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## G = C2×C4.Dic7order 224 = 25·7

### Direct product of C2 and C4.Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C4.Dic7
 Chief series C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8 — C2×C4.Dic7
 Lower central C7 — C14 — C2×C4.Dic7
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C4.Dic7
G = < a,b,c,d | a2=b4=1, c14=b2, d2=b2c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c13 >

Subgroups: 142 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C22×C14, C2×C7⋊C8, C4.Dic7, C22×C28, C2×C4.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, Dic7, D14, C2×M4(2), C2×Dic7, C22×D7, C4.Dic7, C22×Dic7, C2×C4.Dic7

Smallest permutation representation of C2×C4.Dic7
On 112 points
Generators in S112
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 111)(58 112)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)
(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 68 22 61 15 82 8 75)(2 81 23 74 16 67 9 60)(3 66 24 59 17 80 10 73)(4 79 25 72 18 65 11 58)(5 64 26 57 19 78 12 71)(6 77 27 70 20 63 13 84)(7 62 28 83 21 76 14 69)(29 101 50 94 43 87 36 108)(30 86 51 107 44 100 37 93)(31 99 52 92 45 85 38 106)(32 112 53 105 46 98 39 91)(33 97 54 90 47 111 40 104)(34 110 55 103 48 96 41 89)(35 95 56 88 49 109 42 102)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28X order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C2 C4 C4 D7 M4(2) Dic7 D14 Dic7 C4.Dic7 kernel C2×C4.Dic7 C2×C7⋊C8 C4.Dic7 C22×C28 C2×C28 C22×C14 C22×C4 C14 C2×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 3 4 9 9 3 24

Matrix representation of C2×C4.Dic7 in GL3(𝔽113) generated by

 112 0 0 0 112 0 0 0 112
,
 112 0 0 0 15 0 0 0 98
,
 1 0 0 0 81 0 0 0 53
,
 1 0 0 0 0 1 0 15 0
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[112,0,0,0,15,0,0,0,98],[1,0,0,0,81,0,0,0,53],[1,0,0,0,0,15,0,1,0] >;

C2×C4.Dic7 in GAP, Magma, Sage, TeX

C_2\times C_4.{\rm Dic}_7
% in TeX

G:=Group("C2xC4.Dic7");
// GroupNames label

G:=SmallGroup(224,116);
// by ID

G=gap.SmallGroup(224,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,362,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^14=b^2,d^2=b^2*c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^13>;
// generators/relations

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