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G = C24.72D10order 320 = 26·5

12nd non-split extension by C24 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.72D10, (C2×C20)⋊38D4, (C23×C4)⋊6D5, C207D451C2, (C23×C20)⋊10C2, C20.425(C2×D4), (C2×D20)⋊51C22, C225(C4○D20), C242D515C2, C4⋊Dic565C22, C20.48D451C2, (C2×C10).289C24, (C2×C20).887C23, C57(C22.19C24), (C4×Dic5)⋊59C22, (C22×C4).449D10, C10.135(C22×D4), D10⋊C443C22, (C2×Dic10)⋊59C22, C10.D445C22, C23.235(C22×D5), C22.304(C23×D5), C23.21D1013C2, C23.23D1033C2, (C23×C10).111C22, (C22×C20).530C22, (C22×C10).418C23, (C2×Dic5).151C23, (C22×D5).127C23, C23.D5.130C22, (C4×C5⋊D4)⋊51C2, (C2×C4×D5)⋊54C22, (C2×C4○D20)⋊14C2, (C2×C4)⋊17(C5⋊D4), C10.64(C2×C4○D4), C2.72(C2×C4○D20), C4.145(C2×C5⋊D4), (C2×C10)⋊12(C4○D4), C2.8(C22×C5⋊D4), (C2×C10).575(C2×D4), C22.35(C2×C5⋊D4), (C2×C4).740(C22×D5), (C2×C5⋊D4).145C22, SmallGroup(320,1463)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.72D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×C4○D20 — C24.72D10
C5C2×C10 — C24.72D10
C1C2×C4C23×C4

Generators and relations for C24.72D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, faf-1=ad=da, ae=ea, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1022 in 330 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D5 [×2], C10, C10 [×2], C10 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic5 [×6], C20 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], C2×C10 [×14], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×D5 [×2], C22×C10, C22×C10 [×2], C22×C10 [×6], C22.19C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×6], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×6], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C20.48D4 [×2], C23.21D10, C4×C5⋊D4 [×4], C23.23D10 [×2], C207D4 [×2], C242D5 [×2], C2×C4○D20, C23×C20, C24.72D10
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], C5⋊D4 [×4], C22×D5 [×7], C22.19C24, C4○D20 [×4], C2×C5⋊D4 [×6], C23×D5, C2×C4○D20 [×2], C22×C5⋊D4, C24.72D10

Smallest permutation representation of C24.72D10
On 80 points
Generators in S80
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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 73 51 63)(42 62 52 72)(43 71 53 61)(44 80 54 70)(45 69 55 79)(46 78 56 68)(47 67 57 77)(48 76 58 66)(49 65 59 75)(50 74 60 64)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4J4K···4P5A5B10A···10AD20A···20AF
order12222···22244444···44···45510···1020···20
size11112···2202011112···220···20222···22···2

92 irreducible representations

dim1111111112222222
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D5C4○D4D10D10C5⋊D4C4○D20
kernelC24.72D10C20.48D4C23.21D10C4×C5⋊D4C23.23D10C207D4C242D5C2×C4○D20C23×C20C2×C20C23×C4C2×C10C22×C4C24C2×C4C22
# reps1214222114281221632

Matrix representation of C24.72D10 in GL4(𝔽41) generated by

40000
04000
00400
0001
,
1000
04000
0010
00040
,
40000
04000
00400
00040
,
1000
0100
00400
00040
,
40000
04000
0080
0005
,
04000
40000
0005
0080
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,8,0,0,0,0,5],[0,40,0,0,40,0,0,0,0,0,0,8,0,0,5,0] >;

C24.72D10 in GAP, Magma, Sage, TeX

C_2^4._{72}D_{10}
% in TeX

G:=Group("C2^4.72D10");
// GroupNames label

G:=SmallGroup(320,1463);
// by ID

G=gap.SmallGroup(320,1463);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,12550]);
// 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,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*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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