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G = C245D10order 320 = 26·5

4th semidirect product of C24 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C245D10, C10.302+ 1+4, C22≀C27D5, C22⋊C48D10, C23⋊D107C2, (C2×D4).87D10, C242D59C2, (C2×C20).32C23, (C23×D5)⋊8C22, C20.17D413C2, (C2×C10).138C24, (C23×C10)⋊11C22, C51(C24⋊C22), (C4×Dic5)⋊18C22, C23.D518C22, C2.32(D46D10), D10⋊C415C22, Dic5.5D415C2, (C2×Dic10)⋊23C22, (D4×C10).112C22, (C2×Dic5).63C23, (C22×D5).57C23, C22.159(C23×D5), C23.110(C22×D5), (C22×C10).183C23, (C5×C22≀C2)⋊9C2, (C5×C22⋊C4)⋊8C22, (C2×C4).32(C22×D5), (C2×C5⋊D4).22C22, SmallGroup(320,1266)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C245D10
C1C5C10C2×C10C22×D5C23×D5C23⋊D10 — C245D10
C5C2×C10 — C245D10
C1C22C22≀C2

Generators and relations for C245D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1094 in 260 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×26], C5, C2×C4 [×3], C2×C4 [×6], D4 [×9], Q8 [×3], C23, C23 [×3], C23 [×8], D5 [×2], C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×15], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×3], C24, C24, Dic5 [×6], C20 [×3], D10 [×10], C2×C10, C2×C10 [×16], C22≀C2, C22≀C2 [×5], C4.4D4 [×9], Dic10 [×3], C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C22×D5 [×2], C22×D5 [×3], C22×C10, C22×C10 [×3], C22×C10 [×3], C24⋊C22, C4×Dic5 [×3], D10⋊C4 [×6], C23.D5 [×9], C5×C22⋊C4 [×3], C2×Dic10 [×3], C2×C5⋊D4 [×6], D4×C10 [×3], C23×D5, C23×C10, Dic5.5D4 [×6], C20.17D4 [×3], C23⋊D10 [×3], C242D5 [×2], C5×C22≀C2, C245D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×3], C22×D5 [×7], C24⋊C22, C23×D5, D46D10 [×3], C245D10

Smallest permutation representation of C245D10
On 80 points
Generators in S80
(1 58)(2 54)(3 60)(4 56)(5 52)(6 69)(7 65)(8 61)(9 67)(10 63)(11 51)(12 57)(13 53)(14 59)(15 55)(16 68)(17 64)(18 70)(19 66)(20 62)(21 45)(22 74)(23 47)(24 76)(25 49)(26 78)(27 41)(28 80)(29 43)(30 72)(31 71)(32 44)(33 73)(34 46)(35 75)(36 48)(37 77)(38 50)(39 79)(40 42)
(1 21)(2 27)(3 23)(4 29)(5 25)(6 30)(7 26)(8 22)(9 28)(10 24)(11 31)(12 37)(13 33)(14 39)(15 35)(16 36)(17 32)(18 38)(19 34)(20 40)(41 54)(42 62)(43 56)(44 64)(45 58)(46 66)(47 60)(48 68)(49 52)(50 70)(51 71)(53 73)(55 75)(57 77)(59 79)(61 74)(63 76)(65 78)(67 80)(69 72)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 17)(7 18)(8 19)(9 20)(10 16)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 7)(2 8)(3 9)(4 10)(5 6)(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 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 71)(49 72)(50 73)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 19)(12 18)(13 17)(14 16)(15 20)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(41 43)(44 50)(45 49)(46 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2D5D10D10D102+ 1+4D46D10
kernelC245D10Dic5.5D4C20.17D4C23⋊D10C242D5C5×C22≀C2C22≀C2C22⋊C4C2×D4C24C10C2
# reps1633212662312

Matrix representation of C245D10 in GL8(𝔽41)

400000000
040000000
028100000
1328010000
00001000
00000100
00002828400
00003138040
,
177000000
3524000000
001860000
0035230000
0000244000
000011700
00003382336
0000304018
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
0342000000
63438230000
00160000
003560000
00004073927
0000347237
000000035
000000735
,
7340200000
13423380000
00610000
006350000
000034100
000034700
000000356
00000016

G:=sub<GL(8,GF(41))| [40,0,0,13,0,0,0,0,0,40,28,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,28,31,0,0,0,0,0,1,28,38,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[17,35,0,0,0,0,0,0,7,24,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,24,1,3,3,0,0,0,0,40,17,38,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18],[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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,6,0,0,0,0,0,0,34,34,0,0,0,0,0,0,20,38,1,35,0,0,0,0,0,23,6,6,0,0,0,0,0,0,0,0,40,34,0,0,0,0,0,0,7,7,0,0,0,0,0,0,39,2,0,7,0,0,0,0,27,37,35,35],[7,1,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,23,6,6,0,0,0,0,20,38,1,35,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,35,1,0,0,0,0,0,0,6,6] >;

C245D10 in GAP, Magma, Sage, TeX

C_2^4\rtimes_5D_{10}
% in TeX

G:=Group("C2^4:5D10");
// GroupNames label

G:=SmallGroup(320,1266);
// by ID

G=gap.SmallGroup(320,1266);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,f*b*f=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=e^-1>;
// generators/relations

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