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G = C24.42D10order 320 = 26·5

42nd non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.42D10, C10.912+ 1+4, (C5×D4)⋊17D4, D48(C5⋊D4), C202D441C2, C510(D45D4), (D4×Dic5)⋊39C2, C20.253(C2×D4), (C22×D4)⋊12D5, (C2×D4).231D10, C242D514C2, C4⋊Dic545C22, Dic5⋊D442C2, C20.48D437C2, C20.17D429C2, C225(D42D5), (C2×C10).301C24, (C2×C20).546C23, (C4×Dic5)⋊43C22, (C22×C4).273D10, C10.148(C22×D4), C2.94(D46D10), C23.D540C22, D10⋊C473C22, (C2×Dic10)⋊42C22, (D4×C10).312C22, C10.D439C22, (C23×C10).80C22, C22.314(C23×D5), C23.236(C22×D5), C23.18D1030C2, (C22×C10).235C23, (C22×C20).278C22, (C2×Dic5).296C23, (C22×Dic5)⋊35C22, (C22×D5).132C23, (D4×C2×C10)⋊8C2, (C4×C5⋊D4)⋊26C2, (C2×C4×D5)⋊32C22, C4.68(C2×C5⋊D4), (C2×C10).74(C2×D4), (C2×D42D5)⋊27C2, (C2×C10)⋊15(C4○D4), C22.3(C2×C5⋊D4), C10.107(C2×C4○D4), C2.71(C2×D42D5), (C2×C5⋊D4)⋊29C22, (C2×C23.D5)⋊31C2, C2.21(C22×C5⋊D4), (C2×C4).239(C22×D5), SmallGroup(320,1478)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.42D10
C1C5C10C2×C10C22×D5C2×C4×D5C2×D42D5 — C24.42D10
C5C2×C10 — C24.42D10
C1C22C22×D4

Generators and relations for C24.42D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, eae-1=ad=da, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 1078 in 334 conjugacy classes, 115 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×23], C5, C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×11], D5, C10 [×3], C10 [×8], C42, C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×9], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×7], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C5×D4 [×6], C22×D5, C22×C10, C22×C10 [×4], C22×C10 [×10], D45D4, C4×Dic5, C10.D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4, C23.D5, C23.D5 [×10], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×4], C22×C20, D4×C10 [×2], D4×C10 [×2], D4×C10 [×4], C23×C10 [×2], C20.48D4, C4×C5⋊D4, D4×Dic5, C23.18D10 [×2], C20.17D4, C202D4, Dic5⋊D4 [×2], C2×C23.D5 [×2], C242D5 [×2], C2×D42D5, D4×C2×C10, C24.42D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C5⋊D4 [×4], C22×D5 [×7], D45D4, D42D5 [×2], C2×C5⋊D4 [×6], C23×D5, C2×D42D5, D46D10, C22×C5⋊D4, C24.42D10

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

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

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

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

65 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L4A4B4C4D4E4F4G4H···4L5A5B10A···10N10O···10AD20A···20H
order12222···222244444444···45510···1010···1020···20
size11112···244202241010101020···20222···24···44···4

65 irreducible representations

dim1111111111112222222444
type++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D42+ 1+4D42D5D46D10
kernelC24.42D10C20.48D4C4×C5⋊D4D4×Dic5C23.18D10C20.17D4C202D4Dic5⋊D4C2×C23.D5C242D5C2×D42D5D4×C2×C10C5×D4C22×D4C2×C10C22×C4C2×D4C24D4C10C22C2
# reps11112112221142428416144

Matrix representation of C24.42D10 in GL4(𝔽41) generated by

1000
0100
0010
00140
,
1000
04000
00400
00040
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
37000
01000
00139
00140
,
03100
4000
0090
0009
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[37,0,0,0,0,10,0,0,0,0,1,1,0,0,39,40],[0,4,0,0,31,0,0,0,0,0,9,0,0,0,0,9] >;

C24.42D10 in GAP, Magma, Sage, TeX

C_2^4._{42}D_{10}
% in TeX

G:=Group("C2^4.42D10");
// GroupNames label

G:=SmallGroup(320,1478);
// by ID

G=gap.SmallGroup(320,1478);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,675,12550]);
// 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,e*a*e^-1=a*d=d*a,a*f=f*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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