,0,0,0,0,0,26,100,0,0,26,87] >;

C7×D4⋊C4 in GAP, Magma, Sage, TeX

C_7\times D_4\rtimes C_4
% in TeX

G:=Group("C7xD4:C4");
// GroupNames label

G:=SmallGroup(224,51);
// by ID

G=gap.SmallGroup(224,51);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,1689,117]);
// Polycyclic

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

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