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G = C24.D10order 320 = 26·5

1st non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.1D10, C23.8D20, C23.1Dic10, (C22×C20)⋊1C4, C23.8(C4×D5), C23.D511C4, (C22×C4)⋊1Dic5, (C22×C10).6Q8, (C2×C10).39C42, (C22×C10).41D4, C53(C23.9D4), C22.8(C4×Dic5), C10.38(C23⋊C4), C23.46(C5⋊D4), C2.2(C23⋊Dic5), C22.8(C4⋊Dic5), C23.21(C2×Dic5), (C23×C10).22C22, C22.1(C10.D4), C22.24(C23.D5), C22.16(D10⋊C4), C10.22(C2.C42), C2.4(C10.10C42), (C2×C22⋊C4).2D5, (C2×C10).29(C4⋊C4), (C10×C22⋊C4).1C2, (C2×C23.D5).1C2, (C22×C10).96(C2×C4), (C2×C10).73(C22⋊C4), SmallGroup(320,84)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C24.D10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

Subgroups: 518 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×12], C23 [×3], C23 [×4], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×8], C22×C4 [×2], C22×C4 [×2], C24, Dic5 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×4], C2×C10 [×10], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×Dic5 [×8], C2×C20 [×4], C22×C10 [×3], C22×C10 [×4], C22×C10 [×2], C23.9D4, C23.D5 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×2], C22×C20 [×2], C23×C10, C2×C23.D5 [×2], C10×C22⋊C4, C24.D10
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, C10.10C42, C23⋊Dic5 [×2], C24.D10

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

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

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

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

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

dim1111122222222244
type++++-+-+-++
imageC1C2C2C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4C23⋊C4C23⋊Dic5
kernelC24.D10C2×C23.D5C10×C22⋊C4C23.D5C22×C20C22×C10C22×C10C2×C22⋊C4C22×C4C24C23C23C23C23C10C2
# reps1218431242484828

Matrix representation of C24.D10 in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
100000
010000
00244000
0011700
00002440
0000117
,
100000
010000
0040000
0004000
0000400
0000040
,
010000
4000000
0000341
0000400
0034100
0040000
,
0320000
3200000
000001
000010
00402400
0017100

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,34,40,0,0,0,0,1,0,0,0],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,0,40,17,0,0,0,0,24,1,0,0,0,1,0,0,0,0,1,0,0,0] >;

C24.D10 in GAP, Magma, Sage, TeX

C_2^4.D_{10}
% in TeX

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

G:=SmallGroup(320,84);
// by ID

G=gap.SmallGroup(320,84);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;
// generators/relations

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