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## G = C7×C8.26D4order 448 = 26·7

### Direct product of C7 and C8.26D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C8.26D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4≀C2 — C7×C8.26D4
 Lower central C1 — C2 — C4 — C7×C8.26D4
 Upper central C1 — C28 — C2×C56 — C7×C8.26D4

Generators and relations for C7×C8.26D4
G = < a,b,c,d | a7=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >

Subgroups: 154 in 104 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C14, C14, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C28, C28, C2×C14, C2×C14, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C56, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C8.26D4, C4×C28, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C7×C8⋊C4, C7×C4≀C2, C7×C8.C4, C7×C8○D4, C7×C4○D8, C7×C8.26D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×C28, C7×D4, C22×C14, C8.26D4, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8.26D4

Smallest permutation representation of C7×C8.26D4
On 112 points
Generators in S112
(1 59 38 55 23 47 11)(2 60 39 56 24 48 12)(3 61 40 49 17 41 13)(4 62 33 50 18 42 14)(5 63 34 51 19 43 15)(6 64 35 52 20 44 16)(7 57 36 53 21 45 9)(8 58 37 54 22 46 10)(25 76 105 96 85 68 97)(26 77 106 89 86 69 98)(27 78 107 90 87 70 99)(28 79 108 91 88 71 100)(29 80 109 92 81 72 101)(30 73 110 93 82 65 102)(31 74 111 94 83 66 103)(32 75 112 95 84 67 104)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(25 27 29 31)(26 32 30 28)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(65 71 69 67)(66 68 70 72)(73 79 77 75)(74 76 78 80)(81 83 85 87)(82 88 86 84)(89 95 93 91)(90 92 94 96)(97 99 101 103)(98 104 102 100)(105 107 109 111)(106 112 110 108)
(1 103 3 97 5 99 7 101)(2 100 4 102 6 104 8 98)(9 72 11 66 13 68 15 70)(10 69 12 71 14 65 16 67)(17 96 19 90 21 92 23 94)(18 93 20 95 22 89 24 91)(25 63 27 57 29 59 31 61)(26 60 28 62 30 64 32 58)(33 73 35 75 37 77 39 79)(34 78 36 80 38 74 40 76)(41 85 43 87 45 81 47 83)(42 82 44 84 46 86 48 88)(49 105 51 107 53 109 55 111)(50 110 52 112 54 106 56 108)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 7A ··· 7F 8A 8B 8C 8D 8E ··· 8J 14A ··· 14F 14G ··· 14L 14M ··· 14X 28A ··· 28L 28M ··· 28R 28S ··· 28AP 56A ··· 56X 56Y ··· 56BH order 1 2 2 2 2 4 4 4 4 4 4 4 7 ··· 7 8 8 8 8 8 ··· 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 4 4 1 1 2 4 4 4 4 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C7 C14 C14 C14 C14 C14 C28 C28 C28 D4 C4○D4 C7×D4 C7×C4○D4 C8.26D4 C7×C8.26D4 kernel C7×C8.26D4 C7×C8⋊C4 C7×C4≀C2 C7×C8.C4 C7×C8○D4 C7×C4○D8 C7×D8 C7×SD16 C7×Q16 C8.26D4 C8⋊C4 C4≀C2 C8.C4 C8○D4 C4○D8 D8 SD16 Q16 C56 C2×C14 C8 C22 C7 C1 # reps 1 1 2 1 2 1 2 4 2 6 6 12 6 12 6 12 24 12 2 2 12 12 2 12

Matrix representation of C7×C8.26D4 in GL4(𝔽113) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 1 0 0 98 0 0 0 0 0 0 1 0 0 98 0
,
 1 0 0 0 0 112 0 0 0 0 98 0 0 0 0 15
,
 0 0 98 0 0 0 0 15 1 0 0 0 0 112 0 0
G:=sub<GL(4,GF(113))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,98,0,0,1,0,0,0,0,0,0,98,0,0,1,0],[1,0,0,0,0,112,0,0,0,0,98,0,0,0,0,15],[0,0,1,0,0,0,0,112,98,0,0,0,0,15,0,0] >;

C7×C8.26D4 in GAP, Magma, Sage, TeX

C_7\times C_8._{26}D_4
% in TeX

G:=Group("C7xC8.26D4");
// GroupNames label

G:=SmallGroup(448,852);
// by ID

G=gap.SmallGroup(448,852);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,604,9804,4911,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
// generators/relations

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