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G = C23×F8  order 448 = 26·7

Direct product of C23 and F8

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

Aliases: C23×F8, C26⋊1C7, C25⋊2C14, C24⋊(C2×C14), C23⋊(C22×C14), SmallGroup(448,1392)

Series: Derived ►Chief ►Lower central ►Upper central

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

Subgroups: 2969 in 463 conjugacy classes, 48 normal (6 characteristic)
C1, C2 [×7], C2 [×8], C22 [×7], C22 [×92], C7, C23 [×2], C23 [×199], C14 [×7], C24 [×7], C24 [×92], C2×C14 [×7], C25 [×7], C25 [×8], F8, C22×C14, C26, C2×F8 [×7], C22×F8 [×7], C23×F8

Quotients:
C1, C2 [×7], C22 [×7], C7, C23, C14 [×7], C2×C14 [×7], F8, C22×C14, C2×F8 [×7], C22×F8 [×7], C23×F8

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g7=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, gdg-1=fe=ef, geg-1=d, gfg-1=e >

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

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

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

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

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

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

(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)| (1,12)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,43)(22,39)(23,40)(24,41)(25,42)(26,36)(27,37)(28,38)(29,54)(30,55)(31,56)(32,50)(33,51)(34,52)(35,53), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,42)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,35)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(43,52)(44,53)(45,54)(46,55)(47,56)(48,50)(49,51), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,43)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(36,54)(37,55)(38,56)(39,50)(40,51)(41,52)(42,53), (1,19)(2,13)(3,34)(4,44)(5,54)(6,37)(7,28)(8,15)(9,29)(10,27)(11,38)(12,48)(14,52)(16,36)(17,55)(18,31)(20,49)(21,24)(22,32)(23,40)(25,53)(26,45)(30,46)(33,51)(35,42)(39,50)(41,43)(47,56), (1,22)(2,20)(3,14)(4,35)(5,45)(6,55)(7,38)(8,53)(9,16)(10,30)(11,28)(12,39)(13,49)(15,25)(17,37)(18,56)(19,32)(21,43)(23,33)(24,41)(26,54)(27,46)(29,36)(31,47)(34,52)(40,51)(42,44)(48,50), (1,39)(2,23)(3,21)(4,8)(5,29)(6,46)(7,56)(9,54)(10,17)(11,31)(12,22)(13,40)(14,43)(15,44)(16,26)(18,38)(19,50)(20,33)(24,34)(25,42)(27,55)(28,47)(30,37)(32,48)(35,53)(36,45)(41,52)(49,51), (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( (1,12)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,43)(22,39)(23,40)(24,41)(25,42)(26,36)(27,37)(28,38)(29,54)(30,55)(31,56)(32,50)(33,51)(34,52)(35,53), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,42)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,35)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(43,52)(44,53)(45,54)(46,55)(47,56)(48,50)(49,51), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,43)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(36,54)(37,55)(38,56)(39,50)(40,51)(41,52)(42,53), (1,19)(2,13)(3,34)(4,44)(5,54)(6,37)(7,28)(8,15)(9,29)(10,27)(11,38)(12,48)(14,52)(16,36)(17,55)(18,31)(20,49)(21,24)(22,32)(23,40)(25,53)(26,45)(30,46)(33,51)(35,42)(39,50)(41,43)(47,56), (1,22)(2,20)(3,14)(4,35)(5,45)(6,55)(7,38)(8,53)(9,16)(10,30)(11,28)(12,39)(13,49)(15,25)(17,37)(18,56)(19,32)(21,43)(23,33)(24,41)(26,54)(27,46)(29,36)(31,47)(34,52)(40,51)(42,44)(48,50), (1,39)(2,23)(3,21)(4,8)(5,29)(6,46)(7,56)(9,54)(10,17)(11,31)(12,22)(13,40)(14,43)(15,44)(16,26)(18,38)(19,50)(20,33)(24,34)(25,42)(27,55)(28,47)(30,37)(32,48)(35,53)(36,45)(41,52)(49,51), (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([(1,12),(2,13),(3,14),(4,8),(5,9),(6,10),(7,11),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,43),(22,39),(23,40),(24,41),(25,42),(26,36),(27,37),(28,38),(29,54),(30,55),(31,56),(32,50),(33,51),(34,52),(35,53)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,42),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,35),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(43,52),(44,53),(45,54),(46,55),(47,56),(48,50),(49,51)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,43),(22,32),(23,33),(24,34),(25,35),(26,29),(27,30),(28,31),(36,54),(37,55),(38,56),(39,50),(40,51),(41,52),(42,53)], [(1,19),(2,13),(3,34),(4,44),(5,54),(6,37),(7,28),(8,15),(9,29),(10,27),(11,38),(12,48),(14,52),(16,36),(17,55),(18,31),(20,49),(21,24),(22,32),(23,40),(25,53),(26,45),(30,46),(33,51),(35,42),(39,50),(41,43),(47,56)], [(1,22),(2,20),(3,14),(4,35),(5,45),(6,55),(7,38),(8,53),(9,16),(10,30),(11,28),(12,39),(13,49),(15,25),(17,37),(18,56),(19,32),(21,43),(23,33),(24,41),(26,54),(27,46),(29,36),(31,47),(34,52),(40,51),(42,44),(48,50)], [(1,39),(2,23),(3,21),(4,8),(5,29),(6,46),(7,56),(9,54),(10,17),(11,31),(12,22),(13,40),(14,43),(15,44),(16,26),(18,38),(19,50),(20,33),(24,34),(25,42),(27,55),(28,47),(30,37),(32,48),(35,53),(36,45),(41,52),(49,51)], [(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 ⊆ GL9(ℤ)

-100000000
0-10000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
-100000000
010000000
00-1000000
000-100000
0000-10000
00000-1000
000000-100
0000000-10
00000000-1
,
-100000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
010000000
001000000
000-100000
0000-10000
00000-1000
000000100
0000000-10
000000001
,
100000000
010000000
00-1000000
000-100000
0000-10000
000001000
000000-100
000000010
000000001
,
100000000
010000000
00-1000000
000-100000
000010000
00000-1000
000000100
000000010
00000000-1
,
100000000
010000000
000100000
000010000
000001000
000000100
000000010
000000001
001000000

G:=sub<GL(9,Integers())| [-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
 

64 conjugacy classes

class 1 2A···2G2H···2O7A···7F14A···14AP
order12···22···27···714···14
size11···17···78···88···8

64 irreducible representations

dim111177
type++++
imageC1C2C7C14F8C2×F8
kernelC23×F8C22×F8C26C25C23C22
# reps1764217

In GAP, Magma, Sage, TeX 

C_2^3\times F_8
 
% in TeX
 
G:=Group("C2^3xF8");
 
// GroupNames label
 
G:=SmallGroup(448,1392);
 
// by ID
 
G=gap.SmallGroup(448,1392);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,2,2,515,1202,1742]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,g*d*g^-1=f*e=e*f,g*e*g^-1=d,g*f*g^-1=e>;
 
// generators/relations
 

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