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## G = D7×2+ 1+4order 448 = 26·7

### Direct product of D7 and 2+ 1+4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D7×2+ 1+4
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — C2×D4×D7 — D7×2+ 1+4
 Lower central C7 — C14 — D7×2+ 1+4
 Upper central C1 — C2 — 2+ 1+4

Generators and relations for D7×2+ 1+4
G = < a,b,c,d,e,f | a7=b2=c4=d2=f2=1, e2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=c2e >

Subgroups: 4084 in 898 conjugacy classes, 445 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, D7, C14, C14, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, Dic7, C28, D14, D14, D14, C2×C14, C2×C14, C22×D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, C2×2+ 1+4, C2×C4×D7, C2×D28, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, C2×C7⋊D4, D4×C14, C7×C4○D4, C23×D7, C2×D4×D7, D46D14, D7×C4○D4, D48D14, C7×2+ 1+4, D7×2+ 1+4
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, C25, C22×D7, C2×2+ 1+4, C23×D7, D7×C24, D7×2+ 1+4

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

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

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

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

85 conjugacy classes

 class 1 2A 2B ··· 2J 2K 2L 2M ··· 2U 4A ··· 4F 4G ··· 4L 7A 7B 7C 14A 14B 14C 14D ··· 14AD 28A ··· 28R order 1 2 2 ··· 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 14 14 14 ··· 14 28 ··· 28 size 1 1 2 ··· 2 7 7 14 ··· 14 2 ··· 2 14 ··· 14 2 2 2 2 2 2 4 ··· 4 4 ··· 4

85 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 8 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D7 D14 D14 2+ 1+4 D7×2+ 1+4 kernel D7×2+ 1+4 C2×D4×D7 D4⋊6D14 D7×C4○D4 D4⋊8D14 C7×2+ 1+4 2+ 1+4 C2×D4 C4○D4 D7 C1 # reps 1 9 9 6 6 1 3 27 18 2 3

Matrix representation of D7×2+ 1+4 in GL6(𝔽29)

 0 1 0 0 0 0 28 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 28 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 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 0 0 28 0 0 0 1 1 1 2 0 0 1 0 0 0 0 0 28 28 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 28 28 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 28 0 0 0 0 0 28 28 28 27 0 0 1 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 28 28 28 27 0 0 0 0 0 1

G:=sub<GL(6,GF(29))| [0,28,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,28,0,0,0,1,0,28,0,0,28,1,0,0,0,0,0,2,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,1,0,28,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,28,1,0,0,1,0,28,0,0,0,0,0,28,1,0,0,0,0,27,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,28,0,0,0,1,0,28,0,0,0,0,0,28,0,0,0,0,0,27,1] >;

D7×2+ 1+4 in GAP, Magma, Sage, TeX

D_7\times 2_+^{1+4}
% in TeX

G:=Group("D7xES+(2,2)");
// GroupNames label

G:=SmallGroup(448,1379);
// by ID

G=gap.SmallGroup(448,1379);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,851,18822]);
// Polycyclic

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

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