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## G = D7×C22≀C2order 448 = 26·7

### Direct product of D7 and C22≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7×C22≀C2
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C24 — D7×C22≀C2
 Lower central C7 — C2×C14 — D7×C22≀C2
 Upper central C1 — C22 — C22≀C2

Generators and relations for D7×C22≀C2
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=f2=g2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 3916 in 662 conjugacy classes, 131 normal (16 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, D7, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C22≀C2, C22×D4, C25, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22≀C2, D14⋊C4, C23.D7, C7×C22⋊C4, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C23×D7, C23×D7, C23×C14, D7×C22⋊C4, C22⋊D28, C23⋊D14, C24⋊D7, C7×C22≀C2, C2×D4×D7, D7×C24, D7×C22≀C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22≀C2, C22×D4, C22×D7, C2×C22≀C2, D4×D7, C23×D7, C2×D4×D7, D7×C22≀C2

Smallest permutation representation of D7×C22≀C2
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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)
(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)
(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)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(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 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)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 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), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,17)(9,16)(10,15)(11,21)(12,20)(13,19)(14,18)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46), (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), (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), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(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), (1,20)(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), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)>;

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 2N 2O ··· 2T 2U 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A ··· 14I 14J ··· 14AA 14AB 14AC 14AD 28A ··· 28I order 1 2 2 2 2 ··· 2 2 2 2 2 2 2 ··· 2 2 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 size 1 1 1 1 2 ··· 2 4 7 7 7 7 14 ··· 14 28 4 4 4 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 8 ··· 8

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D7 D14 D14 D14 D4×D7 kernel D7×C22≀C2 D7×C22⋊C4 C22⋊D28 C23⋊D14 C24⋊D7 C7×C22≀C2 C2×D4×D7 D7×C24 C22×D7 C22≀C2 C22⋊C4 C2×D4 C24 C22 # reps 1 3 3 3 1 1 3 1 12 3 9 9 3 18

Matrix representation of D7×C22≀C2 in GL6(𝔽29)

 26 1 0 0 0 0 23 21 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
,
 21 28 0 0 0 0 5 8 0 0 0 0 0 0 28 0 0 0 0 0 0 28 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 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 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 1 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 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 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(29))| [26,23,0,0,0,0,1,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,5,0,0,0,0,28,8,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D7×C22≀C2 in GAP, Magma, Sage, TeX

D_7\times C_2^2\wr C_2
% in TeX

G:=Group("D7xC2^2wrC2");
// GroupNames label

G:=SmallGroup(448,1041);
// by ID

G=gap.SmallGroup(448,1041);
# by ID

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

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

׿
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