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G = C4⋊Dic5⋊C4order 320 = 26·5

2nd semidirect product of C4⋊Dic5 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic52C4, C10.14C4≀C2, (C2×Dic10)⋊1C4, (C2×C20).222D4, (C2×C4).104D20, (C2×C10).13Q16, (C2×C10).28SD16, (C22×C4).53D10, C2.5(D204C4), C10.23(C23⋊C4), C10.8(Q8⋊C4), C20.48D4.7C2, C20.55D4.1C2, C2.3(C10.Q16), (C22×C10).175D4, C23.71(C5⋊D4), C53(C23.31D4), C22.4(D4.D5), C22.4(C5⋊Q16), C2.C42.6D5, (C22×C20).90C22, C2.5(C23.1D10), C22.55(D10⋊C4), (C2×C4).9(C4×D5), (C2×C20).194(C2×C4), (C2×C10).98(C22⋊C4), (C5×C2.C42).13C2, SmallGroup(320,10)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4⋊Dic5⋊C4
C1C5C10C2×C10C22×C10C22×C20C20.48D4 — C4⋊Dic5⋊C4
C5C2×C10C2×C20 — C4⋊Dic5⋊C4
C1C22C22×C4C2.C42

Generators and relations for C4⋊Dic5⋊C4
 G = < a,b,c,d | a4=b10=d4=1, c2=b5, ab=ba, cac-1=a-1, dad-1=ab5, cbc-1=b-1, bd=db, dcd-1=a-1b5c >

Subgroups: 286 in 80 conjugacy classes, 29 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C22 [×3], C22 [×2], C5, C8, C2×C4 [×2], C2×C4 [×7], Q8, C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, C22×C4, C22×C4, C2×Q8, Dic5 [×2], C20 [×4], C2×C10 [×3], C2×C10 [×2], C2.C42, C22⋊C8, C22⋊Q8, C52C8, Dic10, C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×5], C22×C10, C23.31D4, C2×C52C8, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C22×C20, C20.55D4, C5×C2.C42, C20.48D4, C4⋊Dic5⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, SD16, Q16, D10, C23⋊C4, Q8⋊C4, C4≀C2, C4×D5, D20, C5⋊D4, C23.31D4, D10⋊C4, D4.D5, C5⋊Q16, D204C4, C23.1D10, C10.Q16, C4⋊Dic5⋊C4

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

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E4A4B4C···4G4H4I5A5B8A8B8C8D10A···10N20A···20X
order122222444···44455888810···1020···20
size111122224···4404022202020202···24···4

59 irreducible representations

dim111111222222222224444
type+++++++-+++--
imageC1C2C2C2C4C4D4D4D5SD16Q16D10C4≀C2C4×D5D20C5⋊D4D204C4C23⋊C4D4.D5C5⋊Q16C23.1D10
kernelC4⋊Dic5⋊C4C20.55D4C5×C2.C42C20.48D4C4⋊Dic5C2×Dic10C2×C20C22×C10C2.C42C2×C10C2×C10C22×C4C10C2×C4C2×C4C23C2C10C22C22C2
# reps1111221122224444161224

Matrix representation of C4⋊Dic5⋊C4 in GL4(𝔽41) generated by

9000
243200
00939
00032
,
10000
53700
00400
00040
,
22400
331900
00330
002738
,
1000
3900
0010
00940
G:=sub<GL(4,GF(41))| [9,24,0,0,0,32,0,0,0,0,9,0,0,0,39,32],[10,5,0,0,0,37,0,0,0,0,40,0,0,0,0,40],[22,33,0,0,4,19,0,0,0,0,3,27,0,0,30,38],[1,3,0,0,0,9,0,0,0,0,1,9,0,0,0,40] >;

C4⋊Dic5⋊C4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,10);
// by ID

G=gap.SmallGroup(320,10);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,1571,570,192,12550]);
// Polycyclic

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

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