Copied to
clipboard

G = (C22×C4)⋊F5order 320 = 26·5

1st semidirect product of C22×C4 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C22×C4)⋊1F5, (C22×C20)⋊5C4, C23.D56C4, C23⋊F5.2C2, C2.4(C23⋊F5), C23.18(C2×F5), C23.F5.2C2, (C2×Dic5).11D4, (C22×D5).11D4, C52(C23.D4), C10.13(C23⋊C4), C22.18(C22⋊F5), C23.23D10.1C2, (C22×C10).45(C2×C4), (C2×C5⋊D4).85C22, (C2×C10).29(C22⋊C4), SmallGroup(320,254)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C22×C4)⋊F5
C1C5C10C2×C10C22×D5C2×C5⋊D4C23⋊F5 — (C22×C4)⋊F5
C5C10C2×C10C22×C10 — (C22×C4)⋊F5
C1C2C22C23C22×C4

Generators and relations for (C22×C4)⋊F5
 G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=abc2, bc=cb, bd=db, ebe-1=bc2, cd=dc, ece-1=abc-1, ede-1=d3 >

Subgroups: 378 in 68 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×4], C5, C8, C2×C4 [×5], D4, C23, C23, D5, C10, C10 [×2], C22⋊C4 [×3], C4⋊C4, M4(2), C22×C4, C2×D4, Dic5 [×2], C20, F5, D10 [×2], C2×C10, C2×C10 [×2], C23⋊C4, C4.D4, C22.D4, C5⋊C8, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20 [×2], C2×F5, C22×D5, C22×C10, C23.D4, C10.D4, D10⋊C4, C23.D5, C22.F5, C22⋊F5, C2×C5⋊D4, C22×C20, C23⋊F5, C23.F5, C23.23D10, (C22×C4)⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C23.D4, C22⋊F5, C23⋊F5, (C22×C4)⋊F5

Character table of (C22×C4)⋊F5

 class 12A2B2C2D4A4B4C4D4E4F58A8B10A10B10C10D10E10F10G20A20B20C20D20E20F20G20H
 size 112420442040404044040444444444444444
ρ111111111111111111111111111111    trivial
ρ211111111-11-11-1-1111111111111111    linear of order 2
ρ311111-1-11-1-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-1-111-111-1-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-i1i1i-i1111111-1-1-1-1-1-1-1-1    linear of order 4
ρ61111-1-1-1-1i1-i1-ii1111111-1-1-1-1-1-1-1-1    linear of order 4
ρ71111-111-1i-1-i1i-i111111111111111    linear of order 4
ρ81111-111-1-i-1i1-ii111111111111111    linear of order 4
ρ9222-2200-200020022-2-2-2-2200000000    orthogonal lifted from D4
ρ10222-2-200200020022-2-2-2-2200000000    orthogonal lifted from D4
ρ1144440440000-100-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1244440-4-40000-100-1-1-1-1-1-1-111111111    orthogonal lifted from C2×F5
ρ1344-4000000004004-40000-400000000    orthogonal lifted from C23⋊C4
ρ14444-40000000-100-1-11111-1-5-55-5-5555    orthogonal lifted from C22⋊F5
ρ15444-40000000-100-1-11111-155-555-5-5-5    orthogonal lifted from C22⋊F5
ρ1644-400000000-100-11-555-5154+2ζ52+153+2ζ5+154+2ζ53+154+2ζ52+153+2ζ5+154+2ζ53+152+2ζ5+152+2ζ5+1    complex lifted from C23⋊F5
ρ1744-400000000-100-11-555-5153+2ζ5+154+2ζ52+152+2ζ5+153+2ζ5+154+2ζ52+152+2ζ5+154+2ζ53+154+2ζ53+1    complex lifted from C23⋊F5
ρ184-4000-2i2i0000400-40000002i2i2i-2i-2i-2i-2i2i    complex lifted from C23.D4
ρ194-40002i-2i0000400-4000000-2i-2i-2i2i2i2i2i-2i    complex lifted from C23.D4
ρ2044-400000000-100-115-5-55152+2ζ5+154+2ζ53+154+2ζ52+152+2ζ5+154+2ζ53+154+2ζ52+153+2ζ5+153+2ζ5+1    complex lifted from C23⋊F5
ρ2144-400000000-100-115-5-55154+2ζ53+152+2ζ5+153+2ζ5+154+2ζ53+152+2ζ5+153+2ζ5+154+2ζ52+154+2ζ52+1    complex lifted from C23⋊F5
ρ224-4000-2i2i0000-1001-553+2ζ5+154+2ζ53+152+2ζ5+154+2ζ52+1543ζ54343ζ544343ζ52434ζ544ζ5444ζ5244ζ53443ζ5343    complex faithful
ρ234-40002i-2i0000-1001552+2ζ5+153+2ζ5+154+2ζ52+154+2ζ53+1-54ζ5244ζ5344ζ54443ζ524343ζ534343ζ544343ζ5434ζ54    complex faithful
ρ244-40002i-2i0000-1001-554+2ζ52+152+2ζ5+154+2ζ53+153+2ζ5+154ζ5444ζ544ζ53443ζ544343ζ54343ζ534343ζ52434ζ524    complex faithful
ρ254-4000-2i2i0000-1001554+2ζ53+154+2ζ52+153+2ζ5+152+2ζ5+1-543ζ534343ζ524343ζ5434ζ5344ζ5244ζ544ζ54443ζ5443    complex faithful
ρ264-40002i-2i0000-1001554+2ζ53+154+2ζ52+153+2ζ5+152+2ζ5+1-54ζ5344ζ5244ζ5443ζ534343ζ524343ζ54343ζ54434ζ544    complex faithful
ρ274-4000-2i2i0000-1001552+2ζ5+153+2ζ5+154+2ζ52+154+2ζ53+1-543ζ524343ζ534343ζ54434ζ5244ζ5344ζ5444ζ5443ζ543    complex faithful
ρ284-4000-2i2i0000-1001-554+2ζ52+152+2ζ5+154+2ζ53+153+2ζ5+1543ζ544343ζ54343ζ53434ζ5444ζ544ζ5344ζ52443ζ5243    complex faithful
ρ294-40002i-2i0000-1001-553+2ζ5+154+2ζ53+152+2ζ5+154+2ζ52+154ζ544ζ5444ζ52443ζ54343ζ544343ζ524343ζ53434ζ534    complex faithful

Smallest permutation representation of (C22×C4)⋊F5
On 80 points
Generators in S80
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 46 6 41)(2 47 7 42)(3 48 8 43)(4 49 9 44)(5 50 10 45)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(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 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 31 26 36)(22 33 30 39)(23 35 29 37)(24 32 28 40)(25 34 27 38)(41 61 51 76)(42 63 55 79)(43 65 54 77)(44 62 53 80)(45 64 52 78)(46 66 56 71)(47 68 60 74)(48 70 59 72)(49 67 58 75)(50 69 57 73)

G:=sub<Sym(80)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,46,6,41)(2,47,7,42)(3,48,8,43)(4,49,9,44)(5,50,10,45)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (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)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,51,76)(42,63,55,79)(43,65,54,77)(44,62,53,80)(45,64,52,78)(46,66,56,71)(47,68,60,74)(48,70,59,72)(49,67,58,75)(50,69,57,73)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,46,6,41)(2,47,7,42)(3,48,8,43)(4,49,9,44)(5,50,10,45)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (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)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,51,76)(42,63,55,79)(43,65,54,77)(44,62,53,80)(45,64,52,78)(46,66,56,71)(47,68,60,74)(48,70,59,72)(49,67,58,75)(50,69,57,73) );

G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,46,6,41),(2,47,7,42),(3,48,8,43),(4,49,9,44),(5,50,10,45),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75)], [(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),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17),(21,31,26,36),(22,33,30,39),(23,35,29,37),(24,32,28,40),(25,34,27,38),(41,61,51,76),(42,63,55,79),(43,65,54,77),(44,62,53,80),(45,64,52,78),(46,66,56,71),(47,68,60,74),(48,70,59,72),(49,67,58,75),(50,69,57,73)])

Matrix representation of (C22×C4)⋊F5 in GL4(𝔽41) generated by

5103219
22273213
2891419
2293136
,
22033
3819380
0381938
33022
,
2213138
325434
7103211
30374021
,
40404040
1000
0100
0010
,
1000
0001
0100
40404040
G:=sub<GL(4,GF(41))| [5,22,28,22,10,27,9,9,32,32,14,31,19,13,19,36],[22,38,0,3,0,19,38,3,3,38,19,0,3,0,38,22],[22,3,7,30,1,25,10,37,31,4,32,40,38,34,11,21],[40,1,0,0,40,0,1,0,40,0,0,1,40,0,0,0],[1,0,0,40,0,0,1,40,0,0,0,40,0,1,0,40] >;

(C22×C4)⋊F5 in GAP, Magma, Sage, TeX

(C_2^2\times C_4)\rtimes F_5
% in TeX

G:=Group("(C2^2xC4):F5");
// GroupNames label

G:=SmallGroup(320,254);
// by ID

G=gap.SmallGroup(320,254);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,675,297,1684,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c^2,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^2,c*d=d*c,e*c*e^-1=a*b*c^-1,e*d*e^-1=d^3>;
// generators/relations

Export

Character table of (C22×C4)⋊F5 in TeX

׿
×
𝔽