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G = C524M4(2)  order 400 = 24·52

3rd semidirect product of C52 and M4(2) acting via M4(2)/C2=C2×C4

metabelian, supersoluble, monomial

Aliases: D10.3F5, Dic5.3F5, C524M4(2), C53(C4.F5), C10.5(C2×F5), C524C82C2, C525C82C2, (D5×C10).4C4, C2.5(D5⋊F5), (D5×Dic5).5C2, (C5×Dic5).2C4, C51(C22.F5), C526C4.5C22, (C5×C10).12(C2×C4), SmallGroup(400,128)

Series: Derived Chief Lower central Upper central

C1C5×C10 — C524M4(2)
C1C5C52C5×C10C526C4C524C8 — C524M4(2)
C52C5×C10 — C524M4(2)
C1C2

Generators and relations for C524M4(2)
 G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=a-1, cbc-1=b2, bd=db, dcd=c5 >

10C2
2C5
2C5
5C4
5C22
25C4
2C10
2C10
2D5
10C10
25C2×C4
25C8
25C8
5C2×C10
5Dic5
5C20
5Dic5
10Dic5
10Dic5
2C5×D5
25M4(2)
5C4×D5
5C2×Dic5
5C5⋊C8
5C5⋊C8
5C5⋊C8
5C5⋊C8
10C5⋊C8
10C5⋊C8
5C22.F5
5C4.F5

Character table of C524M4(2)

 class 12A2B4A4B4C5A5B5C5D8A8B8C8D10A10B10C10D10E10F20A20B
 size 1110102525448850505050448820202020
ρ11111111111111111111111    trivial
ρ211-1-11111111-1-111111-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-111111111    linear of order 2
ρ411-1-1111111-111-11111-1-1-1-1    linear of order 2
ρ5111-1-1-11111-ii-ii111111-1-1    linear of order 4
ρ611-11-1-11111-i-iii1111-1-111    linear of order 4
ρ7111-1-1-11111i-ii-i111111-1-1    linear of order 4
ρ811-11-1-11111ii-i-i1111-1-111    linear of order 4
ρ92-2002i-2i22220000-2-2-2-20000    complex lifted from M4(2)
ρ102-200-2i2i22220000-2-2-2-20000    complex lifted from M4(2)
ρ11444000-14-1-10000-14-1-1-1-100    orthogonal lifted from F5
ρ12440-4004-1-1-100004-1-1-10011    orthogonal lifted from C2×F5
ρ134404004-1-1-100004-1-1-100-1-1    orthogonal lifted from F5
ρ1444-4000-14-1-10000-14-1-11100    orthogonal lifted from C2×F5
ρ154-40000-14-1-100001-411-5500    symplectic lifted from C22.F5, Schur index 2
ρ164-40000-14-1-100001-4115-500    symplectic lifted from C22.F5, Schur index 2
ρ174-400004-1-1-10000-411100-5--5    complex lifted from C4.F5
ρ184-400004-1-1-10000-411100--5-5    complex lifted from C4.F5
ρ19880000-2-23-20000-2-2-230000    orthogonal lifted from D5⋊F5
ρ20880000-2-2-230000-2-23-20000    orthogonal lifted from D5⋊F5
ρ218-80000-2-2-23000022-320000    symplectic faithful, Schur index 2
ρ228-80000-2-23-20000222-30000    symplectic faithful, Schur index 2

Smallest permutation representation of C524M4(2)
On 80 points
Generators in S80
(1 10 28 77 66)(2 78 11 67 29)(3 68 79 30 12)(4 31 69 13 80)(5 14 32 73 70)(6 74 15 71 25)(7 72 75 26 16)(8 27 65 9 76)(17 34 64 50 41)(18 51 35 42 57)(19 43 52 58 36)(20 59 44 37 53)(21 38 60 54 45)(22 55 39 46 61)(23 47 56 62 40)(24 63 48 33 49)
(1 66 77 28 10)(2 78 11 67 29)(3 12 30 79 68)(4 31 69 13 80)(5 70 73 32 14)(6 74 15 71 25)(7 16 26 75 72)(8 27 65 9 76)(17 34 64 50 41)(18 57 42 35 51)(19 43 52 58 36)(20 53 37 44 59)(21 38 60 54 45)(22 61 46 39 55)(23 47 56 62 40)(24 49 33 48 63)
(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 64)(2 61)(3 58)(4 63)(5 60)(6 57)(7 62)(8 59)(9 37)(10 34)(11 39)(12 36)(13 33)(14 38)(15 35)(16 40)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(41 77)(42 74)(43 79)(44 76)(45 73)(46 78)(47 75)(48 80)(49 69)(50 66)(51 71)(52 68)(53 65)(54 70)(55 67)(56 72)

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

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

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

Matrix representation of C524M4(2) in GL10(𝔽41)

1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000000001
0000001000
00000040404040
0000000010
,
1000000000
0100000000
00004000000
00004010000
00014000000
00104000000
0000001000
0000000100
0000000010
0000000001
,
14000000000
02700000000
0000090000
0090000000
0009000000
0000900000
0000001000
0000000010
0000000001
00000040404040
,
0100000000
1000000000
00400000000
00040000000
00004000000
00000400000
0000001000
0000000001
00000040404040
0000000100

G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,40,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[14,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,40],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,40,0] >;

C524M4(2) in GAP, Magma, Sage, TeX

C_5^2\rtimes_4M_4(2)
% in TeX

G:=Group("C5^2:4M4(2)");
// GroupNames label

G:=SmallGroup(400,128);
// by ID

G=gap.SmallGroup(400,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1444,970,496,8645,2897]);
// Polycyclic

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

Export

Subgroup lattice of C524M4(2) in TeX
Character table of C524M4(2) in TeX

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