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## G = C7×C4⋊1D4order 224 = 25·7

### Direct product of C7 and C4⋊1D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C4⋊1D4
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — D4×C14 — C7×C4⋊1D4
 Lower central C1 — C22 — C7×C4⋊1D4
 Upper central C1 — C2×C14 — C7×C4⋊1D4

Generators and relations for C7×C41D4
G = < a,b,c,d | a7=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, D4, C23, C14, C14, C42, C2×D4, C28, C2×C14, C2×C14, C41D4, C2×C28, C7×D4, C22×C14, C4×C28, D4×C14, C7×C41D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C41D4, C7×D4, C22×C14, D4×C14, C7×C41D4

Smallest permutation representation of C7×C41D4
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)
(1 106 43 9)(2 107 44 10)(3 108 45 11)(4 109 46 12)(5 110 47 13)(6 111 48 14)(7 112 49 8)(15 35 24 39)(16 29 25 40)(17 30 26 41)(18 31 27 42)(19 32 28 36)(20 33 22 37)(21 34 23 38)(50 91 63 78)(51 85 57 79)(52 86 58 80)(53 87 59 81)(54 88 60 82)(55 89 61 83)(56 90 62 84)(64 104 76 92)(65 105 77 93)(66 99 71 94)(67 100 72 95)(68 101 73 96)(69 102 74 97)(70 103 75 98)
(1 63 35 66)(2 57 29 67)(3 58 30 68)(4 59 31 69)(5 60 32 70)(6 61 33 64)(7 62 34 65)(8 90 21 93)(9 91 15 94)(10 85 16 95)(11 86 17 96)(12 87 18 97)(13 88 19 98)(14 89 20 92)(22 104 111 83)(23 105 112 84)(24 99 106 78)(25 100 107 79)(26 101 108 80)(27 102 109 81)(28 103 110 82)(36 75 47 54)(37 76 48 55)(38 77 49 56)(39 71 43 50)(40 72 44 51)(41 73 45 52)(42 74 46 53)
(1 91)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 62)(9 63)(10 57)(11 58)(12 59)(13 60)(14 61)(15 66)(16 67)(17 68)(18 69)(19 70)(20 64)(21 65)(22 76)(23 77)(24 71)(25 72)(26 73)(27 74)(28 75)(29 95)(30 96)(31 97)(32 98)(33 92)(34 93)(35 94)(36 103)(37 104)(38 105)(39 99)(40 100)(41 101)(42 102)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)

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

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

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

98 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 7A ··· 7F 14A ··· 14R 14S ··· 14AP 28A ··· 28AJ order 1 2 2 2 2 2 2 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 4 4 4 4 2 ··· 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2

98 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C7 C14 C14 D4 C7×D4 kernel C7×C4⋊1D4 C4×C28 D4×C14 C4⋊1D4 C42 C2×D4 C28 C4 # reps 1 1 6 6 6 36 6 36

Matrix representation of C7×C41D4 in GL4(𝔽29) generated by

 24 0 0 0 0 24 0 0 0 0 25 0 0 0 0 25
,
 28 0 0 0 0 28 0 0 0 0 0 1 0 0 28 0
,
 0 1 0 0 28 0 0 0 0 0 0 1 0 0 28 0
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 28
G:=sub<GL(4,GF(29))| [24,0,0,0,0,24,0,0,0,0,25,0,0,0,0,25],[28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[0,28,0,0,1,0,0,0,0,0,0,28,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,28] >;

C7×C41D4 in GAP, Magma, Sage, TeX

C_7\times C_4\rtimes_1D_4
% in TeX

G:=Group("C7xC4:1D4");
// GroupNames label

G:=SmallGroup(224,162);
// by ID

G=gap.SmallGroup(224,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,343,2090,518]);
// Polycyclic

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

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