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G = D107M4(2)  order 320 = 26·5

1st semidirect product of D10 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D107M4(2), (C2×C8)⋊18D10, C22⋊C811D5, C4.195(D4×D5), (C2×C40)⋊20C22, (C4×D5).102D4, C20.354(C2×D4), (C2×C10)⋊5M4(2), D101C817C2, (C23×D5).8C4, C23.47(C4×D5), C223(C8⋊D5), C54(C24.4C4), C2.12(D5×M4(2)), C20.55D423C2, (C2×C20).821C23, (C22×C4).304D10, C10.39(C2×M4(2)), D10.35(C22⋊C4), (C22×Dic5).16C4, Dic5.36(C22⋊C4), (C22×C20).338C22, (C2×C4×D5).22C4, C2.9(C2×C8⋊D5), C2.9(D5×C22⋊C4), (C2×C8⋊D5)⋊11C2, (C5×C22⋊C8)⋊15C2, (C2×C4).132(C4×D5), (D5×C22×C4).17C2, C22.103(C2×C4×D5), (C2×C20).327(C2×C4), (C2×C52C8)⋊27C22, C10.49(C2×C22⋊C4), (C2×C4×D5).343C22, (C22×D5).99(C2×C4), (C2×C4).763(C22×D5), (C22×C10).107(C2×C4), (C2×C10).177(C22×C4), (C2×Dic5).140(C2×C4), SmallGroup(320,353)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D107M4(2)
C1C5C10C20C2×C20C2×C4×D5D5×C22×C4 — D107M4(2)
C5C2×C10 — D107M4(2)
C1C2×C4C22⋊C8

Generators and relations for D107M4(2)
 G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a3b, dbd=a8b, dcd=c5 >

Subgroups: 734 in 190 conjugacy classes, 61 normal (33 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 [×2], C2×C8 [×2], 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, C22⋊C8 [×3], C2×M4(2) [×2], C23×C4, C52C8 [×2], C40 [×2], 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, C8⋊D5 [×4], C2×C52C8 [×2], C2×C40 [×2], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C23×D5, D101C8 [×2], C20.55D4, C5×C22⋊C8, C2×C8⋊D5 [×2], D5×C22×C4, D107M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C2×M4(2) [×2], C4×D5 [×2], C22×D5, C24.4C4, C8⋊D5 [×2], C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, C2×C8⋊D5, D5×M4(2), D107M4(2)

Smallest permutation representation of D107M4(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 20)(12 19)(13 18)(14 17)(15 16)(21 23)(24 30)(25 29)(26 28)(31 39)(32 38)(33 37)(34 36)(41 43)(44 50)(45 49)(46 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 74)(72 73)(75 80)(76 79)(77 78)
(1 73 45 70 25 16 38 53)(2 72 46 69 26 15 39 52)(3 71 47 68 27 14 40 51)(4 80 48 67 28 13 31 60)(5 79 49 66 29 12 32 59)(6 78 50 65 30 11 33 58)(7 77 41 64 21 20 34 57)(8 76 42 63 22 19 35 56)(9 75 43 62 23 18 36 55)(10 74 44 61 24 17 37 54)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 16)(12 15)(13 14)(17 20)(18 19)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 70)(66 69)(67 68)(71 80)(72 79)(73 78)(74 77)(75 76)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222222224444444444558888888810···101010101020···202020202040···40
size1111221010101011112210101010224444202020202···244442···244444···4

68 irreducible representations

dim11111111122222222244
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5M4(2)M4(2)D10D10C4×D5C4×D5C8⋊D5D4×D5D5×M4(2)
kernelD107M4(2)D101C8C20.55D4C5×C22⋊C8C2×C8⋊D5D5×C22×C4C2×C4×D5C22×Dic5C23×D5C4×D5C22⋊C8D10C2×C10C2×C8C22×C4C2×C4C23C22C4C2
# reps121121422424442441644

Matrix representation of D107M4(2) in GL6(𝔽41)

4000000
0400000
007700
00344000
0000400
0000040
,
100000
32400000
007700
00403400
000010
00002540
,
32390000
3690000
0032000
0022900
0000937
00002032
,
100000
32400000
001000
00344000
0000400
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,25,0,0,0,0,0,40],[32,36,0,0,0,0,39,9,0,0,0,0,0,0,32,22,0,0,0,0,0,9,0,0,0,0,0,0,9,20,0,0,0,0,37,32],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

D107M4(2) in GAP, Magma, Sage, TeX

D_{10}\rtimes_7M_4(2)
% in TeX

G:=Group("D10:7M4(2)");
// GroupNames label

G:=SmallGroup(320,353);
// by ID

G=gap.SmallGroup(320,353);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^3*b,d*b*d=a^8*b,d*c*d=c^5>;
// generators/relations

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