Copied to
clipboard

G = C7×F8  order 392 = 23·72

Direct product of C7 and F8

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

Aliases: C7×F8, C23⋊C72, (C22×C14)⋊C7, SmallGroup(392,39)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C7×F8
Chief series
C1 — C23 — F8 — C7×F8
Lower central
C23 — C7×F8
Upper central
C1 — C7

Generators and relations for C7×F8
 G = < a,b,c,d,e | a7=b2=c2=d2=e7=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=dc=cd, ece-1=b, ede-1=c >

C1
7C2
C7
8C7
8C7
8C7
8C7
8C7
8C7
8C7
7C22
7C14
8C72
C23
7C2×C14
F8
C22×C14
F8
F8
F8
F8
F8
F8
C7×F8

Smallest permutation representation of C7×F8
►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 31)(2 32)(3 33)(4 34)(5 35)(6 29)(7 30)(8 54)(9 55)(10 56)(11 50)(12 51)(13 52)(14 53)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(36 44)(37 45)(38 46)(39 47)(40 48)(41 49)(42 43)

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

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

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

56 conjugacy classes

class 1  2 7A···7F7G···7AV14A···14F
order127···77···714···14
size171···18···87···7

56 irreducible representations

dim11177
type++
imageC1C7C7F8C7×F8
kernelC7×F8F8C22×C14C7C1
# reps142616

Matrix representation of C7×F8 ►in GL7(𝔽29)

25000000
02500000
00250000
00025000
00002500
00000250
00000025
,
28000000
28010000
28100000
28000100
28001000
28000001
28000010
,
00012800
00002801
00002810
10002800
00002800
00102800
01002800
,
00000128
00001028
00010028
00100028
01000028
10000028
00000028
,
0100000
0001000
0000010
0010000
1000000
0000001
0000100

G:=sub<GL(7,GF(29))| [25,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,25],[28,28,28,28,28,28,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,28,28,28,28,28,28,28,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,28,28,28,28,28,28,28],[0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
 

C7×F8 in GAP, Magma, Sage, TeX 

C_7\times F_8
 
% in TeX
 
G:=Group("C7xF8");
 
// GroupNames label
 
G:=SmallGroup(392,39);
 
// by ID
 
G=gap.SmallGroup(392,39);
 
# by ID
 
G:=PCGroup([5,-7,-7,-2,2,2,1472,3923,6129]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^2=e^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=d*c=c*d,e*c*e^-1=b,e*d*e^-1=c>;
 
// generators/relations
 

Export

Subgroup lattice of C7×F8 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁