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G = (C2×C28).D4order 448 = 26·7

2nd non-split extension by C2×C28 of D4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14 — (C2×C28).D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — D4×C14 — C23.18D14 — (C2×C28).D4
 Lower central C7 — C14 — C2×C14 — C22×C14 — (C2×C28).D4
 Upper central C1 — C2 — C22 — C2×D4 — C23⋊C4

Generators and relations for (C2×C28).D4
G = < a,b,c,d | a2=b4=c28=1, d2=b-1, ab=ba, cac-1=dad-1=ab2, cbc-1=ab-1, bd=db, dcd-1=bc-1 >

Subgroups: 348 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, Dic7, C28, C2×C14, C2×C14, C23⋊C4, C4.D4, C22.D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C23.D4, C4.Dic7, Dic7⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C22×Dic7, D4×C14, C28.D4, C7×C23⋊C4, C23.18D14, (C2×C28).D4
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C23.D4, D14⋊C4, C23.1D14, (C2×C28).D4

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

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

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

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

46 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 14A 14B 14C 14D ··· 14L 14M 14N 14O 28A ··· 28O order 1 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 14 14 14 14 ··· 14 14 14 14 28 ··· 28 size 1 1 2 4 4 4 8 8 28 28 56 2 2 2 56 56 2 2 2 4 ··· 4 8 8 8 8 ··· 8

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + - image C1 C2 C2 C2 C4 C4 D4 D4 D7 D14 D28 C4×D7 C7⋊D4 C23⋊C4 C23.D4 C23.1D14 (C2×C28).D4 kernel (C2×C28).D4 C28.D4 C7×C23⋊C4 C23.18D14 C23.D7 C22×Dic7 C2×C28 C22×C14 C23⋊C4 C2×D4 C2×C4 C23 C23 C14 C7 C2 C1 # reps 1 1 1 1 2 2 1 1 3 3 6 6 6 1 2 6 3

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 96 27 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 85 98 0 0 0 0 0 0 98 0 0 0 39 66 15 15
,
 0 104 0 0 0 0 88 79 0 0 0 0 0 0 0 0 1 0 0 0 46 20 81 81 0 0 98 105 0 0 0 0 57 73 45 93
,
 112 80 0 0 0 0 0 1 0 0 0 0 0 0 29 47 98 83 0 0 101 39 28 28 0 0 1 106 0 0 0 0 13 46 3 45

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,96,0,0,0,112,0,27,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,85,0,39,0,0,0,98,0,66,0,0,0,0,98,15,0,0,0,0,0,15],[0,88,0,0,0,0,104,79,0,0,0,0,0,0,0,46,98,57,0,0,0,20,105,73,0,0,1,81,0,45,0,0,0,81,0,93],[112,0,0,0,0,0,80,1,0,0,0,0,0,0,29,101,1,13,0,0,47,39,106,46,0,0,98,28,0,3,0,0,83,28,0,45] >;

(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(448,29);
// by ID

G=gap.SmallGroup(448,29);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,346,297,851,18822]);
// Polycyclic

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

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