Copied to
clipboard

G = M4(2).22D10order 320 = 26·5

5th non-split extension by M4(2) of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.24D10, M4(2).22D10, C4≀C26D5, D4⋊D57C4, Q8⋊D57C4, D4.D57C4, D4.3(C4×D5), C5⋊Q167C4, Q8.3(C4×D5), C55(C8.26D4), C10.68(C4×D4), C4.202(D4×D5), C52C8.50D4, D204C46C2, C4○D4.20D10, D20.20(C2×C4), C20.361(C2×D4), D4.Dic51C2, D20.2C48C2, C20.53D45C2, C20.54(C22×C4), (C4×C20).50C22, C42.D52C2, (C2×C20).263C23, Dic10.21(C2×C4), D4.8D10.1C2, C4○D20.12C22, C4.Dic5.8C22, C22.8(D42D5), C2.22(Dic54D4), (C5×M4(2)).16C22, (C5×C4≀C2)⋊7C2, C4.19(C2×C4×D5), C52C8.2(C2×C4), (C5×D4).20(C2×C4), (C5×Q8).21(C2×C4), (C5×C4○D4).4C22, (C2×C10).34(C4○D4), (C2×C52C8).50C22, (C2×C4).369(C22×D5), SmallGroup(320,450)

Series: Derived Chief Lower central Upper central

C1C20 — M4(2).22D10
C1C5C10C20C2×C20C4○D20D4.8D10 — M4(2).22D10
C5C10C20 — M4(2).22D10
C1C4C2×C4C4≀C2

Generators and relations for M4(2).22D10
 G = < a,b,c,d | a8=b2=c10=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 326 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×2], C5, C8 [×6], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], C42, C2×C8 [×4], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5, C20 [×2], C20 [×2], D10, C2×C10, C2×C10, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4 [×2], C4○D8, C52C8 [×4], C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C8.26D4, C8×D5, C8⋊D5, C2×C52C8 [×2], C2×C52C8, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4×C20, C5×M4(2), C4○D20, C5×C4○D4, C42.D5, D204C4, C20.53D4, C5×C4≀C2, D20.2C4, D4.Dic5, D4.8D10, M4(2).22D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C8.26D4, C2×C4×D5, D4×D5, D42D5, Dic54D4, M4(2).22D10

Smallest permutation representation of M4(2).22D10
On 80 points
Generators in S80
(1 51 15 6 33 20 56 38)(2 39 16 52 34 7 57 11)(3 53 17 8 35 12 58 40)(4 31 18 54 36 9 59 13)(5 55 19 10 37 14 60 32)(21 46 41 64 69 73 78 26)(22 27 42 47 70 65 79 74)(23 48 43 66 61 75 80 28)(24 29 44 49 62 67 71 76)(25 50 45 68 63 77 72 30)
(1 33)(3 35)(5 37)(7 39)(9 31)(11 52)(13 54)(15 56)(17 58)(19 60)(21 69)(23 61)(25 63)(27 65)(29 67)(41 78)(43 80)(45 72)(47 74)(49 76)
(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 67 15 76 33 29 56 49)(2 48 57 28 34 75 16 66)(3 65 17 74 35 27 58 47)(4 46 59 26 36 73 18 64)(5 63 19 72 37 25 60 45)(6 44 51 24 38 71 20 62)(7 61 11 80 39 23 52 43)(8 42 53 22 40 79 12 70)(9 69 13 78 31 21 54 41)(10 50 55 30 32 77 14 68)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H8I8J10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order1222244444445588888888881010101010102020202020···20202040404040
size112420112444202244101010102020202022448822224···4888888

50 irreducible representations

dim111111111111222222224444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D5C4○D4D10D10D10C4×D5C4×D5C8.26D4D4×D5D42D5M4(2).22D10
kernelM4(2).22D10C42.D5D204C4C20.53D4C5×C4≀C2D20.2C4D4.Dic5D4.8D10D4⋊D5D4.D5Q8⋊D5C5⋊Q16C52C8C4≀C2C2×C10C42M4(2)C4○D4D4Q8C5C4C22C1
# reps111111112222222222442228

Matrix representation of M4(2).22D10 in GL6(𝔽41)

900000
090000
0011700
00274000
00004024
0000141
,
4000000
0400000
0040000
0017100
000010
00002440
,
1000000
0370000
00402400
000100
0000930
00001132
,
040000
3100000
00004024
000001
0093000
00113200

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,27,0,0,0,0,17,40,0,0,0,0,0,0,40,14,0,0,0,0,24,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,17,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,0,40],[10,0,0,0,0,0,0,37,0,0,0,0,0,0,40,0,0,0,0,0,24,1,0,0,0,0,0,0,9,11,0,0,0,0,30,32],[0,31,0,0,0,0,4,0,0,0,0,0,0,0,0,0,9,11,0,0,0,0,30,32,0,0,40,0,0,0,0,0,24,1,0,0] >;

M4(2).22D10 in GAP, Magma, Sage, TeX

M_4(2)._{22}D_{10}
% in TeX

G:=Group("M4(2).22D10");
// GroupNames label

G:=SmallGroup(320,450);
// by ID

G=gap.SmallGroup(320,450);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,219,58,136,1684,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=d*a*d^-1=a^-1*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;
// generators/relations

׿
×
𝔽