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G = C24.36D10  order 320 = 26·5

36th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.36D10, C10.312+ 1+4, C22≀C2⋊8D5, C20⋊D4⋊13C2, C20⋊2D4⋊15C2, (C2×D4).88D10, C22⋊C4.3D10, D10⋊D4⋊15C2, Dic5⋊D4⋊6C2, (C2×D20)⋊21C22, C24⋊2D5⋊10C2, (C2×C20).33C23, C4⋊Dic5⋊28C22, (C2×C10).139C24, (C4×Dic5)⋊19C22, D10.12D4⋊15C2, C23.D5⋊19C22, C2.33(D4⋊6D10), D10⋊C4⋊16C22, C5⋊1(C22.54C24), (D4×C10).113C22, C23.D10⋊13C2, C10.D4⋊13C22, C23.18D10⋊6C2, (C23×C10).71C22, (C2×Dic5).64C23, (C22×D5).58C23, C22.160(C23×D5), C23.111(C22×D5), (C22×C10).184C23, (C22×Dic5)⋊17C22, (C2×C4×D5)⋊11C22, (C5×C22≀C2)⋊10C2, (C2×C5⋊D4)⋊11C22, (C2×C4).33(C22×D5), (C5×C22⋊C4).4C22, SmallGroup(320,1267)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C24.36D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C24.36D10
Lower central
C5 — C2×C10 — C24.36D10
Upper central
C1 — C22 — C22≀C2

Generators and relations for C24.36D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 998 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22×C10, C22.54C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, D4×C10, D4×C10, C23×C10, C23.D10, D10.12D4, D10⋊D4, C23.18D10, C20⋊2D4, Dic5⋊D4, C20⋊D4, C24⋊2D5, C5×C22≀C2, C24.36D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C22.54C24, C23×D5, D4⋊6D10, C24.36D10

Smallest permutation representation of C24.36D10
►On 80 points
Generators in S80
(2 33)(4 35)(6 37)(8 39)(10 21)(12 23)(14 25)(16 27)(18 29)(20 31)(41 65)(42 52)(43 67)(44 54)(45 69)(46 56)(47 71)(48 58)(49 73)(50 60)(51 75)(53 77)(55 79)(57 61)(59 63)(62 72)(64 74)(66 76)(68 78)(70 80)

(1 11)(3 13)(5 15)(7 17)(9 19)(22 32)(24 34)(26 36)(28 38)(30 40)(41 65)(42 76)(43 67)(44 78)(45 69)(46 80)(47 71)(48 62)(49 73)(50 64)(51 75)(52 66)(53 77)(54 68)(55 79)(56 70)(57 61)(58 72)(59 63)(60 74)

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D···4I5A5B10A···10F10G···10R10S10T20A···20F
order12222222224444···45510···1010···10101020···20
size11114444202044420···20222···24···4888···8

47 irreducible representations

dim1111111111222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D5D10D10D102+ 1+4D4⋊6D10
kernelC24.36D10C23.D10D10.12D4D10⋊D4C23.18D10C20⋊2D4Dic5⋊D4C20⋊D4C24⋊2D5C5×C22≀C2C22≀C2C22⋊C4C2×D4C24C10C2
# reps12221221212662312

Matrix representation of C24.36D10 ►in GL8(𝔽41)

10000000
040000000
004000000
00010000
00001000
00000100
000000400
000000040
,
400000000
01000000
004000000
00010000
000040000
00000100
00000010
000000040
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
018000000
230000000
000160000
002500000
00000100
000040000
00000001
000000400
,
000160000
002500000
018000000
230000000
00000001
000000400
00000100
000040000

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40],[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],[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],[0,23,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,0,0,23,0,0,0,0,0,0,18,0,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0] >;
 

C24.36D10 in GAP, Magma, Sage, TeX 

C_2^4._{36}D_{10}
 
% in TeX
 
G:=Group("C2^4.36D10");
 
// GroupNames label
 
G:=SmallGroup(320,1267);
 
// by ID
 
G=gap.SmallGroup(320,1267);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
 
// generators/relations
 

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