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G = C24.2D10order 320 = 26·5

2nd non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.2D10, C23.32D20, C23.2Dic10, C22⋊C43Dic5, C23.D512C4, C23.42(C4×D5), (C22×C10).7Q8, (C2×C10).23C42, (C22×Dic5)⋊3C4, C54(C23.9D4), C22.1(C4×Dic5), C23.5(C2×Dic5), C10.36(C23⋊C4), (C22×C10).177D4, C23.15(C5⋊D4), C22.1(C4⋊Dic5), (C23×C10).23C22, C22.9(C23.D5), C2.3(C23.1D10), C22.2(C10.D4), C22.37(D10⋊C4), C2.5(C10.10C42), C10.23(C2.C42), (C5×C22⋊C4)⋊11C4, (C2×C22⋊C4).3D5, (C2×C10).30(C4⋊C4), (C10×C22⋊C4).2C2, (C2×C23.D5).2C2, (C22×C10).97(C2×C4), (C2×C10).114(C22⋊C4), SmallGroup(320,85)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.2D10
C1C5C10C2×C10C22×C10C23×C10C2×C23.D5 — C24.2D10
C5C10C2×C10 — C24.2D10
C1C22C24C2×C22⋊C4

Generators and relations for C24.2D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=b, f2=abcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde9 >

Subgroups: 518 in 142 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22 [×7], C22 [×10], C5, C2×C4 [×12], C23 [×7], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×2], C22⋊C4 [×6], C22×C4 [×4], C24, Dic5 [×4], C20 [×2], C2×C10 [×7], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×Dic5 [×8], C2×C20 [×4], C22×C10 [×7], C22×C10 [×2], C23.9D4, C23.D5 [×2], C23.D5 [×3], C5×C22⋊C4 [×2], C5×C22⋊C4, C22×Dic5 [×2], C22×Dic5, C22×C20, C23×C10, C2×C23.D5 [×2], C10×C22⋊C4, C24.2D10
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, C23⋊C4 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C23.9D4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C23.1D10 [×2], C10.10C42, C24.2D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4L5A5B10A···10N10O···10V20A···20P
order12222···244444···45510···1010···1020···20
size11112···2444420···20222···24···44···4

62 irreducible representations

dim11111122222222244
type++++-+-+-++
imageC1C2C2C4C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4C23⋊C4C23.1D10
kernelC24.2D10C2×C23.D5C10×C22⋊C4C23.D5C5×C22⋊C4C22×Dic5C22×C10C22×C10C2×C22⋊C4C22⋊C4C24C23C23C23C23C10C2
# reps12144431242484828

Matrix representation of C24.2D10 in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
4000000
0400000
00177635
003524035
0000186
00003523
,
4000000
0400000
002434239
0061720
0000186
00003523
,
100000
010000
0040000
0004000
0000400
0000040
,
0400000
100000
000362613
003391214
001439406
002728833
,
3200000
090000
00734317
00134418
00002018
00002121

G:=sub<GL(6,GF(41))| [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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,17,35,0,0,0,0,7,24,0,0,0,0,6,0,18,35,0,0,35,35,6,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,24,6,0,0,0,0,34,17,0,0,0,0,2,2,18,35,0,0,39,0,6,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,33,14,27,0,0,36,9,39,28,0,0,26,12,40,8,0,0,13,14,6,33],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,7,1,0,0,0,0,34,34,0,0,0,0,31,4,20,21,0,0,7,18,18,21] >;

C24.2D10 in GAP, Magma, Sage, TeX

C_2^4._2D_{10}
% in TeX

G:=Group("C2^4.2D10");
// GroupNames label

G:=SmallGroup(320,85);
// by ID

G=gap.SmallGroup(320,85);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,851,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=b,f^2=a*b*c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^9>;
// generators/relations

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