Copied to
clipboard

?

G = C2×C4×F8  order 448 = 26·7

Direct product of C2×C4 and F8

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

Aliases: C2×C4×F8, C24⋊C28, C25.C14, (C24×C4)⋊C7, C23⋊(C2×C28), C22.(C2×F8), (C22×F8).C2, (C23×C4)⋊2C14, C2.1(C22×F8), C24.1(C2×C14), (C2×F8).1C22, SmallGroup(448,1362)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C2×C4×F8
Chief series
C1 — C23 — C24 — C2×F8 — C22×F8 — C2×C4×F8
Lower central
C23 — C2×C4×F8

Subgroups: 753 in 127 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×22], C7, C2×C4, C2×C4 [×17], C23, C23 [×22], C14 [×3], C22×C4 [×20], C24, C24 [×2], C24 [×4], C28 [×2], C2×C14, C23×C4 [×2], C23×C4 [×4], C25, C2×C28, F8, C24×C4, C2×F8, C2×F8 [×2], C4×F8 [×2], C22×F8, C2×C4×F8

Quotients:
C1, C2 [×3], C4 [×2], C22, C7, C2×C4, C14 [×3], C28 [×2], C2×C14, C2×C28, F8, C2×F8 [×3], C4×F8 [×2], C22×F8, C2×C4×F8

Generators and relations
 G = < a,b,c,d,e,f | a2=b4=c2=d2=e2=f7=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=ed=de, fdf-1=c, fef-1=d >

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽29)

10000000
028000000
002800000
000280000
000028000
000002800
000000280
000000028
,
120000000
017000000
001700000
000170000
000017000
000001700
000000170
000000017
,
10000000
01000000
002800000
00010000
00001000
000002800
000000280
000000028
,
10000000
028000000
00100000
00010000
000028000
000002800
000000280
00000001
,
10000000
01000000
00100000
000280000
000028000
000002800
00000010
000000028
,
10000000
00100000
00010000
00001000
00000100
00000010
00000001
01000000

G:=sub<GL(8,GF(29))| [1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;
 

64 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H7A···7F14A···14R28A···28X
order12222222444444447···714···1428···28
size11117777111177778···88···88···8

64 irreducible representations

dim111111117777
type++++++
imageC1C2C2C4C7C14C14C28F8C2×F8C2×F8C4×F8
kernelC2×C4×F8C4×F8C22×F8C2×F8C24×C4C23×C4C25C24C2×C4C4C22C2
# reps12146126241214

In GAP, Magma, Sage, TeX 

C_2\times C_4\times F_8
 
% in TeX
 
G:=Group("C2xC4xF8");
 
// GroupNames label
 
G:=SmallGroup(448,1362);
 
// by ID
 
G=gap.SmallGroup(448,1362);
 
# by ID
 
G:=PCGroup([7,-2,-2,-7,-2,-2,2,2,204,998,2371,3450]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^4=c^2=d^2=e^2=f^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=e*d=d*e,f*d*f^-1=c,f*e*f^-1=d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀