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G = C4⋊C4⋊Dic5order 320 = 26·5

1st semidirect product of C4⋊C4 and Dic5 acting via Dic5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C41Dic5, (C2×C10).3D8, C10.28C4≀C2, (D4×C10)⋊13C4, (C2×D4)⋊1Dic5, C4⋊D4.1D5, (C2×C20).229D4, C22.2(D4⋊D5), (C2×C10).10SD16, (C22×C10).44D4, (C22×C4).59D10, C55(C22.SD16), C20.55D428C2, C10.41(C23⋊C4), C2.3(D4⋊Dic5), C23.48(C5⋊D4), C22.2(D4.D5), C2.5(C23⋊Dic5), C10.38(D4⋊C4), C2.4(D42Dic5), C10.10C4241C2, (C22×C20).371C22, C22.37(C23.D5), (C5×C4⋊C4)⋊8C4, (C2×C4).7(C2×Dic5), (C2×C20).340(C2×C4), (C5×C4⋊D4).10C2, (C2×C4).163(C5⋊D4), (C2×C10).159(C22⋊C4), SmallGroup(320,95)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4⋊C4⋊Dic5
C1C5C10C2×C10C22×C10C22×C20C10.10C42 — C4⋊C4⋊Dic5
C5C2×C10C2×C20 — C4⋊C4⋊Dic5
C1C22C22×C4C4⋊D4

Generators and relations for C4⋊C4⋊Dic5
 G = < a,b,c,d | a4=b4=c10=1, d2=c5, bab-1=a-1, ac=ca, dad-1=a-1b2, cbc-1=b-1, dbd-1=a-1b, dcd-1=c-1 >

Subgroups: 334 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C2×C10, C2×C10, C2.C42, C22⋊C8, C4⋊D4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22.SD16, C2×C52C8, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, D4×C10, D4×C10, C20.55D4, C10.10C42, C5×C4⋊D4, C4⋊C4⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, Dic5, D10, C23⋊C4, D4⋊C4, C4≀C2, C2×Dic5, C5⋊D4, C22.SD16, D4⋊D5, D4.D5, C23.D5, D4⋊Dic5, C23⋊Dic5, D42Dic5, C4⋊C4⋊Dic5

Smallest permutation representation of C4⋊C4⋊Dic5
On 80 points
Generators in S80
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 26 16 24)(12 27 17 25)(13 28 18 21)(14 29 19 22)(15 30 20 23)(41 55 78 63)(42 56 79 64)(43 57 80 65)(44 58 71 66)(45 59 72 67)(46 60 73 68)(47 51 74 69)(48 52 75 70)(49 53 76 61)(50 54 77 62)
(1 55 14 60)(2 51 15 56)(3 57 11 52)(4 53 12 58)(5 59 13 54)(6 65 16 70)(7 61 17 66)(8 67 18 62)(9 63 19 68)(10 69 20 64)(21 77 33 72)(22 73 34 78)(23 79 35 74)(24 75 31 80)(25 71 32 76)(26 48 36 43)(27 44 37 49)(28 50 38 45)(29 46 39 41)(30 42 40 47)
(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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 40)(22 39)(23 38)(24 37)(25 36)(26 32)(27 31)(28 35)(29 34)(30 33)(41 63 46 68)(42 62 47 67)(43 61 48 66)(44 70 49 65)(45 69 50 64)(51 77 56 72)(52 76 57 71)(53 75 58 80)(54 74 59 79)(55 73 60 78)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222224444444455888810···10101010101010101020···2020202020
size111122822482020202022202020202···2444488884···48888

47 irreducible representations

dim1111112222222222244444
type++++++++-+-++-
imageC1C2C2C2C4C4D4D4D5D8SD16Dic5D10Dic5C4≀C2C5⋊D4C5⋊D4C23⋊C4D4⋊D5D4.D5C23⋊Dic5D42Dic5
kernelC4⋊C4⋊Dic5C20.55D4C10.10C42C5×C4⋊D4C5×C4⋊C4D4×C10C2×C20C22×C10C4⋊D4C2×C10C2×C10C4⋊C4C22×C4C2×D4C10C2×C4C23C10C22C22C2C2
# reps1111221122222244412244

Matrix representation of C4⋊C4⋊Dic5 in GL6(𝔽41)

3200000
2690000
001000
000100
0000937
0000032
,
3210000
21380000
0040000
0004000
000037
00003438
,
100000
29400000
000100
00403400
000010
000001
,
100000
22320000
001000
00344000
000010
00002540

G:=sub<GL(6,GF(41))| [32,26,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,37,32],[3,21,0,0,0,0,21,38,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,34,0,0,0,0,7,38],[1,29,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,22,0,0,0,0,0,32,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,25,0,0,0,0,0,40] >;

C4⋊C4⋊Dic5 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C4:C4:Dic5");
// GroupNames label

G:=SmallGroup(320,95);
// by ID

G=gap.SmallGroup(320,95);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,136,12550]);
// Polycyclic

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

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