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G = D40.6C4order 320 = 26·5

4th non-split extension by D40 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.7D4, D40.6C4, C8.17D20, C20.48D8, C8.2(C4×D5), C40.63(C2×C4), C8.C41D5, (C2×C8).43D10, (C2×C20).95D4, C20.4C85C2, C4.21(D4⋊D5), (C2×D40).12C2, C52(M5(2)⋊C2), (C2×C10).4SD16, C22.3(Q8⋊D5), C4.4(D10⋊C4), C2.9(D206C4), C20.51(C22⋊C4), (C2×C40).100C22, C10.22(D4⋊C4), (C5×C8.C4)⋊9C2, (C2×C4).18(C5⋊D4), SmallGroup(320,53)

Series: Derived Chief Lower central Upper central

C1C40 — D40.6C4
C1C5C10C20C2×C20C2×C40C2×D40 — D40.6C4
C5C10C20C40 — D40.6C4
C1C2C2×C4C2×C8C8.C4

Generators and relations for D40.6C4
 G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a11, cbc-1=a15b >

Subgroups: 414 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, D5, C10, C10, C16, C2×C8, M4(2), D8, C2×D4, C20, D10, C2×C10, C8.C4, M5(2), C2×D8, C40, C40, D20, C2×C20, C22×D5, M5(2)⋊C2, C52C16, D40, D40, C2×C40, C5×M4(2), C2×D20, C20.4C8, C5×C8.C4, C2×D40, D40.6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, M5(2)⋊C2, D10⋊C4, D4⋊D5, Q8⋊D5, D206C4, D40.6C4

Smallest permutation representation of D40.6C4
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 10)(2 9)(3 8)(4 7)(5 6)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(72 80)(73 79)(74 78)(75 77)
(1 69 11 59 21 49 31 79)(2 80 12 70 22 60 32 50)(3 51 13 41 23 71 33 61)(4 62 14 52 24 42 34 72)(5 73 15 63 25 53 35 43)(6 44 16 74 26 64 36 54)(7 55 17 45 27 75 37 65)(8 66 18 56 28 46 38 76)(9 77 19 67 29 57 39 47)(10 48 20 78 30 68 40 58)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D4A4B5A5B8A8B8C8D8E10A10B10C10D16A16B16C16D20A20B20C20D20E20F40A···40H40I···40P
order12222445588888101010101616161620202020202040···4040···40
size11240402222224882244202020202222444···48···8

44 irreducible representations

dim111112222222224444
type++++++++++++++
imageC1C2C2C2C4D4D4D5D8SD16D10C4×D5D20C5⋊D4M5(2)⋊C2D4⋊D5Q8⋊D5D40.6C4
kernelD40.6C4C20.4C8C5×C8.C4C2×D40D40C40C2×C20C8.C4C20C2×C10C2×C8C8C8C2×C4C5C4C22C1
# reps111141122224442228

Matrix representation of D40.6C4 in GL4(𝔽241) generated by

2918500
7619400
71825627
35172214228
,
2272700
2071400
13224190190
9410924051
,
002401
12619623951
1961450
21958450
G:=sub<GL(4,GF(241))| [29,76,7,35,185,194,182,172,0,0,56,214,0,0,27,228],[227,207,13,94,27,14,224,109,0,0,190,240,0,0,190,51],[0,126,19,219,0,196,61,58,240,239,45,45,1,51,0,0] >;

D40.6C4 in GAP, Magma, Sage, TeX

D_{40}._6C_4
% in TeX

G:=Group("D40.6C4");
// GroupNames label

G:=SmallGroup(320,53);
// by ID

G=gap.SmallGroup(320,53);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,184,675,794,192,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

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