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## G = C2×Dic14⋊C4order 448 = 26·7

### Direct product of C2 and Dic14⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×Dic14⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4○D28 — C2×C4○D28 — C2×Dic14⋊C4
 Lower central C7 — C14 — C28 — C2×Dic14⋊C4
 Upper central C1 — C2×C4 — C22×C4 — C2×C42

Generators and relations for C2×Dic14⋊C4
G = < a,b,c,d | a2=b28=d4=1, c2=b14, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b7c >

Subgroups: 772 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×C4≀C2, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×C28, C4×C28, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, C22×C28, Dic14⋊C4, C2×C4.Dic7, C2×C4×C28, C2×C4○D28, C2×Dic14⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4≀C2, C2×C22⋊C4, C4×D7, D28, C7⋊D4, C22×D7, C2×C4≀C2, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, Dic14⋊C4, C2×D14⋊C4, C2×Dic14⋊C4

Smallest permutation representation of C2×Dic14⋊C4
On 112 points
Generators in S112
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)(28 35)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 99)(65 100)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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)
(1 100 15 86)(2 99 16 85)(3 98 17 112)(4 97 18 111)(5 96 19 110)(6 95 20 109)(7 94 21 108)(8 93 22 107)(9 92 23 106)(10 91 24 105)(11 90 25 104)(12 89 26 103)(13 88 27 102)(14 87 28 101)(29 72 43 58)(30 71 44 57)(31 70 45 84)(32 69 46 83)(33 68 47 82)(34 67 48 81)(35 66 49 80)(36 65 50 79)(37 64 51 78)(38 63 52 77)(39 62 53 76)(40 61 54 75)(41 60 55 74)(42 59 56 73)
(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)

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O 4P 7A 7B 7C 8A 8B 8C 8D 14A ··· 14U 28A ··· 28BT order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 28 28 1 1 1 1 2 ··· 2 28 28 2 2 2 28 28 28 28 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D7 D14 D14 C4≀C2 C4×D7 D28 C7⋊D4 C7⋊D4 Dic14⋊C4 kernel C2×Dic14⋊C4 Dic14⋊C4 C2×C4.Dic7 C2×C4×C28 C2×C4○D28 C2×Dic14 C2×D28 C4○D28 C2×C28 C22×C14 C2×C42 C42 C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 2 2 4 3 1 3 6 3 8 12 12 6 6 48

Matrix representation of C2×Dic14⋊C4 in GL3(𝔽113) generated by

 112 0 0 0 1 0 0 0 1
,
 1 0 0 0 56 0 0 23 111
,
 112 0 0 0 67 3 0 10 46
,
 98 0 0 0 1 0 0 57 98
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[1,0,0,0,56,23,0,0,111],[112,0,0,0,67,10,0,3,46],[98,0,0,0,1,57,0,0,98] >;

C2×Dic14⋊C4 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{14}\rtimes C_4
% in TeX

G:=Group("C2xDic14:C4");
// GroupNames label

G:=SmallGroup(448,461);
// by ID

G=gap.SmallGroup(448,461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,58,1123,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=d^4=1,c^2=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

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