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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

C1C2 — C7×C22⋊C4
C1C2C22C2×C14C2×C28 — C7×C22⋊C4
C1C2 — C7×C22⋊C4
C1C2×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 >

2C2
2C2
2C4
2C22
2C4
2C22
2C14
2C14
2C28
2C28
2C2×C14
2C2×C14

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

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

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

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

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 2A2B2C2D2E4A4B4C4D7A···7F14A···14R14S···14AD28A···28X
order12222244447···714···1414···1428···28
size11112222221···11···12···22···2

70 irreducible representations

dim1111111122
type++++
imageC1C2C2C4C7C14C14C28D4C7×D4
kernelC7×C22⋊C4C2×C28C22×C14C2×C14C22⋊C4C2×C4C23C22C14C2
# reps1214612624212

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

100
070
007
,
100
0280
001
,
100
0280
0028
,
1200
001
0280
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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Subgroup lattice of C7×C22⋊C4 in TeX

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