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## G = C7×C22⋊C4order 112 = 24·7

### Direct product of C7 and C22⋊C4

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

Aliases: C7×C22⋊C4, C22⋊C28, C23.C14, C14.12D4, (C2×C4)⋊1C14, (C2×C14)⋊1C4, (C2×C28)⋊2C2, C2.1(C7×D4), C2.1(C2×C28), C14.10(C2×C4), (C22×C14).1C2, C22.2(C2×C14), (C2×C14).13C22, SmallGroup(112,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C22⋊C4
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C22⋊C4
 Lower central C1 — C2 — C7×C22⋊C4
 Upper central C1 — C2×C14 — C7×C22⋊C4

Generators and relations for C7×C22⋊C4
G = < a,b,c,d | a7=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Smallest permutation representation of C7×C22⋊C4
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 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 56)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)
(1 9 35 15)(2 10 29 16)(3 11 30 17)(4 12 31 18)(5 13 32 19)(6 14 33 20)(7 8 34 21)(22 48 55 36)(23 49 56 37)(24 43 50 38)(25 44 51 39)(26 45 52 40)(27 46 53 41)(28 47 54 42)

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

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

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

C7×C22⋊C4 is a maximal subgroup of
C23.1D14  C23.11D14  C22⋊Dic14  C23.D14  Dic74D4  C22⋊D28  D14.D4  D14⋊D4  Dic7.D4  C22.D28  D4×C28

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A ··· 7F 14A ··· 14R 14S ··· 14AD 28A ··· 28X order 1 2 2 2 2 2 4 4 4 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C4 C7 C14 C14 C28 D4 C7×D4 kernel C7×C22⋊C4 C2×C28 C22×C14 C2×C14 C22⋊C4 C2×C4 C23 C22 C14 C2 # reps 1 2 1 4 6 12 6 24 2 12

Matrix representation of C7×C22⋊C4 in GL3(𝔽29) generated by

 1 0 0 0 7 0 0 0 7
,
 1 0 0 0 28 0 0 0 1
,
 1 0 0 0 28 0 0 0 28
,
 12 0 0 0 0 1 0 28 0
G:=sub<GL(3,GF(29))| [1,0,0,0,7,0,0,0,7],[1,0,0,0,28,0,0,0,1],[1,0,0,0,28,0,0,0,28],[12,0,0,0,0,28,0,1,0] >;

C7×C22⋊C4 in GAP, Magma, Sage, TeX

C_7\times C_2^2\rtimes C_4
% in TeX

G:=Group("C7xC2^2:C4");
// GroupNames label

G:=SmallGroup(112,20);
// by ID

G=gap.SmallGroup(112,20);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-2,280,301]);
// Polycyclic

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

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