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G = Q8.10D10order 160 = 25·5

1st non-split extension by Q8 of D10 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8.10D10, C512- 1+4, C20.24C23, C10.10C24, D10.5C23, D20.13C22, Dic5.6C23, Dic10.13C22, (C2×Q8)⋊5D5, (Q8×D5)⋊4C2, C4○D206C2, (Q8×C10)⋊7C2, Q82D54C2, (C2×C4).24D10, (C4×D5).5C22, C4.24(C22×D5), C2.11(C23×D5), C5⋊D4.2C22, (C2×C10).68C23, (C2×C20).48C22, (C5×Q8).10C22, C22.7(C22×D5), SmallGroup(160,222)

Series: Derived Chief Lower central Upper central

C1C10 — Q8.10D10
C1C5C10D10C4×D5Q8×D5 — Q8.10D10
C5C10 — Q8.10D10
C1C2C2×Q8

Generators and relations for Q8.10D10
 G = < a,b,c,d | a4=1, b2=c10=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c9 >

Subgroups: 392 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2 [×5], C4 [×6], C4 [×4], C22, C22 [×4], C5, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], D5 [×4], C10, C10, C2×Q8, C2×Q8 [×4], C4○D4 [×10], Dic5 [×4], C20 [×6], D10 [×4], C2×C10, 2- 1+4, Dic10 [×6], C4×D5 [×12], D20 [×6], C5⋊D4 [×4], C2×C20 [×3], C5×Q8 [×4], C4○D20 [×6], Q8×D5 [×4], Q82D5 [×4], Q8×C10, Q8.10D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2- 1+4, C22×D5 [×7], C23×D5, Q8.10D10

Smallest permutation representation of Q8.10D10
On 80 points
Generators in S80
(1 43 11 53)(2 44 12 54)(3 45 13 55)(4 46 14 56)(5 47 15 57)(6 48 16 58)(7 49 17 59)(8 50 18 60)(9 51 19 41)(10 52 20 42)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(1 21 11 31)(2 32 12 22)(3 23 13 33)(4 34 14 24)(5 25 15 35)(6 36 16 26)(7 27 17 37)(8 38 18 28)(9 29 19 39)(10 40 20 30)(41 69 51 79)(42 80 52 70)(43 71 53 61)(44 62 54 72)(45 73 55 63)(46 64 56 74)(47 75 57 65)(48 66 58 76)(49 77 59 67)(50 68 60 78)
(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 80 11 70)(2 69 12 79)(3 78 13 68)(4 67 14 77)(5 76 15 66)(6 65 16 75)(7 74 17 64)(8 63 18 73)(9 72 19 62)(10 61 20 71)(21 42 31 52)(22 51 32 41)(23 60 33 50)(24 49 34 59)(25 58 35 48)(26 47 36 57)(27 56 37 46)(28 45 38 55)(29 54 39 44)(30 43 40 53)

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

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

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

Q8.10D10 is a maximal subgroup of
D20.4D4  D20.5D4  D20.14D4  D20.15D4  D20.29D4  D20.30D4  C40.C23  D20.44D4  C10.C25  D5×2- 1+4  D20.39C23  SL2(𝔽3).11D10  D20.38D6  D20.29D6  C30.33C24  Q8.15D30
Q8.10D10 is a maximal quotient of
C10.12- 1+4  C10.82+ 1+4  C10.2- 1+4  C10.2+ 1+4  C10.52- 1+4  C10.62- 1+4  C42.122D10  Q85Dic10  C42.125D10  C42.126D10  Q85D20  C42.132D10  C42.133D10  C42.134D10  C10.152- 1+4  C10.162- 1+4  C10.172- 1+4  D2022D4  Dic1022D4  C10.202- 1+4  C10.212- 1+4  C10.222- 1+4  C10.232- 1+4  C10.242- 1+4  C10.582+ 1+4  C10.262- 1+4  C42.147D10  C42.150D10  C42.151D10  C42.154D10  C42.157D10  C42.158D10  Dic108Q8  C42.171D10  D2012D4  D208Q8  C42.174D10  C42.176D10  C42.177D10  C42.178D10  C42.180D10  C10.422- 1+4  Q8×C5⋊D4  C10.442- 1+4  C10.452- 1+4  D20.38D6  D20.29D6  C30.33C24  Q8.15D30

37 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G4H4I4J5A5B10A···10F20A···20L
order12222224···444445510···1020···20
size112101010102···210101010222···24···4

37 irreducible representations

dim1111122244
type++++++++-
imageC1C2C2C2C2D5D10D102- 1+4Q8.10D10
kernelQ8.10D10C4○D20Q8×D5Q82D5Q8×C10C2×Q8C2×C4Q8C5C1
# reps1644126814

Matrix representation of Q8.10D10 in GL4(𝔽41) generated by

1072
013934
1740400
124040
,
288120
3313012
3901333
039828
,
3382727
324142
77383
34403817
,
2626365
815405
6352626
3435815
G:=sub<GL(4,GF(41))| [1,0,17,1,0,1,40,24,7,39,40,0,2,34,0,40],[28,33,39,0,8,13,0,39,12,0,13,8,0,12,33,28],[3,3,7,34,38,24,7,40,27,14,38,38,27,2,3,17],[26,8,6,34,26,15,35,35,36,40,26,8,5,5,26,15] >;

Q8.10D10 in GAP, Magma, Sage, TeX

Q_8._{10}D_{10}
% in TeX

G:=Group("Q8.10D10");
// GroupNames label

G:=SmallGroup(160,222);
// by ID

G=gap.SmallGroup(160,222);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,86,579,4613]);
// Polycyclic

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

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