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G = M4(2).21D10order 320 = 26·5

4th non-split extension by M4(2) of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).21D10, (C2×D20).7C4, (C4×D5).92D4, C20.96(C2×D4), C4.151(D4×D5), C4.10D46D5, (D5×M4(2))⋊7C2, (C2×C20).8C23, (C2×Q8).95D10, C20.10D44C2, C20.46D47C2, (Q8×C10).6C22, (C2×D20).44C22, C4.Dic5.5C22, D10.22(C22⋊C4), Dic5.55(C22⋊C4), (C5×M4(2)).13C22, C54(M4(2).8C22), (C2×C4×D5).3C4, (C2×C4).7(C4×D5), (C2×C4×D5).8C22, C22.17(C2×C4×D5), (C2×C20).21(C2×C4), C2.16(D5×C22⋊C4), (C2×C4).8(C22×D5), (C5×C4.10D4)⋊6C2, C10.56(C2×C22⋊C4), (C2×Q82D5).1C2, (C22×D5).3(C2×C4), (C2×C10).112(C22×C4), (C2×Dic5).141(C2×C4), SmallGroup(320,378)

Series: Derived Chief Lower central Upper central

C1C2×C10 — M4(2).21D10
C1C5C10C20C2×C20C2×C4×D5C2×Q82D5 — M4(2).21D10
C5C10C2×C10 — M4(2).21D10
C1C2C2×C4C4.10D4

Generators and relations for M4(2).21D10
 G = < a,b,c,d | a8=b2=d2=1, c10=a4, bab=a5, cac-1=ab, dad=a5b, bc=cb, bd=db, dcd=a4c9 >

Subgroups: 622 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10, C10, C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, C4.D4 [×2], C4.10D4, C4.10D4, C2×M4(2) [×2], C2×C4○D4, C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], M4(2).8C22, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5 [×2], C5×M4(2) [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q82D5 [×4], Q8×C10, C20.46D4 [×2], C20.10D4, C5×C4.10D4, D5×M4(2) [×2], C2×Q82D5, M4(2).21D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], C22×D5, M4(2).8C22, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, M4(2).21D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12222224444444558888888810101010202020202020202040···40
size1121010202022445510224444202020202244444488888···8

44 irreducible representations

dim1111111122222448
type++++++++++++
imageC1C2C2C2C2C2C4C4D4D5D10D10C4×D5M4(2).8C22D4×D5M4(2).21D10
kernelM4(2).21D10C20.46D4C20.10D4C5×C4.10D4D5×M4(2)C2×Q82D5C2×C4×D5C2×D20C4×D5C4.10D4M4(2)C2×Q8C2×C4C5C4C1
# reps1211214442428242

Matrix representation of M4(2).21D10 in GL8(𝔽41)

90000000
932000000
00900000
009320000
00009313639
000032050
00008133221
00002025160
,
400000000
040000000
004000000
000400000
00001000
000004000
00000010
0000903640
,
4027270000
010340000
34147270000
070340000
0000400370
00009313639
000021010
000021302510
,
139000000
040000000
7274020000
034010000
00001040
0000321052
000000400
000029121631

G:=sub<GL(8,GF(41))| [9,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,32,8,20,0,0,0,0,31,0,13,25,0,0,0,0,36,5,32,16,0,0,0,0,39,0,21,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,9,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40],[40,0,34,0,0,0,0,0,2,1,14,7,0,0,0,0,7,0,7,0,0,0,0,0,27,34,27,34,0,0,0,0,0,0,0,0,40,9,21,21,0,0,0,0,0,31,0,30,0,0,0,0,37,36,1,25,0,0,0,0,0,39,0,10],[1,0,7,0,0,0,0,0,39,40,27,34,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,32,0,29,0,0,0,0,0,10,0,12,0,0,0,0,4,5,40,16,0,0,0,0,0,2,0,31] >;

M4(2).21D10 in GAP, Magma, Sage, TeX

M_4(2)._{21}D_{10}
% in TeX

G:=Group("M4(2).21D10");
// GroupNames label

G:=SmallGroup(320,378);
// by ID

G=gap.SmallGroup(320,378);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,219,58,570,136,438,12550]);
// Polycyclic

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

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