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G = M4(2)⋊4Dic5order 320 = 26·5

4th semidirect product of M4(2) and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊4Dic5, C20.44(C4⋊C4), (C2×C20).13Q8, (C2×C4).130D20, (C2×C20).112D4, C4.4(C4⋊Dic5), (C2×C4).7Dic10, C23.10(C4×D5), C23.D5.9C4, (C5×M4(2))⋊13C4, (C2×C10).26C42, (C22×C4).64D10, C22.4(C4×Dic5), C56(M4(2)⋊4C4), (C2×M4(2)).10D5, C4.19(C23.D5), C4.51(D10⋊C4), C20.135(C22⋊C4), C4.13(C10.D4), (C10×M4(2)).14C2, (C22×C20).128C22, C22.21(D10⋊C4), C10.37(C2.C42), C23.21D10.11C2, C22.14(C10.D4), C2.18(C10.10C42), (C2×C52C8)⋊3C4, (C2×C4).142(C4×D5), (C2×C10).69(C4⋊C4), (C2×C20).238(C2×C4), (C2×C4).15(C2×Dic5), (C2×C4).235(C5⋊D4), (C2×C4.Dic5).13C2, (C2×C10).76(C22⋊C4), (C22×C10).101(C2×C4), SmallGroup(320,117)

Series: Derived Chief Lower central Upper central

C1C2×C10 — M4(2)⋊4Dic5
C1C5C10C20C2×C20C22×C20C2×C4.Dic5 — M4(2)⋊4Dic5
C5C10C2×C10 — M4(2)⋊4Dic5
C1C4C22×C4C2×M4(2)

Generators and relations for M4(2)⋊4Dic5
 G = < a,b,c,d | a8=b2=c10=1, d2=c5, bab=cac-1=a5, dad-1=a5b, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 262 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×3], C4 [×4], C4 [×2], C22 [×3], C22, C5, C8 [×4], C2×C4 [×6], C2×C4 [×2], C23, C10, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×3], M4(2) [×2], M4(2) [×3], C22×C4, Dic5 [×2], C20 [×4], C2×C10 [×3], C2×C10, C42⋊C2, C2×M4(2), C2×M4(2), C52C8 [×2], C40 [×2], C2×Dic5 [×2], C2×C20 [×6], C22×C10, M4(2)⋊4C4, C2×C52C8 [×2], C4.Dic5 [×2], C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C40, C5×M4(2) [×2], C5×M4(2), C22×C20, C2×C4.Dic5, C23.21D10, C10×M4(2), M4(2)⋊4Dic5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], M4(2)⋊4C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C10.10C42, M4(2)⋊4Dic5

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222444444444558888888810···101010101020···202020202040···40
size112221122220202020224444202020202···244442···244444···4

62 irreducible representations

dim1111111222222222244
type+++++-+-+-+
imageC1C2C2C2C4C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4C4×D5M4(2)⋊4C4M4(2)⋊4Dic5
kernelM4(2)⋊4Dic5C2×C4.Dic5C23.21D10C10×M4(2)C2×C52C8C23.D5C5×M4(2)C2×C20C2×C20C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C2×C4C23C5C1
# reps1111444312424448428

Matrix representation of M4(2)⋊4Dic5 in GL4(𝔽41) generated by

00186
003523
9000
0900
,
18600
352300
002335
00618
,
6100
40000
003540
0010
,
391600
28200
003228
0009
G:=sub<GL(4,GF(41))| [0,0,9,0,0,0,0,9,18,35,0,0,6,23,0,0],[18,35,0,0,6,23,0,0,0,0,23,6,0,0,35,18],[6,40,0,0,1,0,0,0,0,0,35,1,0,0,40,0],[39,28,0,0,16,2,0,0,0,0,32,0,0,0,28,9] >;

M4(2)⋊4Dic5 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_4{\rm Dic}_5
% in TeX

G:=Group("M4(2):4Dic5");
// GroupNames label

G:=SmallGroup(320,117);
// by ID

G=gap.SmallGroup(320,117);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,1123,136,851,12550]);
// Polycyclic

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

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