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G = D7×C2×C14order 392 = 23·72

Direct product of C2×C14 and D7

direct product, metabelian, supersoluble, monomial, A-group

Aliases: D7×C2×C14, C1424C2, C722C23, C14⋊(C2×C14), C7⋊(C22×C14), (C2×C14)⋊3C14, (C7×C14)⋊2C22, SmallGroup(392,42)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C2×C14
C1C7C72C7×D7D7×C14 — D7×C2×C14
C7 — D7×C2×C14
C1C2×C14

Generators and relations for D7×C2×C14
 G = < a,b,c,d | a2=b14=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 226 in 79 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C7, C7, C23, D7, C14, C14, D14, C2×C14, C2×C14, C72, C22×D7, C22×C14, C7×D7, C7×C14, D7×C14, C142, D7×C2×C14
Quotients: C1, C2, C22, C7, C23, D7, C14, D14, C2×C14, C22×D7, C22×C14, C7×D7, D7×C14, D7×C2×C14

Smallest permutation representation of D7×C2×C14
On 56 points
Generators in S56
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(29 53)(30 54)(31 55)(32 56)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)
(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 5 9 13 3 7 11)(2 6 10 14 4 8 12)(15 19 23 27 17 21 25)(16 20 24 28 18 22 26)(29 39 35 31 41 37 33)(30 40 36 32 42 38 34)(43 53 49 45 55 51 47)(44 54 50 46 56 52 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 43)(27 44)(28 45)

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

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

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

140 conjugacy classes

class 1 2A2B2C2D2E2F2G7A···7F7G···7AA14A···14R14S···14CC14CD···14DA
order122222227···77···714···1414···1414···14
size111177771···12···21···12···27···7

140 irreducible representations

dim1111112222
type+++++
imageC1C2C2C7C14C14D7D14C7×D7D7×C14
kernelD7×C2×C14D7×C14C142C22×D7D14C2×C14C2×C14C14C22C2
# reps1616366391854

Matrix representation of D7×C2×C14 in GL3(𝔽29) generated by

100
0280
0028
,
400
0220
0022
,
100
0230
02524
,
100
056
02524
G:=sub<GL(3,GF(29))| [1,0,0,0,28,0,0,0,28],[4,0,0,0,22,0,0,0,22],[1,0,0,0,23,25,0,0,24],[1,0,0,0,5,25,0,6,24] >;

D7×C2×C14 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_{14}
% in TeX

G:=Group("D7xC2xC14");
// GroupNames label

G:=SmallGroup(392,42);
// by ID

G=gap.SmallGroup(392,42);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,8404]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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