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G = C23.F8  order 448 = 26·7

2nd non-split extension by C23 of F8 acting via F8/C23=C7

non-abelian, soluble, monomial

Aliases: C23.2F8, C23.84C23⋊C7, SmallGroup(448,179)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C23.84C23 — C23.F8
Chief series
C1 — C23 — C23.84C23 — C23.F8
Lower central
C23.84C23 — C23.F8
Upper central
C1

Generators and relations for C23.F8
 G = < a,b,c,d,e,f,g | a2=b2=c2=g7=1, d2=ba=ab, e2=gag-1=abc, f2=gcg-1=a, gbg-1=ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, fef-1=ce=ec, cf=fc, fdf-1=bcd, gdg-1=abef, geg-1=acd, gfg-1=abe >

C1
7C2
64C7
7C22
28C4
C23
14C2×C4
14C2×C4
14C2×C4
7C22×C4
8F8
7C2.C42
C23.84C23
C23.F8

Character table of C23.F8

 class 124A4B7A7B7C7D7E7F
 size 172828646464646464
ρ11111111111    trivial
ρ21111ζ74ζ76ζ72ζ75ζ7ζ73    linear of order 7
ρ31111ζ72ζ73ζ7ζ76ζ74ζ75    linear of order 7
ρ41111ζ75ζ74ζ76ζ7ζ73ζ72    linear of order 7
ρ51111ζ73ζ7ζ75ζ72ζ76ζ74    linear of order 7
ρ61111ζ7ζ75ζ74ζ73ζ72ζ76    linear of order 7
ρ71111ζ76ζ72ζ73ζ74ζ75ζ7    linear of order 7
ρ877-1-1000000    orthogonal lifted from F8
ρ914-2-2i2i000000    complex faithful
ρ1014-22i-2i000000    complex faithful

Smallest permutation representation of C23.F8
►On 56 points
Generators in S56
(1 31)(4 34)(5 35)(6 29)(8 54)(9 55)(11 50)(14 53)(16 25)(17 26)(18 27)(20 22)(36 44)(37 45)(38 46)(40 48)

(1 31)(2 32)(3 33)(5 35)(8 54)(11 50)(12 51)(13 52)(15 24)(17 26)(20 22)(21 23)(37 45)(40 48)(41 49)(42 43)

(2 32)(5 35)(6 29)(7 30)(8 54)(9 55)(10 56)(12 51)(17 26)(18 27)(19 28)(21 23)(37 45)(38 46)(39 47)(41 49)

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

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

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

Matrix representation of C23.F8 ►in GL14(𝔽29)

280000000000000
028000000000000
00100000000000
00010000000000
000028000000000
000002800000000
000000280000000
000000028000000
0023100002800000
00131100000280000
2622007211111001000
1311007211111000100
40002723822000010
120002723822000001
,
280000000000000
028000000000000
00100000000000
00010000000000
00001000000000
00000100000000
000000280000000
000000028000000
17250000021100000
26170000210010000
001312228000028000
00139228000002800
00282726000000280
00282226000000028
,
10000000000000
01000000000000
002800000000000
000280000000000
000028000000000
000002800000000
000000280000000
000000028000000
006281711021100000
0016181117210010000
0016177211111001000
0016207211111000100
2500000000000280
2890000000000028
,
10000000000000
028000000000000
000120000000000
001700000000000
00000100000000
000028000000000
00000001000000
00000010000000
2131166144101200000
58122815231940170000
719242492117170001200
19713692117170012000
22018614238120000120
201419714238120000012
,
017000000000000
120000000000000
00010000000000
002800000000000
00000100000000
00001000000000
000000170000000
000000012000000
612318172110190120000
0161615211710101200000
2241462010130017000
1121172010130001700
27241513441520000280
1116432272324000001
,
01000000000000
280000000000000
000280000000000
002800000000000
000017000000000
000001200000000
000000017000000
000000170000000
131121261010101200000
5218016010100120000
2427222152828001000
2817227242317170002800
23232420142630000017
514242750660000120
,
00100000000000
00010000000000
00001000000000
00000100000000
00000010000000
00000001000000
1242311218082700000
31213111812800270000
25181101000211000
11702001002100100
21287110120011110010
5928190250011110001
20262420027008220000
6111110010008220000

G:=sub<GL(14,GF(29))| [28,0,0,0,0,0,0,0,0,0,26,13,4,1,0,28,0,0,0,0,0,0,0,0,22,11,0,20,0,0,1,0,0,0,0,0,23,13,0,0,0,0,0,0,0,1,0,0,0,0,1,11,0,0,0,0,0,0,0,0,28,0,0,0,0,0,7,7,27,27,0,0,0,0,0,28,0,0,0,0,21,21,23,23,0,0,0,0,0,0,28,0,0,0,11,11,8,8,0,0,0,0,0,0,0,28,0,0,11,11,22,22,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,17,26,0,0,0,0,0,28,0,0,0,0,0,0,25,17,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,13,28,28,0,0,0,1,0,0,0,0,0,0,12,9,27,22,0,0,0,0,1,0,0,0,0,0,22,22,2,2,0,0,0,0,0,1,0,0,0,0,8,8,6,6,0,0,0,0,0,0,28,0,0,21,0,0,0,0,0,0,0,0,0,0,0,28,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,0,0,0,25,28,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,28,0,0,0,0,0,6,16,16,16,0,0,0,0,0,28,0,0,0,0,28,18,17,20,0,0,0,0,0,0,28,0,0,0,17,11,7,7,0,0,0,0,0,0,0,28,0,0,11,17,21,21,0,0,0,0,0,0,0,0,28,0,0,21,11,11,0,0,0,0,0,0,0,0,0,28,21,0,11,11,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,21,5,7,19,22,20,0,28,0,0,0,0,0,0,3,8,19,7,0,14,0,0,0,17,0,0,0,0,1,12,24,13,18,19,0,0,12,0,0,0,0,0,16,28,24,6,6,7,0,0,0,0,0,28,0,0,6,15,9,9,14,14,0,0,0,0,1,0,0,0,14,23,21,21,23,23,0,0,0,0,0,0,0,1,4,19,17,17,8,8,0,0,0,0,0,0,1,0,10,4,17,17,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,6,0,2,1,27,11,17,0,0,0,0,0,0,0,1,16,24,12,24,16,0,0,0,28,0,0,0,0,23,16,14,1,15,4,0,0,1,0,0,0,0,0,18,15,6,17,13,3,0,0,0,0,0,1,0,0,17,21,20,20,4,2,0,0,0,0,1,0,0,0,21,17,1,1,4,27,0,0,0,0,0,0,17,0,10,10,0,0,15,23,0,0,0,0,0,0,0,12,19,10,13,13,2,24,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,28,0,0,0,0,0,0,13,5,24,28,23,5,1,0,0,0,0,0,0,0,11,2,27,17,23,14,0,0,0,28,0,0,0,0,21,18,2,2,24,24,0,0,28,0,0,0,0,0,26,0,2,27,2,27,0,0,0,0,17,0,0,0,1,16,2,24,0,5,0,0,0,0,0,12,0,0,0,0,15,23,14,0,0,0,0,0,0,0,0,17,10,10,28,17,26,6,0,0,0,0,0,0,17,0,10,10,28,17,3,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,17,0],[0,0,0,0,0,0,12,3,25,11,21,5,20,6,0,0,0,0,0,0,4,12,1,7,28,9,26,11,1,0,0,0,0,0,23,13,8,0,7,28,24,11,0,1,0,0,0,0,1,11,11,20,11,19,20,10,0,0,1,0,0,0,12,18,0,0,0,0,0,0,0,0,0,1,0,0,18,12,1,1,12,25,27,10,0,0,0,0,1,0,0,8,0,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,21,11,11,8,8,0,0,0,0,0,0,0,27,21,0,11,11,22,22,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

C23.F8 in GAP, Magma, Sage, TeX 

C_2^3.F_8
 
% in TeX
 
G:=Group("C2^3.F8");
 
// GroupNames label
 
G:=SmallGroup(448,179);
 
// by ID
 
G=gap.SmallGroup(448,179);
 
# by ID
 
G:=PCGroup([7,-7,-2,2,2,-2,2,2,197,792,590,219,268,983,570,521,80,7844,11765,5494]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^7=1,d^2=b*a=a*b,e^2=g*a*g^-1=a*b*c,f^2=g*c*g^-1=a,g*b*g^-1=a*c=c*a,e*d*e^-1=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,f*e*f^-1=c*e=e*c,c*f=f*c,f*d*f^-1=b*c*d,g*d*g^-1=a*b*e*f,g*e*g^-1=a*c*d,g*f*g^-1=a*b*e>;
 
// generators/relations
 

Export

Subgroup lattice of C23.F8 in TeX
Character table of C23.F8 in TeX

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