Copied to
clipboard

G = M4(2).Dic5order 320 = 26·5

1st non-split extension by M4(2) of Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.7Dic10, M4(2).1Dic5, C40.47(C2×C4), C4.87(C2×D20), (C2×C8).76D10, C20.53(C4⋊C4), (C2×C20).29Q8, C8.4(C2×Dic5), (C2×C20).169D4, (C2×C4).150D20, C20.305(C2×D4), C4.7(C4⋊Dic5), C40.6C413C2, (C2×C40).62C22, (C2×C4).17Dic10, (C2×M4(2)).2D5, (C5×M4(2)).5C4, (C22×C10).17Q8, C54(M4(2).C4), (C2×C20).796C23, C20.233(C22×C4), (C22×C4).135D10, (C10×M4(2)).2C2, C22.7(C4⋊Dic5), C22.9(C2×Dic10), C4.28(C22×Dic5), C4.Dic5.36C22, (C22×C20).184C22, C10.76(C2×C4⋊C4), C2.15(C2×C4⋊Dic5), (C2×C10).41(C2×Q8), (C2×C10).45(C4⋊C4), (C2×C20).277(C2×C4), (C2×C4).22(C2×Dic5), (C2×C4).721(C22×D5), (C2×C4.Dic5).25C2, SmallGroup(320,752)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).Dic5
C1C5C10C20C2×C20C4.Dic5C2×C4.Dic5 — M4(2).Dic5
C5C10C20 — M4(2).Dic5
C1C4C22×C4C2×M4(2)

Generators and relations for M4(2).Dic5
 G = < a,b,c,d | a8=b2=1, c10=a4, d2=c5, bab=a5, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 238 in 102 conjugacy classes, 67 normal (21 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C22, C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×3], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×6], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10, C8.C4 [×4], C2×M4(2), C2×M4(2) [×2], C52C8 [×4], C40 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, M4(2).C4, C2×C52C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C2×C40 [×2], C5×M4(2) [×4], C22×C20, C40.6C4 [×4], C2×C4.Dic5 [×2], C10×M4(2), M4(2).Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, M4(2).C4, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic5, M4(2).Dic5

Smallest permutation representation of M4(2).Dic5
On 80 points
Generators in S80
(1 22 6 27 11 32 16 37)(2 23 7 28 12 33 17 38)(3 24 8 29 13 34 18 39)(4 25 9 30 14 35 19 40)(5 26 10 31 15 36 20 21)(41 61 56 76 51 71 46 66)(42 62 57 77 52 72 47 67)(43 63 58 78 53 73 48 68)(44 64 59 79 54 74 49 69)(45 65 60 80 55 75 50 70)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(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 46 6 51 11 56 16 41)(2 55 7 60 12 45 17 50)(3 44 8 49 13 54 18 59)(4 53 9 58 14 43 19 48)(5 42 10 47 15 52 20 57)(21 62 26 67 31 72 36 77)(22 71 27 76 32 61 37 66)(23 80 28 65 33 70 38 75)(24 69 29 74 34 79 39 64)(25 78 30 63 35 68 40 73)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B8A8B8C8D8E···8L10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444445588888···810···101010101020···202020202040···40
size112221122222444420···202···244442···244444···4

62 irreducible representations

dim11111222222222244
type+++++--++-+-+-
imageC1C2C2C2C4D4Q8Q8D5D10Dic5D10Dic10D20Dic10M4(2).C4M4(2).Dic5
kernelM4(2).Dic5C40.6C4C2×C4.Dic5C10×M4(2)C5×M4(2)C2×C20C2×C20C22×C10C2×M4(2)C2×C8M4(2)C22×C4C2×C4C2×C4C23C5C1
# reps14218211248248428

Matrix representation of M4(2).Dic5 in GL4(𝔽41) generated by

0100
32000
362909
15010
,
1000
04000
0410
370040
,
33000
03300
206360
621036
,
331470
14807
260827
0262733
G:=sub<GL(4,GF(41))| [0,32,36,15,1,0,29,0,0,0,0,1,0,0,9,0],[1,0,0,37,0,40,4,0,0,0,1,0,0,0,0,40],[33,0,20,6,0,33,6,21,0,0,36,0,0,0,0,36],[33,14,26,0,14,8,0,26,7,0,8,27,0,7,27,33] >;

M4(2).Dic5 in GAP, Magma, Sage, TeX

M_4(2).{\rm Dic}_5
% in TeX

G:=Group("M4(2).Dic5");
// GroupNames label

G:=SmallGroup(320,752);
// by ID

G=gap.SmallGroup(320,752);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^10=a^4,d^2=c^5,b*a*b=a^5,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations

׿
×
𝔽