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## G = C7×C4≀C2order 224 = 25·7

### Direct product of C7 and C4≀C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×M4(2) — C7×C4≀C2
 Lower central C1 — C2 — C4 — C7×C4≀C2
 Upper central C1 — C28 — C2×C28 — C7×C4≀C2

Generators and relations for C7×C4≀C2
G = < a,b,c,d | a7=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Smallest permutation representation of C7×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 43 35 38)(2 44 29 39)(3 45 30 40)(4 46 31 41)(5 47 32 42)(6 48 33 36)(7 49 34 37)(8 23 21 56)(9 24 15 50)(10 25 16 51)(11 26 17 52)(12 27 18 53)(13 28 19 54)(14 22 20 55)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 49)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 38)(16 39)(17 40)(18 41)(19 42)(20 36)(21 37)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 50)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 23 21 56)(9 24 15 50)(10 25 16 51)(11 26 17 52)(12 27 18 53)(13 28 19 54)(14 22 20 55)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)

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

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

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

C7×C4≀C2 is a maximal subgroup of   C42⋊D14  D44D28  M4(2).22D14  C42.196D14  M4(2)⋊D14  D4.9D28  D4.10D28

98 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C ··· 4G 4H 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14L 14M ··· 14R 28A ··· 28L 28M ··· 28AP 28AQ ··· 28AV 56A ··· 56L order 1 2 2 2 4 4 4 ··· 4 4 7 ··· 7 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 4 1 1 2 ··· 2 4 1 ··· 1 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 D4 C4≀C2 C7×D4 C7×D4 C7×C4≀C2 kernel C7×C4≀C2 C4×C28 C7×M4(2) C7×C4○D4 C7×D4 C7×Q8 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C28 C2×C14 C7 C4 C22 C1 # reps 1 1 1 1 2 2 6 6 6 6 12 12 1 1 4 6 6 24

Matrix representation of C7×C4≀C2 in GL2(𝔽29) generated by

 25 0 0 25
,
 12 0 0 17
,
 0 9 13 0
,
 17 0 0 1
G:=sub<GL(2,GF(29))| [25,0,0,25],[12,0,0,17],[0,13,9,0],[17,0,0,1] >;

C7×C4≀C2 in GAP, Magma, Sage, TeX

C_7\times C_4\wr C_2
% in TeX

G:=Group("C7xC4wrC2");
// GroupNames label

G:=SmallGroup(224,53);
// by ID

G=gap.SmallGroup(224,53);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,3363,1689,117,88]);
// Polycyclic

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

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