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

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

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

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

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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