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## G = D7×C4≀C2order 448 = 26·7

### Direct product of D7 and C4≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×C4≀C2
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C4○D4 — D7×C4≀C2
 Lower central C7 — C14 — C28 — D7×C4≀C2
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for D7×C4≀C2
G = < a,b,c,d,e | a7=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c-1d >

Subgroups: 892 in 170 conjugacy classes, 53 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, D7, C14, C14, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C4≀C2, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C2×C4≀C2, C8×D7, C8⋊D7, C4.Dic7, C4×Dic7, C4×Dic7, C4×C28, C7×M4(2), C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D42D7, D42D7, Q8×D7, Q82D7, C7×C4○D4, Dic14⋊C4, D284C4, D42Dic7, C7×C4≀C2, D7×C42, D7×M4(2), D7×C4○D4, D7×C4≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C4≀C2, C2×C22⋊C4, C4×D7, C22×D7, C2×C4≀C2, C2×C4×D7, D4×D7, D7×C22⋊C4, D7×C4≀C2

Smallest permutation representation of D7×C4≀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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)
(1 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)(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 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)
(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 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)

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 D4 D7 D14 D14 D14 C4≀C2 C4×D7 C4×D7 D4×D7 D4×D7 D7×C4≀C2 kernel D7×C4≀C2 Dic14⋊C4 D28⋊4C4 D4⋊2Dic7 C7×C4≀C2 D7×C42 D7×M4(2) D7×C4○D4 D4×D7 D4⋊2D7 Q8×D7 Q8⋊2D7 C4×D7 C2×Dic7 C22×D7 C4≀C2 C42 M4(2) C4○D4 D7 D4 Q8 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 3 3 3 3 8 6 6 3 3 12

Matrix representation of D7×C4≀C2 in GL4(𝔽113) generated by

 104 1 0 0 41 33 0 0 0 0 1 0 0 0 0 1
,
 33 112 0 0 71 80 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 98 0 0 0 28 15
,
 112 0 0 0 0 112 0 0 0 0 1 106 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 112 0 0 0 30 15
G:=sub<GL(4,GF(113))| [104,41,0,0,1,33,0,0,0,0,1,0,0,0,0,1],[33,71,0,0,112,80,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,28,0,0,0,15],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,106,112],[1,0,0,0,0,1,0,0,0,0,112,30,0,0,0,15] >;

D7×C4≀C2 in GAP, Magma, Sage, TeX

D_7\times C_4\wr C_2
% in TeX

G:=Group("D7xC4wrC2");
// GroupNames label

G:=SmallGroup(448,354);
// by ID

G=gap.SmallGroup(448,354);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,219,58,136,851,438,102,18822]);
// Polycyclic

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

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