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## G = C2×D14⋊C4order 224 = 25·7

### Direct product of C2 and D14⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D14⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C2×D14⋊C4
 Lower central C7 — C14 — C2×D14⋊C4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D14⋊C4
G = < a,b,c,d | a2=b14=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b7c >

Subgroups: 590 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, D14⋊C4, C22×Dic7, C22×C28, C23×D7, C2×D14⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C2×D14⋊C4

Smallest permutation representation of C2×D14⋊C4
On 112 points
Generators in S112
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 57)(28 58)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)
(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 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 70)(25 69)(26 68)(27 67)(28 66)(29 99)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 101)(42 100)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)
(1 54 15 40)(2 55 16 41)(3 56 17 42)(4 43 18 29)(5 44 19 30)(6 45 20 31)(7 46 21 32)(8 47 22 33)(9 48 23 34)(10 49 24 35)(11 50 25 36)(12 51 26 37)(13 52 27 38)(14 53 28 39)(57 101 76 86)(58 102 77 87)(59 103 78 88)(60 104 79 89)(61 105 80 90)(62 106 81 91)(63 107 82 92)(64 108 83 93)(65 109 84 94)(66 110 71 95)(67 111 72 96)(68 112 73 97)(69 99 74 98)(70 100 75 85)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14U 28A ··· 28X order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 14 14 14 14 2 2 2 2 14 14 14 14 2 2 2 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D7 D14 D14 C4×D7 D28 C7⋊D4 kernel C2×D14⋊C4 D14⋊C4 C22×Dic7 C22×C28 C23×D7 C22×D7 C2×C14 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 8 4 3 6 3 12 12 12

Matrix representation of C2×D14⋊C4 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 8 21 0 0 8 3
,
 1 0 0 0 0 1 0 0 0 0 1 3 0 0 0 28
,
 17 0 0 0 0 28 0 0 0 0 24 16 0 0 13 5
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,8,8,0,0,21,3],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,3,28],[17,0,0,0,0,28,0,0,0,0,24,13,0,0,16,5] >;

C2×D14⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{14}\rtimes C_4
% in TeX

G:=Group("C2xD14:C4");
// GroupNames label

G:=SmallGroup(224,122);
// by ID

G=gap.SmallGroup(224,122);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,362,50,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

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