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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⋊2D4⋊15C2, C20⋊D4⋊13C2, (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, C10.D4⋊13C22, C23.18D10⋊6C2, C23.D10⋊13C2, (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

Subgroups: 998 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×12], C23 [×2], C23 [×2], C23 [×5], D5 [×2], C10, C10 [×2], C10 [×4], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×9], C24, Dic5 [×6], C20 [×3], D10 [×6], C2×C10, C2×C10 [×16], C22≀C2, C22≀C2 [×2], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, C4×D5 [×2], D20, C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×3], C22×D5 [×2], C22×C10 [×2], C22×C10 [×2], C22×C10 [×3], C22.54C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5, C23.D5 [×6], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×C4×D5 [×2], C2×D20, C22×Dic5, C2×C5⋊D4 [×8], D4×C10, D4×C10 [×2], C23×C10, C23.D10 [×2], D10.12D4 [×2], D10⋊D4 [×2], C23.18D10, C20⋊2D4 [×2], Dic5⋊D4 [×2], C20⋊D4, C24⋊2D5 [×2], C5×C22≀C2, C24.36D10

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4) [×3], C22×D5 [×7], C22.54C24, C23×D5, D4⋊6D10 [×3], C24.36D10

Generators and relations
 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 >

Smallest permutation representation
►On 80 points
Generators in S80
(2 59)(4 41)(6 43)(8 45)(10 47)(12 49)(14 51)(16 53)(18 55)(20 57)(21 80)(22 32)(23 62)(24 34)(25 64)(26 36)(27 66)(28 38)(29 68)(30 40)(31 70)(33 72)(35 74)(37 76)(39 78)(61 71)(63 73)(65 75)(67 77)(69 79)

(1 11)(3 13)(5 15)(7 17)(9 19)(21 80)(22 71)(23 62)(24 73)(25 64)(26 75)(27 66)(28 77)(29 68)(30 79)(31 70)(32 61)(33 72)(34 63)(35 74)(36 65)(37 76)(38 67)(39 78)(40 69)(42 52)(44 54)(46 56)(48 58)(50 60)

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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