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G = C22⋊C4⋊F5order 320 = 26·5

2nd semidirect product of C22⋊C4 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22⋊C42F5, C23⋊F5.1C2, C23.2(C2×F5), C10.6(C23⋊C4), C23.F5.1C2, (C2×Dic5).10D4, (C22×Dic5)⋊6C4, (C22×D5).10D4, C51(C23.D4), C2.9(D10.D4), C22.D20.1C2, C22.15(C22⋊F5), (C5×C22⋊C4)⋊2C4, (C22×C10).9(C2×C4), (C2×C5⋊D4).84C22, (C2×C10).15(C22⋊C4), SmallGroup(320,203)

Series: Derived Chief Lower central Upper central

C1C22×C10 — C22⋊C4⋊F5
C1C5C10C2×C10C2×Dic5C2×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, cac-1=ab=ba, ad=da, eae-1=abc2, bc=cb, bd=db, be=eb, cd=dc, ece-1=ac, ede-1=d3 >

Subgroups: 394 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, C22⋊C4 [×2], 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 [×2], C5⋊D4, C2×C20, C2×F5, C22×D5, C22×C10, C23.D4, C4⋊Dic5, D10⋊C4, C5×C22⋊C4, C22.F5, C22⋊F5, C22×Dic5, C2×C5⋊D4, C23⋊F5, C23.F5, C22.D20, 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, D10.D4, C22⋊C4⋊F5

Character table of C22⋊C4⋊F5

 class 12A2B2C2D4A4B4C4D4E4F58A8B10A10B10C10D10E20A20B20C20D
 size 1124208202020404044040444888888
ρ111111111111111111111111    trivial
ρ2111111111-1-11-1-1111111111    linear of order 2
ρ311111-11-1-1-1-111111111-1-1-1-1    linear of order 2
ρ411111-11-1-1111-1-111111-1-1-1-1    linear of order 2
ρ51111-1-1-111i-i1-ii11111-1-1-1-1    linear of order 4
ρ61111-1-1-111-ii1i-i11111-1-1-1-1    linear of order 4
ρ71111-11-1-1-1-ii1-ii111111111    linear of order 4
ρ81111-11-1-1-1i-i1i-i111111111    linear of order 4
ρ9222-2-2020000200222-2-20000    orthogonal lifted from D4
ρ10222-220-20000200222-2-20000    orthogonal lifted from D4
ρ1144440-400000-100-1-1-1-1-11111    orthogonal lifted from C2×F5
ρ1244-400000000400-44-4000000    orthogonal lifted from C23⋊C4
ρ1344440400000-100-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1444-400000000-1001-115-54ζ53+2ζ4ζ5443ζ52+2ζ43ζ54343ζ54+2ζ43ζ53434ζ54+2ζ4ζ524    orthogonal lifted from D10.D4
ρ1544-400000000-1001-115-54ζ54+2ζ4ζ52443ζ54+2ζ43ζ534343ζ52+2ζ43ζ5434ζ53+2ζ4ζ54    orthogonal lifted from D10.D4
ρ16444-40000000-100-1-1-111-555-5    orthogonal lifted from C22⋊F5
ρ17444-40000000-100-1-1-1115-5-55    orthogonal lifted from C22⋊F5
ρ1844-400000000-1001-11-5543ζ52+2ζ43ζ5434ζ54+2ζ4ζ5244ζ53+2ζ4ζ5443ζ54+2ζ43ζ5343    orthogonal lifted from D10.D4
ρ1944-400000000-1001-11-5543ζ54+2ζ43ζ53434ζ53+2ζ4ζ544ζ54+2ζ4ζ52443ζ52+2ζ43ζ543    orthogonal lifted from D10.D4
ρ204-400000-2i2i004000-40000000    complex lifted from C23.D4
ρ214-4000002i-2i004000-40000000    complex lifted from C23.D4
ρ228-8000000000-200252-25000000    symplectic faithful, Schur index 2
ρ238-8000000000-200-25225000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C22⋊C4⋊F5 in GL8(𝔽41)

00010000
00100000
01000000
10000000
0000193038
0000022338
0000383220
0000380319
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
162516160000
251616160000
252525160000
252516250000
0000377357
000033311
00004040388
0000346344
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
10000000
040000000
00010000
004000000
000000400
000040000
000000040
000004000

G:=sub<GL(8,GF(41))| [0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,19,0,38,38,0,0,0,0,3,22,3,0,0,0,0,0,0,3,22,3,0,0,0,0,38,38,0,19],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,25,25,25,0,0,0,0,25,16,25,25,0,0,0,0,16,16,25,16,0,0,0,0,16,16,16,25,0,0,0,0,0,0,0,0,37,33,40,34,0,0,0,0,7,3,40,6,0,0,0,0,35,1,38,34,0,0,0,0,7,1,8,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0] >;

C22⋊C4⋊F5 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\rtimes F_5
% in TeX

G:=Group("C2^2:C4:F5");
// GroupNames label

G:=SmallGroup(320,203);
// by ID

G=gap.SmallGroup(320,203);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,219,184,675,570,297,1684,6278,3156]);
// Polycyclic

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

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Character table of C22⋊C4⋊F5 in TeX

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