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G = C22⋊C4.F5order 320 = 26·5

1st non-split extension by C22⋊C4 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22⋊C4.1F5, C22.F51C4, C23.6(C2×F5), C22.7(C4×F5), (C2×C10).2C42, (C2×Dic5).9Q8, C22.9(C4⋊F5), C23.D5.1C4, Dic5.2(C4⋊C4), C51(M4(2)⋊4C4), (C2×Dic5).254D4, C2.6(D10.3Q8), C22.17(C22⋊F5), C10.4(C2.C42), Dic5.29(C22⋊C4), C23.11D10.4C2, (C22×Dic5).170C22, (C2×C5⋊C8)⋊1C4, (C2×C10).2(C4⋊C4), (C5×C22⋊C4).1C4, (C2×C22.F5).1C2, (C22×C10).11(C2×C4), (C2×Dic5).41(C2×C4), (C2×C10).17(C22⋊C4), SmallGroup(320,205)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C22⋊C4.F5
C1C5C10Dic5C2×Dic5C22×Dic5C2×C22.F5 — C22⋊C4.F5
C5C10C2×C10 — C22⋊C4.F5
C1C2C23C22⋊C4

Generators and relations for C22⋊C4.F5
 G = < a,b,c,d,e | a2=b2=c5=e4=1, d4=b, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, dcd-1=c3, ce=ec, ede-1=abd >

Subgroups: 322 in 90 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×3], C4 [×6], C22 [×3], C22, C5, C8 [×4], C2×C4 [×8], C23, C10, C10 [×3], C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8 [×3], M4(2) [×5], C22×C4, Dic5 [×4], Dic5, C20, C2×C10 [×3], C2×C10, C42⋊C2, C2×M4(2) [×2], C5⋊C8 [×4], C2×Dic5 [×6], C2×Dic5, C2×C20, C22×C10, M4(2)⋊4C4, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8, C22.F5 [×2], C22.F5 [×3], C22×Dic5, C23.11D10, C2×C22.F5 [×2], C22⋊C4.F5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C2×F5, M4(2)⋊4C4, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C22⋊C4.F5

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I 5 8A···8H10A10B10C10D10E20A20B20C20D
order1222244444444458···8101010101020202020
size1122244551010102020420···20444888888

32 irreducible representations

dim1111111224444448
type++++-+++-
imageC1C2C2C4C4C4C4D4Q8F5C2×F5M4(2)⋊4C4C4×F5C4⋊F5C22⋊F5C22⋊C4.F5
kernelC22⋊C4.F5C23.11D10C2×C22.F5C23.D5C5×C22⋊C4C2×C5⋊C8C22.F5C2×Dic5C2×Dic5C22⋊C4C23C5C22C22C22C1
# reps1122244311122222

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

10000000
39401600000
00100000
0021400000
00001000
00000100
00000010
00000001
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000040100
000033700
000000346
000000340
,
992320000
00090000
37373200000
39401600000
000000400
000000040
000034100
000034700
,
3232010000
09000000
0032360000
001690000
000040000
000004000
000000400
000000040

G:=sub<GL(8,GF(41))| [1,39,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,16,1,21,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,33,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,6,0],[9,0,37,39,0,0,0,0,9,0,37,40,0,0,0,0,2,0,32,16,0,0,0,0,32,9,0,0,0,0,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,1,7,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0],[32,0,0,0,0,0,0,0,32,9,0,0,0,0,0,0,0,0,32,16,0,0,0,0,1,0,36,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

C22⋊C4.F5 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4.F_5
% in TeX

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

G:=SmallGroup(320,205);
// by ID

G=gap.SmallGroup(320,205);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,851,6278,3156]);
// Polycyclic

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

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