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G = C22.58C24⋊C5  order 320 = 26·5

The semidirect product of C22.58C24 and C5 acting faithfully

non-abelian, soluble

Aliases: C22.58C24⋊C5, C22.2(C24⋊C5), C2.(2- (1+4)⋊C5), SmallGroup(320,1012)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C22.58C24 — C22.58C24⋊C5
Chief series
C1 — C2 — C22 — C22.58C24 — C22.58C24⋊C5
Lower central
C22.58C24 — C22.58C24⋊C5

Subgroups: 182 in 28 conjugacy classes, 7 normal (4 characteristic)
C1, C2 [×3], C4 [×3], C22, C5, C2×C4 [×3], C10 [×3], C42, C4⋊C4 [×6], C2×C10, C42.C2 [×3], C22.58C24, C22.58C24⋊C5

Quotients:
C1, C5, C24⋊C5, 2- (1+4)⋊C5 [×3], C22.58C24⋊C5

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=g5=1, c2=f2=a, d2=e2=b, ab=ba, dcd-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=gfg-1=bc=cb, bd=db, be=eb, bf=fb, bg=gb, fcf-1=abc, gcg-1=abcde, ede-1=abd, gdg-1=abcd, ef=fe, geg-1=bcdef >

Smallest permutation representation
►On 64 points
Generators in S64
(1 2)(3 4)(5 31)(6 32)(7 33)(8 34)(9 30)(10 22)(11 23)(12 24)(13 20)(14 21)(15 54)(16 50)(17 51)(18 52)(19 53)(25 42)(26 43)(27 44)(28 40)(29 41)(35 60)(36 61)(37 62)(38 63)(39 64)(45 57)(46 58)(47 59)(48 55)(49 56)

(1 4)(2 3)(5 24)(6 20)(7 21)(8 22)(9 23)(10 34)(11 30)(12 31)(13 32)(14 33)(15 44)(16 40)(17 41)(18 42)(19 43)(25 52)(26 53)(27 54)(28 50)(29 51)(35 49)(36 45)(37 46)(38 47)(39 48)(55 64)(56 60)(57 61)(58 62)(59 63)

(1 48 2 55)(3 64 4 39)(5 43 31 26)(6 45 32 57)(7 10 33 22)(8 21 34 14)(9 37 30 62)(11 58 23 46)(12 53 24 19)(13 61 20 36)(15 18 54 52)(16 63 50 38)(17 49 51 56)(25 44 42 27)(28 47 40 59)(29 60 41 35)

(1 8 4 22)(2 34 3 10)(5 25 24 52)(6 16 20 40)(7 55 21 64)(9 49 23 35)(11 60 30 56)(12 18 31 42)(13 28 32 50)(14 39 33 48)(15 26 44 53)(17 46 41 37)(19 54 43 27)(29 62 51 58)(36 47 45 38)(57 63 61 59)

(1 6 4 20)(2 32 3 13)(5 58 24 62)(7 47 21 38)(8 28 22 50)(9 19 23 43)(10 16 34 40)(11 26 30 53)(12 37 31 46)(14 63 33 59)(15 49 44 35)(17 52 41 25)(18 29 42 51)(27 60 54 56)(36 39 45 48)(55 61 64 57)

(1 35 2 60)(3 56 4 49)(5 47 31 59)(6 15 32 54)(7 37 33 62)(8 30 34 9)(10 23 22 11)(12 63 24 38)(13 27 20 44)(14 58 21 46)(16 43 50 26)(17 55 51 48)(18 36 52 61)(19 28 53 40)(25 57 42 45)(29 39 41 64)

(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 57 58 59)(60 61 62 63 64)
 
G:=sub<Sym(64)| (1,2)(3,4)(5,31)(6,32)(7,33)(8,34)(9,30)(10,22)(11,23)(12,24)(13,20)(14,21)(15,54)(16,50)(17,51)(18,52)(19,53)(25,42)(26,43)(27,44)(28,40)(29,41)(35,60)(36,61)(37,62)(38,63)(39,64)(45,57)(46,58)(47,59)(48,55)(49,56), (1,4)(2,3)(5,24)(6,20)(7,21)(8,22)(9,23)(10,34)(11,30)(12,31)(13,32)(14,33)(15,44)(16,40)(17,41)(18,42)(19,43)(25,52)(26,53)(27,54)(28,50)(29,51)(35,49)(36,45)(37,46)(38,47)(39,48)(55,64)(56,60)(57,61)(58,62)(59,63), (1,48,2,55)(3,64,4,39)(5,43,31,26)(6,45,32,57)(7,10,33,22)(8,21,34,14)(9,37,30,62)(11,58,23,46)(12,53,24,19)(13,61,20,36)(15,18,54,52)(16,63,50,38)(17,49,51,56)(25,44,42,27)(28,47,40,59)(29,60,41,35), (1,8,4,22)(2,34,3,10)(5,25,24,52)(6,16,20,40)(7,55,21,64)(9,49,23,35)(11,60,30,56)(12,18,31,42)(13,28,32,50)(14,39,33,48)(15,26,44,53)(17,46,41,37)(19,54,43,27)(29,62,51,58)(36,47,45,38)(57,63,61,59), (1,6,4,20)(2,32,3,13)(5,58,24,62)(7,47,21,38)(8,28,22,50)(9,19,23,43)(10,16,34,40)(11,26,30,53)(12,37,31,46)(14,63,33,59)(15,49,44,35)(17,52,41,25)(18,29,42,51)(27,60,54,56)(36,39,45,48)(55,61,64,57), (1,35,2,60)(3,56,4,49)(5,47,31,59)(6,15,32,54)(7,37,33,62)(8,30,34,9)(10,23,22,11)(12,63,24,38)(13,27,20,44)(14,58,21,46)(16,43,50,26)(17,55,51,48)(18,36,52,61)(19,28,53,40)(25,57,42,45)(29,39,41,64), (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)(55,56,57,58,59)(60,61,62,63,64)>;
 
G:=Group( (1,2)(3,4)(5,31)(6,32)(7,33)(8,34)(9,30)(10,22)(11,23)(12,24)(13,20)(14,21)(15,54)(16,50)(17,51)(18,52)(19,53)(25,42)(26,43)(27,44)(28,40)(29,41)(35,60)(36,61)(37,62)(38,63)(39,64)(45,57)(46,58)(47,59)(48,55)(49,56), (1,4)(2,3)(5,24)(6,20)(7,21)(8,22)(9,23)(10,34)(11,30)(12,31)(13,32)(14,33)(15,44)(16,40)(17,41)(18,42)(19,43)(25,52)(26,53)(27,54)(28,50)(29,51)(35,49)(36,45)(37,46)(38,47)(39,48)(55,64)(56,60)(57,61)(58,62)(59,63), (1,48,2,55)(3,64,4,39)(5,43,31,26)(6,45,32,57)(7,10,33,22)(8,21,34,14)(9,37,30,62)(11,58,23,46)(12,53,24,19)(13,61,20,36)(15,18,54,52)(16,63,50,38)(17,49,51,56)(25,44,42,27)(28,47,40,59)(29,60,41,35), (1,8,4,22)(2,34,3,10)(5,25,24,52)(6,16,20,40)(7,55,21,64)(9,49,23,35)(11,60,30,56)(12,18,31,42)(13,28,32,50)(14,39,33,48)(15,26,44,53)(17,46,41,37)(19,54,43,27)(29,62,51,58)(36,47,45,38)(57,63,61,59), (1,6,4,20)(2,32,3,13)(5,58,24,62)(7,47,21,38)(8,28,22,50)(9,19,23,43)(10,16,34,40)(11,26,30,53)(12,37,31,46)(14,63,33,59)(15,49,44,35)(17,52,41,25)(18,29,42,51)(27,60,54,56)(36,39,45,48)(55,61,64,57), (1,35,2,60)(3,56,4,49)(5,47,31,59)(6,15,32,54)(7,37,33,62)(8,30,34,9)(10,23,22,11)(12,63,24,38)(13,27,20,44)(14,58,21,46)(16,43,50,26)(17,55,51,48)(18,36,52,61)(19,28,53,40)(25,57,42,45)(29,39,41,64), (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)(55,56,57,58,59)(60,61,62,63,64) );
 
G=PermutationGroup([(1,2),(3,4),(5,31),(6,32),(7,33),(8,34),(9,30),(10,22),(11,23),(12,24),(13,20),(14,21),(15,54),(16,50),(17,51),(18,52),(19,53),(25,42),(26,43),(27,44),(28,40),(29,41),(35,60),(36,61),(37,62),(38,63),(39,64),(45,57),(46,58),(47,59),(48,55),(49,56)], [(1,4),(2,3),(5,24),(6,20),(7,21),(8,22),(9,23),(10,34),(11,30),(12,31),(13,32),(14,33),(15,44),(16,40),(17,41),(18,42),(19,43),(25,52),(26,53),(27,54),(28,50),(29,51),(35,49),(36,45),(37,46),(38,47),(39,48),(55,64),(56,60),(57,61),(58,62),(59,63)], [(1,48,2,55),(3,64,4,39),(5,43,31,26),(6,45,32,57),(7,10,33,22),(8,21,34,14),(9,37,30,62),(11,58,23,46),(12,53,24,19),(13,61,20,36),(15,18,54,52),(16,63,50,38),(17,49,51,56),(25,44,42,27),(28,47,40,59),(29,60,41,35)], [(1,8,4,22),(2,34,3,10),(5,25,24,52),(6,16,20,40),(7,55,21,64),(9,49,23,35),(11,60,30,56),(12,18,31,42),(13,28,32,50),(14,39,33,48),(15,26,44,53),(17,46,41,37),(19,54,43,27),(29,62,51,58),(36,47,45,38),(57,63,61,59)], [(1,6,4,20),(2,32,3,13),(5,58,24,62),(7,47,21,38),(8,28,22,50),(9,19,23,43),(10,16,34,40),(11,26,30,53),(12,37,31,46),(14,63,33,59),(15,49,44,35),(17,52,41,25),(18,29,42,51),(27,60,54,56),(36,39,45,48),(55,61,64,57)], [(1,35,2,60),(3,56,4,49),(5,47,31,59),(6,15,32,54),(7,37,33,62),(8,30,34,9),(10,23,22,11),(12,63,24,38),(13,27,20,44),(14,58,21,46),(16,43,50,26),(17,55,51,48),(18,36,52,61),(19,28,53,40),(25,57,42,45),(29,39,41,64)], [(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),(55,56,57,58,59),(60,61,62,63,64)])
 

Matrix representation ►G ⊆ GL8(𝔽41)

400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
03129180000
10018120000
29180310000
18121000000
00002313110
00001318011
00001101828
00000112823
,
701130000
073300000
30383400000
38110340000
000007431
00003403137
0000371007
0000104340
,
004000000
000400000
400000000
040000000
000000400
000000040
00001000
00000100
,
0416130000
37013250000
1613040000
13253700000
0000251203
00001216380
00000382512
0000301216
,
10000000
29180310000
000400000
10018120000
00001000
00003002313
00000001
00002823030

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,10,29,18,0,0,0,0,31,0,18,12,0,0,0,0,29,18,0,10,0,0,0,0,18,12,31,0,0,0,0,0,0,0,0,0,23,13,11,0,0,0,0,0,13,18,0,11,0,0,0,0,11,0,18,28,0,0,0,0,0,11,28,23],[7,0,30,38,0,0,0,0,0,7,38,11,0,0,0,0,11,3,34,0,0,0,0,0,3,30,0,34,0,0,0,0,0,0,0,0,0,34,37,10,0,0,0,0,7,0,10,4,0,0,0,0,4,31,0,34,0,0,0,0,31,37,7,0],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[0,37,16,13,0,0,0,0,4,0,13,25,0,0,0,0,16,13,0,37,0,0,0,0,13,25,4,0,0,0,0,0,0,0,0,0,25,12,0,3,0,0,0,0,12,16,38,0,0,0,0,0,0,38,25,12,0,0,0,0,3,0,12,16],[1,29,0,10,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,31,40,12,0,0,0,0,0,0,0,0,1,30,0,28,0,0,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,0,0,13,1,30] >;
 

Character table of C22.58C24⋊C5

 class 12A2B2C4A4B4C5A5B5C5D10A10B10C10D10E10F10G10H10I10J10K10L
 size 111120202016161616161616161616161616161616
ρ111111111111111111111111    trivial
ρ21111111ζ5ζ53ζ54ζ52ζ53ζ5ζ5ζ5ζ53ζ54ζ54ζ54ζ53ζ52ζ52ζ52    linear of order 5
ρ31111111ζ53ζ54ζ52ζ5ζ54ζ53ζ53ζ53ζ54ζ52ζ52ζ52ζ54ζ5ζ5ζ5    linear of order 5
ρ41111111ζ54ζ52ζ5ζ53ζ52ζ54ζ54ζ54ζ52ζ5ζ5ζ5ζ52ζ53ζ53ζ53    linear of order 5
ρ51111111ζ52ζ5ζ53ζ54ζ5ζ52ζ52ζ52ζ5ζ53ζ53ζ53ζ5ζ54ζ54ζ54    linear of order 5
ρ644-4-4000-1-1-1-1111-1111-1-111-1    symplectic lifted from 2- (1+4)⋊C5, Schur index 2
ρ74-4-44000-1-1-1-1-1-1111-1111-111    symplectic lifted from 2- (1+4)⋊C5, Schur index 2
ρ84-44-4000-1-1-1-111-11-11-1111-11    symplectic lifted from 2- (1+4)⋊C5, Schur index 2
ρ94-44-4000-ζ52-ζ5-ζ53-ζ54ζ5ζ52-ζ52ζ52-ζ5ζ53-ζ53ζ53ζ5ζ54-ζ54ζ54    complex lifted from 2- (1+4)⋊C5
ρ104-4-44000-ζ5-ζ53-ζ54-ζ52-ζ53-ζ5ζ5ζ5ζ53-ζ54ζ54ζ54ζ53-ζ52ζ52ζ52    complex lifted from 2- (1+4)⋊C5
ρ114-4-44000-ζ52-ζ5-ζ53-ζ54-ζ5-ζ52ζ52ζ52ζ5-ζ53ζ53ζ53ζ5-ζ54ζ54ζ54    complex lifted from 2- (1+4)⋊C5
ρ124-44-4000-ζ54-ζ52-ζ5-ζ53ζ52ζ54-ζ54ζ54-ζ52ζ5-ζ5ζ5ζ52ζ53-ζ53ζ53    complex lifted from 2- (1+4)⋊C5
ρ134-4-44000-ζ53-ζ54-ζ52-ζ5-ζ54-ζ53ζ53ζ53ζ54-ζ52ζ52ζ52ζ54-ζ5ζ5ζ5    complex lifted from 2- (1+4)⋊C5
ρ1444-4-4000-ζ5-ζ53-ζ54-ζ52ζ53ζ5ζ5-ζ5ζ53ζ54ζ54-ζ54-ζ53ζ52ζ52-ζ52    complex lifted from 2- (1+4)⋊C5
ρ1544-4-4000-ζ54-ζ52-ζ5-ζ53ζ52ζ54ζ54-ζ54ζ52ζ5ζ5-ζ5-ζ52ζ53ζ53-ζ53    complex lifted from 2- (1+4)⋊C5
ρ164-44-4000-ζ53-ζ54-ζ52-ζ5ζ54ζ53-ζ53ζ53-ζ54ζ52-ζ52ζ52ζ54ζ5-ζ5ζ5    complex lifted from 2- (1+4)⋊C5
ρ174-44-4000-ζ5-ζ53-ζ54-ζ52ζ53ζ5-ζ5ζ5-ζ53ζ54-ζ54ζ54ζ53ζ52-ζ52ζ52    complex lifted from 2- (1+4)⋊C5
ρ1844-4-4000-ζ53-ζ54-ζ52-ζ5ζ54ζ53ζ53-ζ53ζ54ζ52ζ52-ζ52-ζ54ζ5ζ5-ζ5    complex lifted from 2- (1+4)⋊C5
ρ194-4-44000-ζ54-ζ52-ζ5-ζ53-ζ52-ζ54ζ54ζ54ζ52-ζ5ζ5ζ5ζ52-ζ53ζ53ζ53    complex lifted from 2- (1+4)⋊C5
ρ2044-4-4000-ζ52-ζ5-ζ53-ζ54ζ5ζ52ζ52-ζ52ζ5ζ53ζ53-ζ53-ζ5ζ54ζ54-ζ54    complex lifted from 2- (1+4)⋊C5
ρ2155551-310000000000000000    orthogonal lifted from C24⋊C5
ρ22555511-30000000000000000    orthogonal lifted from C24⋊C5
ρ235555-3110000000000000000    orthogonal lifted from C24⋊C5

In GAP, Magma, Sage, TeX 

C_2^2._{58}C_2^4\rtimes C_5
 
% in TeX
 
G:=Group("C2^2.58C2^4:C5");
 
// GroupNames label
 
G:=SmallGroup(320,1012);
 
// by ID
 
G=gap.SmallGroup(320,1012);
 
# by ID
 
G:=PCGroup([7,-5,-2,2,2,2,-2,-2,561,456,947,387,184,1543,1466,745,360,2629,851,718,375,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=g^5=1,c^2=f^2=a,d^2=e^2=b,a*b=b*a,d*c*d^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=g*f*g^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*c*f^-1=a*b*c,g*c*g^-1=a*b*c*d*e,e*d*e^-1=a*b*d,g*d*g^-1=a*b*c*d,e*f=f*e,g*e*g^-1=b*c*d*e*f>;
 
// generators/relations
 

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