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## G = D7×C8⋊C22order 448 = 26·7

### Direct product of D7 and C8⋊C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×C8⋊C22
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×D4×D7 — D7×C8⋊C22
 Lower central C7 — C14 — C28 — D7×C8⋊C22
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for D7×C8⋊C22
G = < a,b,c,d,e | a7=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 1772 in 298 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C8⋊C22, C22×D4, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C2×C8⋊C22, C8×D7, C8⋊D7, C56⋊C2, D56, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7×M4(2), C7×D8, C7×SD16, C2×C4×D7, C2×C4×D7, C2×D28, C4○D28, C4○D28, D4×D7, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q82D7, C2×C7⋊D4, D4×C14, C7×C4○D4, C23×D7, D7×M4(2), C8⋊D14, D7×D8, D8⋊D7, D7×SD16, D56⋊C2, D4.D14, D4⋊D14, C7×C8⋊C22, C2×D4×D7, D7×C4○D4, D7×C8⋊C22
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8⋊C22, C22×D4, C22×D7, C2×C8⋊C22, D4×D7, C23×D7, C2×D4×D7, D7×C8⋊C22

Smallest permutation representation of D7×C8⋊C22
On 56 points
Generators in S56
(1 37 55 14 17 45 27)(2 38 56 15 18 46 28)(3 39 49 16 19 47 29)(4 40 50 9 20 48 30)(5 33 51 10 21 41 31)(6 34 52 11 22 42 32)(7 35 53 12 23 43 25)(8 36 54 13 24 44 26)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(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)
(1 7)(3 5)(4 8)(9 13)(10 16)(12 14)(17 23)(19 21)(20 24)(25 27)(26 30)(29 31)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(49 51)(50 54)(53 55)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)

G:=sub<Sym(56)| (1,37,55,14,17,45,27)(2,38,56,15,18,46,28)(3,39,49,16,19,47,29)(4,40,50,9,20,48,30)(5,33,51,10,21,41,31)(6,34,52,11,22,42,32)(7,35,53,12,23,43,25)(8,36,54,13,24,44,26), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,7)(3,5)(4,8)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)>;

G:=Group( (1,37,55,14,17,45,27)(2,38,56,15,18,46,28)(3,39,49,16,19,47,29)(4,40,50,9,20,48,30)(5,33,51,10,21,41,31)(6,34,52,11,22,42,32)(7,35,53,12,23,43,25)(8,36,54,13,24,44,26), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(17,55)(18,56)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (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), (1,7)(3,5)(4,8)(9,13)(10,16)(12,14)(17,23)(19,21)(20,24)(25,27)(26,30)(29,31)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(49,51)(50,54)(53,55), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(26,30)(28,32)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56) );

G=PermutationGroup([[(1,37,55,14,17,45,27),(2,38,56,15,18,46,28),(3,39,49,16,19,47,29),(4,40,50,9,20,48,30),(5,33,51,10,21,41,31),(6,34,52,11,22,42,32),(7,35,53,12,23,43,25),(8,36,54,13,24,44,26)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(17,55),(18,56),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(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)], [(1,7),(3,5),(4,8),(9,13),(10,16),(12,14),(17,23),(19,21),(20,24),(25,27),(26,30),(29,31),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(49,51),(50,54),(53,55)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(26,30),(28,32),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56)]])

55 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 14D 14E 14F 14G ··· 14O 28A ··· 28F 28G 28H 28I 56A ··· 56F order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 14 14 14 14 ··· 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 4 4 4 7 7 14 28 28 28 2 2 4 14 14 28 2 2 2 4 4 28 28 2 2 2 4 4 4 8 ··· 8 4 ··· 4 8 8 8 8 ··· 8

55 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 D14 C8⋊C22 D4×D7 D4×D7 D7×C8⋊C22 kernel D7×C8⋊C22 D7×M4(2) C8⋊D14 D7×D8 D8⋊D7 D7×SD16 D56⋊C2 D4.D14 D4⋊D14 C7×C8⋊C22 C2×D4×D7 D7×C4○D4 C4×D7 C2×Dic7 C22×D7 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 D7 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 1 1 3 3 6 6 3 3 2 3 3 3

Matrix representation of D7×C8⋊C22 in GL8(𝔽113)

 79 1 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 79 1 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 88 25 0 0 0 0 0 0 79 25 0 0 0 0 0 0 0 0 88 25 0 0 0 0 0 0 79 25 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 0 0 112 22 91 2 0 0 0 0 0 1 111 72 0 0 0 0 0 1 112 72 0 0 0 0 112 22 0 1
,
 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 112 22 0 0 0 0 0 0 0 1 0 0 0 0 0 0 41 1 1 0 0 0 0 0 112 0 22 112
,
 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 112 0 0 0 0 0 2 91 0 112

G:=sub<GL(8,GF(113))| [79,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,79,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[88,79,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,88,79,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,112,0,0,0,0,22,1,1,22,0,0,0,0,91,111,112,0,0,0,0,0,2,72,72,1],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,41,112,0,0,0,0,22,1,1,0,0,0,0,0,0,0,1,22,0,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,72,2,0,0,0,0,0,1,0,91,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112] >;

D7×C8⋊C22 in GAP, Magma, Sage, TeX

D_7\times C_8\rtimes C_2^2
% in TeX

G:=Group("D7xC8:C2^2");
// GroupNames label

G:=SmallGroup(448,1225);
// by ID

G=gap.SmallGroup(448,1225);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,570,185,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^8=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
// generators/relations

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