Copied to
clipboard

G = D1010M4(2)  order 320 = 26·5

4th semidirect product of D10 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D1010M4(2), D10⋊C86C2, (C4×D5).112D4, (C2×C10)⋊3M4(2), C222(C4.F5), C23.43(C2×F5), (C22×C4).21F5, (C22×C20).23C4, C52(C24.4C4), (C23×D5).16C4, C4.31(C22⋊F5), C23.2F57C2, C20.30(C22⋊C4), Dic5.106(C2×D4), C10.22(C2×M4(2)), D10.39(C22⋊C4), C22.85(C22×F5), C2.16(D5⋊M4(2)), (C2×Dic5).347C23, (C22×Dic5).275C22, (C2×C5⋊C8)⋊2C22, (C2×C4×D5).37C4, (C2×C4.F5)⋊13C2, C2.11(C2×C4.F5), C2.9(C2×C22⋊F5), C10.6(C2×C22⋊C4), (C2×C4).143(C2×F5), (D5×C22×C4).28C2, (C2×C20).109(C2×C4), (C2×C4×D5).398C22, (C22×C10).63(C2×C4), (C2×C10).63(C22×C4), (C2×Dic5).185(C2×C4), (C22×D5).126(C2×C4), SmallGroup(320,1094)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D1010M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C23.2F5 — D1010M4(2)
C5C2×C10 — D1010M4(2)
C1C22C22×C4

Generators and relations for D1010M4(2)
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=ab, dbd=a8b, dcd=c5 >

Subgroups: 762 in 190 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×18], C5, C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×9], C24, Dic5 [×2], Dic5, C20 [×2], C20, D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C24.4C4, C4.F5 [×4], C2×C5⋊C8 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D10⋊C8 [×2], C23.2F5 [×2], C2×C4.F5 [×2], D5×C22×C4, D1010M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×M4(2) [×2], C2×F5 [×3], C24.4C4, C4.F5 [×2], C22⋊F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), C2×C22⋊F5, D1010M4(2)

Smallest permutation representation of D1010M4(2)
On 80 points
Generators in S80
(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 4)(6 10)(7 9)(11 16)(12 15)(13 14)(17 20)(18 19)(22 30)(23 29)(24 28)(25 27)(31 37)(32 36)(33 35)(38 40)(41 45)(42 44)(46 50)(47 49)(51 56)(52 55)(53 54)(57 60)(58 59)(61 68)(62 67)(63 66)(64 65)(69 70)(71 80)(72 79)(73 78)(74 77)(75 76)
(1 14 41 59 24 71 32 65)(2 17 50 56 25 74 31 62)(3 20 49 53 26 77 40 69)(4 13 48 60 27 80 39 66)(5 16 47 57 28 73 38 63)(6 19 46 54 29 76 37 70)(7 12 45 51 30 79 36 67)(8 15 44 58 21 72 35 64)(9 18 43 55 22 75 34 61)(10 11 42 52 23 78 33 68)
(1 6)(2 5)(3 4)(7 10)(8 9)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 80)(21 22)(23 30)(24 29)(25 28)(26 27)(31 38)(32 37)(33 36)(34 35)(39 40)(41 46)(42 45)(43 44)(47 50)(48 49)(51 68)(52 67)(53 66)(54 65)(55 64)(56 63)(57 62)(58 61)(59 70)(60 69)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J 5 8A···8H10A···10G20A···20H
order1222222222444444444458···810···1020···20
size11112210101010222255551010420···204···44···4

44 irreducible representations

dim11111111222444444
type++++++++++
imageC1C2C2C2C2C4C4C4D4M4(2)M4(2)F5C2×F5C2×F5C22⋊F5C4.F5D5⋊M4(2)
kernelD1010M4(2)D10⋊C8C23.2F5C2×C4.F5D5×C22×C4C2×C4×D5C22×C20C23×D5C4×D5D10C2×C10C22×C4C2×C4C23C4C22C2
# reps12221422444121444

Matrix representation of D1010M4(2) in GL6(𝔽41)

4000000
0400000
0035352828
00640032
0000408
0000407
,
4000000
3710000
00406013
000190
0000400
0000401
,
32250000
590000
000402426
000403019
001323135
0093200
,
100000
010000
006402621
0035351317
0000733
0000634

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,28,0,40,40,0,0,28,32,8,7],[40,37,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,9,40,40,0,0,13,0,0,1],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,0,0,13,9,0,0,40,40,23,32,0,0,24,30,1,0,0,0,26,19,35,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,35,0,0,0,0,40,35,0,0,0,0,26,13,7,6,0,0,21,17,33,34] >;

D1010M4(2) in GAP, Magma, Sage, TeX

D_{10}\rtimes_{10}M_4(2)
% in TeX

G:=Group("D10:10M4(2)");
// GroupNames label

G:=SmallGroup(320,1094);
// by ID

G=gap.SmallGroup(320,1094);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a*b,d*b*d=a^8*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽