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G = C10.29C4≀C2order 320 = 26·5

5th non-split extension by C10 of C4≀C2 acting via C4≀C2/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C42Dic5, C10.29C4≀C2, (Q8×C10)⋊11C4, (C2×Q8)⋊1Dic5, (C2×C10).3Q16, C22⋊Q8.1D5, (C2×C20).230D4, (C2×C10).11SD16, (C22×C10).45D4, (C22×C4).60D10, C22.5(Q8⋊D5), C10.42(C23⋊C4), C2.3(Q8⋊Dic5), C55(C23.31D4), C23.49(C5⋊D4), C2.6(C23⋊Dic5), C22.2(C5⋊Q16), C10.18(Q8⋊C4), C2.5(D42Dic5), C20.55D4.16C2, (C22×C20).372C22, C22.38(C23.D5), C10.10C42.35C2, (C5×C4⋊C4)⋊9C4, (C2×C4).8(C2×Dic5), (C2×C20).341(C2×C4), (C5×C22⋊Q8).10C2, (C2×C4).164(C5⋊D4), (C2×C10).160(C22⋊C4), SmallGroup(320,96)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C10.29C4≀C2
C1C5C10C2×C10C22×C10C22×C20C10.10C42 — C10.29C4≀C2
C5C2×C10C2×C20 — C10.29C4≀C2
C1C22C22×C4C22⋊Q8

Generators and relations for C10.29C4≀C2
 G = < a,b,c,d | a10=b4=d4=1, c2=a5, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, dbd-1=a5b, dcd-1=a5b-1c >

Subgroups: 286 in 80 conjugacy classes, 31 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, C4⋊C4, 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, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×3], C5×Q8, C22×C10, C23.31D4, C2×C52C8, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, Q8×C10, C20.55D4, C10.10C42, C5×C22⋊Q8, C10.29C4≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, SD16, Q16, Dic5 [×2], D10, C23⋊C4, Q8⋊C4, C4≀C2, C2×Dic5, C5⋊D4 [×2], C23.31D4, Q8⋊D5, C5⋊Q16, C23.D5, C23⋊Dic5, Q8⋊Dic5, D42Dic5, C10.29C4≀C2

Smallest permutation representation of C10.29C4≀C2
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 50 25 33)(2 41 26 34)(3 42 27 35)(4 43 28 36)(5 44 29 37)(6 45 30 38)(7 46 21 39)(8 47 22 40)(9 48 23 31)(10 49 24 32)(11 70 78 53)(12 61 79 54)(13 62 80 55)(14 63 71 56)(15 64 72 57)(16 65 73 58)(17 66 74 59)(18 67 75 60)(19 68 76 51)(20 69 77 52)
(1 70 6 65)(2 61 7 66)(3 62 8 67)(4 63 9 68)(5 64 10 69)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 59 26 54)(22 60 27 55)(23 51 28 56)(24 52 29 57)(25 53 30 58)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 33 30 45)(2 32 21 44)(3 31 22 43)(4 40 23 42)(5 39 24 41)(6 38 25 50)(7 37 26 49)(8 36 27 48)(9 35 28 47)(10 34 29 46)(11 16)(12 15)(13 14)(17 20)(18 19)(51 55)(52 54)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)(71 80)(72 79)(73 78)(74 77)(75 76)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20P
order12222244444444455888810···101010101020···2020···20
size111122224882020202022202020202···244444···48···8

47 irreducible representations

dim1111112222222222244444
type+++++++--+-++-
imageC1C2C2C2C4C4D4D4D5SD16Q16Dic5D10Dic5C4≀C2C5⋊D4C5⋊D4C23⋊C4Q8⋊D5C5⋊Q16C23⋊Dic5D42Dic5
kernelC10.29C4≀C2C20.55D4C10.10C42C5×C22⋊Q8C5×C4⋊C4Q8×C10C2×C20C22×C10C22⋊Q8C2×C10C2×C10C4⋊C4C22×C4C2×Q8C10C2×C4C23C10C22C22C2C2
# reps1111221122222244412244

Matrix representation of C10.29C4≀C2 in GL6(𝔽41)

100000
010000
0040000
0004000
0000407
0000347
,
900000
0320000
009200
0003200
0000400
0000040
,
010000
100000
00141100
00382700
00002440
0000117
,
3200000
010000
009200
0013200
0000734
0000134

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,2,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,38,0,0,0,0,11,27,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[32,0,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,2,32,0,0,0,0,0,0,7,1,0,0,0,0,34,34] >;

C10.29C4≀C2 in GAP, Magma, Sage, TeX

C_{10}._{29}C_4\wr C_2
% in TeX

G:=Group("C10.29C4wrC2");
// GroupNames label

G:=SmallGroup(320,96);
// by ID

G=gap.SmallGroup(320,96);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,136,12550]);
// Polycyclic

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

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