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## G = C2×C28.D4order 448 = 26·7

### Direct product of C2 and C28.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C28.D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C2×C4.Dic7 — C2×C28.D4
 Lower central C7 — C14 — C2×C14 — C2×C28.D4
 Upper central C1 — C22 — C22×C4 — C22×D4

Generators and relations for C2×C28.D4
G = < a,b,c,d | a2=b28=1, c4=b14, d2=b21, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b13, dcd-1=b7c3 >

Subgroups: 532 in 186 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C28, C2×C14, C2×C14, C4.D4, C2×M4(2), C22×D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C4.D4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C28.D4, C2×C4.Dic7, D4×C2×C14, C2×C28.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C4.D4, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C2×C4.D4, C23.D7, C22×Dic7, C2×C7⋊D4, C28.D4, C2×C23.D7, C2×C28.D4

Smallest permutation representation of C2×C28.D4
On 112 points
Generators in S112
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 106)(30 107)(31 108)(32 109)(33 110)(34 111)(35 112)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)
(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 92 70 50 15 106 84 36)(2 91 71 49 16 105 57 35)(3 90 72 48 17 104 58 34)(4 89 73 47 18 103 59 33)(5 88 74 46 19 102 60 32)(6 87 75 45 20 101 61 31)(7 86 76 44 21 100 62 30)(8 85 77 43 22 99 63 29)(9 112 78 42 23 98 64 56)(10 111 79 41 24 97 65 55)(11 110 80 40 25 96 66 54)(12 109 81 39 26 95 67 53)(13 108 82 38 27 94 68 52)(14 107 83 37 28 93 69 51)
(1 99 22 92 15 85 8 106)(2 112 23 105 16 98 9 91)(3 97 24 90 17 111 10 104)(4 110 25 103 18 96 11 89)(5 95 26 88 19 109 12 102)(6 108 27 101 20 94 13 87)(7 93 28 86 21 107 14 100)(29 77 50 70 43 63 36 84)(30 62 51 83 44 76 37 69)(31 75 52 68 45 61 38 82)(32 60 53 81 46 74 39 67)(33 73 54 66 47 59 40 80)(34 58 55 79 48 72 41 65)(35 71 56 64 49 57 42 78)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 7A 7B 7C 8A ··· 8H 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 2 2 2 28 ··· 28 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + - + - + image C1 C2 C2 C2 C4 C4 D4 D7 D14 Dic7 D14 Dic7 C7⋊D4 C4.D4 C28.D4 kernel C2×C28.D4 C28.D4 C2×C4.Dic7 D4×C2×C14 D4×C14 C23×C14 C2×C28 C22×D4 C22×C4 C2×D4 C2×D4 C24 C2×C4 C14 C2 # reps 1 4 2 1 4 4 4 3 3 6 6 6 24 2 12

Matrix representation of C2×C28.D4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 85 0 0 0 0 0 100 4 0 0 0 0 0 0 83 14 0 0 0 0 49 30 0 0 0 0 24 83 0 49 0 0 40 83 64 0
,
 3 10 0 0 0 0 112 110 0 0 0 0 0 0 1 0 8 0 0 0 0 0 1 112 0 0 0 112 112 0 0 0 28 0 112 0
,
 3 10 0 0 0 0 112 110 0 0 0 0 0 0 1 0 0 8 0 0 0 0 112 1 0 0 0 0 0 112 0 0 28 1 0 112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[85,100,0,0,0,0,0,4,0,0,0,0,0,0,83,49,24,40,0,0,14,30,83,83,0,0,0,0,0,64,0,0,0,0,49,0],[3,112,0,0,0,0,10,110,0,0,0,0,0,0,1,0,0,28,0,0,0,0,112,0,0,0,8,1,112,112,0,0,0,112,0,0],[3,112,0,0,0,0,10,110,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,1,0,0,0,112,0,0,0,0,8,1,112,112] >;

C2×C28.D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}.D_4
% in TeX

G:=Group("C2xC28.D4");
// GroupNames label

G:=SmallGroup(448,750);
// by ID

G=gap.SmallGroup(448,750);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=1,c^4=b^14,d^2=b^21,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^7*c^3>;
// generators/relations

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