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G = C23xF7  order 336 = 24·3·7

Direct product of C23 and F7

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

Aliases: C23xF7, C7:C3:C24, C7:(C23xC6), C14:(C22xC6), D7:(C22xC6), D14:4(C2xC6), (C23xD7):2C3, (C22xC14):3C6, (C22xD7):6C6, (C2xC7:C3):C23, (C2xC14):5(C2xC6), (C23xC7:C3):2C2, (C22xC7:C3):3C22, SmallGroup(336,216)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7 — C23xF7
Chief series
C1 — C7 — C7:C3 — F7 — C2xF7 — C22xF7 — C23xF7
Lower central
C7 — C23xF7
Upper central
C1 — C23

Generators and relations for C23xF7
 G = < a,b,c,d,e | a2=b2=c2=d7=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 976 in 268 conjugacy classes, 150 normal (8 characteristic)
C1, C2, C2, C3, C22, C22, C6, C7, C23, C23, C2xC6, D7, C14, C24, C7:C3, C22xC6, D14, C2xC14, F7, C2xC7:C3, C23xC6, C22xD7, C22xC14, C2xF7, C22xC7:C3, C23xD7, C22xF7, C23xC7:C3, C23xF7
Quotients: C1, C2, C3, C22, C6, C23, C2xC6, C24, C22xC6, F7, C23xC6, C2xF7, C22xF7, C23xF7

Smallest permutation representation of C23xF7
►On 56 points
Generators in S56
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)

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

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

56 conjugacy classes

class 1 2A···2G2H···2O3A3B6A···6AD 7 14A···14G
order12···22···2336···6714···14
size11···17···7777···766···6

56 irreducible representations

dim11111166
type+++++
imageC1C2C2C3C6C6F7C2xF7
kernelC23xF7C22xF7C23xC7:C3C23xD7C22xD7C22xC14C23C22
# reps1141228217

Matrix representation of C23xF7 ►in GL8(F43)

420000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
,
420000000
01000000
004200000
000420000
000042000
000004200
000000420
000000042
,
420000000
042000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00424242424242
00100000
00010000
00001000
00000100
00000010
,
70000000
07000000
004200000
000000042
000004200
000420000
00111111
000000420

G:=sub<GL(8,GF(43))| [42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,42],[42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,1,0,0,0,0,0,0,42,0,1,0,0,0,0,0,42,0,0,1,0,0,0,0,42,0,0,0,1,0,0,0,42,0,0,0,0,1,0,0,42,0,0,0,0,0],[7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,42,0,0,0,1,0,0,0,0,0,0,42,1,0,0,0,0,0,0,0,1,0,0,0,0,0,42,0,1,0,0,0,0,0,0,0,1,42,0,0,0,42,0,0,1,0] >;
 

C23xF7 in GAP, Magma, Sage, TeX 

C_2^3\times F_7
 
% in TeX
 
G:=Group("C2^3xF7");
 
// GroupNames label
 
G:=SmallGroup(336,216);
 
// by ID
 
G=gap.SmallGroup(336,216);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,10373,461]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=e^6=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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
 
// generators/relations
 

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