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G = C25.2D5order 320 = 26·5

1st non-split extension by C25 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C25.2D5, C244Dic5, C24.66D10, (C23×C10)⋊12C4, C53(C243C4), (C24×C10).3C2, C10.86C22≀C2, (C22×C10).202D4, C2.3(C242D5), C23.90(C5⋊D4), C222(C23.D5), C23.34(C2×Dic5), (C22×Dic5)⋊4C22, C23.308(C22×D5), (C22×C10).372C23, (C23×C10).101C22, C22.55(C22×Dic5), (C2×C10).569(C2×D4), (C2×C23.D5)⋊13C2, (C2×C10)⋊11(C22⋊C4), C22.98(C2×C5⋊D4), C2.24(C2×C23.D5), C10.129(C2×C22⋊C4), (C22×C10).206(C2×C4), (C2×C10).304(C22×C4), SmallGroup(320,874)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C25.2D5
C1C5C10C2×C10C22×C10C22×Dic5C2×C23.D5 — C25.2D5
C5C2×C10 — C25.2D5
C1C23C25

Generators and relations for C25.2D5
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=1, g2=c, ab=ba, ac=ca, ad=da, gag-1=ae=ea, af=fa, bc=cb, gbg-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg-1=f-1 >

Subgroups: 1198 in 506 conjugacy classes, 119 normal (9 characteristic)
C1, C2 [×7], C2 [×12], C4 [×4], C22, C22 [×18], C22 [×68], C5, C2×C4 [×12], C23, C23 [×18], C23 [×68], C10 [×7], C10 [×12], C22⋊C4 [×12], C22×C4 [×4], C24 [×7], C24 [×12], Dic5 [×4], C2×C10, C2×C10 [×18], C2×C10 [×68], C2×C22⋊C4 [×6], C25, C2×Dic5 [×12], C22×C10, C22×C10 [×18], C22×C10 [×68], C243C4, C23.D5 [×12], C22×Dic5 [×4], C23×C10 [×7], C23×C10 [×12], C2×C23.D5 [×6], C24×C10, C25.2D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, D5, C22⋊C4 [×12], C22×C4, C2×D4 [×6], Dic5 [×4], D10 [×3], C2×C22⋊C4 [×3], C22≀C2 [×4], C2×Dic5 [×6], C5⋊D4 [×12], C22×D5, C243C4, C23.D5 [×12], C22×Dic5, C2×C5⋊D4 [×6], C2×C23.D5 [×3], C242D5 [×4], C25.2D5

Smallest permutation representation of C25.2D5
On 80 points
Generators in S80
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(1 14)(2 15)(3 11)(4 12)(5 13)(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 9)(2 10)(3 6)(4 7)(5 8)(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 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 65 24 45)(2 64 25 44)(3 63 21 43)(4 62 22 42)(5 61 23 41)(6 68 26 48)(7 67 27 47)(8 66 28 46)(9 70 29 50)(10 69 30 49)(11 73 31 53)(12 72 32 52)(13 71 33 51)(14 75 34 55)(15 74 35 54)(16 78 36 58)(17 77 37 57)(18 76 38 56)(19 80 39 60)(20 79 40 59)

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

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

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

92 conjugacy classes

class 1 2A···2G2H···2S4A···4H5A5B10A···10BJ
order12···22···24···45510···10
size11···12···220···20222···2

92 irreducible representations

dim111122222
type+++++-+
imageC1C2C2C4D4D5Dic5D10C5⋊D4
kernelC25.2D5C2×C23.D5C24×C10C23×C10C22×C10C25C24C24C23
# reps16181228648

Matrix representation of C25.2D5 in GL5(𝔽41)

10000
040000
00100
000400
000211
,
400000
01000
004000
000400
000040
,
400000
040000
004000
00010
00001
,
10000
040000
004000
00010
00001
,
10000
040000
004000
000400
000040
,
10000
018000
001600
000370
0002410
,
320000
001600
023000
000832
000733

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40,21,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,18,0,0,0,0,0,16,0,0,0,0,0,37,24,0,0,0,0,10],[32,0,0,0,0,0,0,23,0,0,0,16,0,0,0,0,0,0,8,7,0,0,0,32,33] >;

C25.2D5 in GAP, Magma, Sage, TeX

C_2^5._2D_5
% in TeX

G:=Group("C2^5.2D5");
// GroupNames label

G:=SmallGroup(320,874);
// by ID

G=gap.SmallGroup(320,874);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,422,12550]);
// Polycyclic

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

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