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G = C24.56D10  order 320 = 26·5

14th non-split extension by C24 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.56D10, C22≀C2⋊9D5, (D4×Dic5)⋊10C2, (C2×Dic5)⋊20D4, C24⋊2D5⋊4C2, C22.39(D4×D5), Dic5⋊D4⋊1C2, Dic5⋊4D4⋊1C2, (C2×D4).148D10, (C2×C20).25C23, C4⋊Dic5⋊23C22, C22⋊C4.44D10, Dic5.84(C2×D4), (C23×Dic5)⋊5C2, C10.53(C22×D4), D10⋊C4⋊8C22, C22⋊3(D4⋊2D5), (C2×C10).130C24, C5⋊3(C22.19C24), (C4×Dic5)⋊12C22, C22.D20⋊8C2, C10.D4⋊6C22, (C22×C10).7C23, C23.D5⋊11C22, (C2×Dic10)⋊18C22, (D4×C10).109C22, C23.18D10⋊2C2, C23.11D10⋊1C2, (C23×C10).66C22, (C22×D5).52C23, C22.151(C23×D5), C23.175(C22×D5), Dic5.14D4⋊11C2, (C2×Dic5).229C23, (C22×Dic5)⋊10C22, C2.26(C2×D4×D5), (C2×C4×D5)⋊4C22, (C5×C22≀C2)⋊2C2, (C2×D4⋊2D5)⋊5C2, (C2×C10)⋊9(C4○D4), C10.75(C2×C4○D4), (C2×C10).52(C2×D4), (C2×C5⋊D4)⋊6C22, C2.26(C2×D4⋊2D5), (C2×C4).25(C22×D5), (C5×C22⋊C4).1C22, SmallGroup(320,1258)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C24.56D10
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C24.56D10
Lower central
C5 — C2×C10 — C24.56D10

Subgroups: 1070 in 330 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×12], C22, C22 [×6], C22 [×20], C5, C2×C4, C2×C4 [×2], C2×C4 [×25], D4 [×14], Q8 [×2], C23 [×2], C23 [×2], C23 [×7], D5, C10, C10 [×2], C10 [×7], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C22×C4 [×12], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic5 [×4], Dic5 [×5], C20 [×3], D10 [×3], C2×C10, C2×C10 [×6], C2×C10 [×17], C42⋊C2, C4×D4 [×4], C22≀C2, C22≀C2, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×8], C2×Dic5 [×12], C5⋊D4 [×8], C2×C20, C2×C20 [×2], C5×D4 [×6], C22×D5, C22×C10 [×2], C22×C10 [×2], C22×C10 [×6], C22.19C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5, C23.D5 [×4], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×Dic10, C2×C4×D5, D4⋊2D5 [×4], C22×Dic5 [×3], C22×Dic5 [×4], C22×Dic5 [×4], C2×C5⋊D4 [×2], C2×C5⋊D4 [×2], D4×C10, D4×C10 [×2], C23×C10, C23.11D10, Dic5.14D4 [×2], Dic5⋊4D4 [×2], C22.D20, D4×Dic5 [×2], C23.18D10, Dic5⋊D4 [×2], C24⋊2D5, C5×C22≀C2, C2×D4⋊2D5, C23×Dic5, C24.56D10

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.19C24, D4×D5 [×2], D4⋊2D5 [×4], C23×D5, C2×D4×D5, C2×D4⋊2D5 [×2], C24.56D10

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

40230000
010000
001000
000100
000010
0000040
,
40230000
010000
001000
000100
0000400
0000040
,
4000000
0400000
001000
000100
000010
000001
,
4000000
0400000
001000
000100
0000400
0000040
,
100000
9400000
0003400
0063500
000001
000010
,
900000
40320000
00131400
00292800
000009
000090

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,9,0,0,0,0,0,40,0,0,0,0,0,0,0,6,0,0,0,0,34,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,40,0,0,0,0,0,32,0,0,0,0,0,0,13,29,0,0,0,0,14,28,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;
 

56 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E4F4G4H···4M4N4O4P5A5B10A···10F10G···10R10S10T20A···20F
order12222···22244444444···44445510···1010···10101020···20
size11112···2420444555510···10202020222···24···4888···8

56 irreducible representations

dim11111111111122222244
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D4×D5D4⋊2D5
kernelC24.56D10C23.11D10Dic5.14D4Dic5⋊4D4C22.D20D4×Dic5C23.18D10Dic5⋊D4C24⋊2D5C5×C22≀C2C2×D4⋊2D5C23×Dic5C2×Dic5C22≀C2C2×C10C22⋊C4C2×D4C24C22C22
# reps11221212111142866248

In GAP, Magma, Sage, TeX 

C_2^4._{56}D_{10}
 
% in TeX
 
G:=Group("C2^4.56D10");
 
// GroupNames label
 
G:=SmallGroup(320,1258);
 
// by ID
 
G=gap.SmallGroup(320,1258);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,570,185,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,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^-1>;
 
// generators/relations
 

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