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G = (D4×C10).29C4order 320 = 26·5

10th non-split extension by D4×C10 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D4×C10).29C4, C20.214(C2×D4), (C2×C20).199D4, (C2×D4).9Dic5, (C2×D4).203D10, C20.D413C2, (C22×C20).13C4, (C2×Q8).172D10, C20.10D413C2, C23.3(C2×Dic5), (C22×C4).6Dic5, (C2×C20).483C23, (C22×C4).163D10, C4.34(C23.D5), C20.138(C22⋊C4), (D4×C10).244C22, (Q8×C10).207C22, C4.Dic5.47C22, C22.7(C23.D5), C22.8(C22×Dic5), (C22×C20).209C22, C56(M4(2).8C22), (C2×C4○D4).6D5, C4.96(C2×C5⋊D4), (C10×C4○D4).6C2, (C2×C4).5(C2×Dic5), (C2×C20).189(C2×C4), (C2×C4).91(C5⋊D4), (C2×C4.Dic5)⋊23C2, C2.22(C2×C23.D5), C10.127(C2×C22⋊C4), (C2×C4).131(C22×D5), (C2×C10).92(C22⋊C4), (C22×C10).143(C2×C4), (C2×C10).302(C22×C4), SmallGroup(320,864)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (D4×C10).29C4
C1C5C10C20C2×C20C4.Dic5C2×C4.Dic5 — (D4×C10).29C4
C5C10C2×C10 — (D4×C10).29C4
C1C4C22×C4C2×C4○D4

Generators and relations for (D4×C10).29C4
 G = < a,b,c,d | a10=b4=c2=1, d4=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, dbd-1=a5b, dcd-1=b2c >

Subgroups: 350 in 150 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×5], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×4], D4 [×6], Q8 [×2], C23, C23 [×2], C10, C10 [×5], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×5], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C52C8 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×C10, C22×C10 [×2], M4(2).8C22, C2×C52C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C20.D4 [×2], C20.10D4 [×2], C2×C4.Dic5 [×2], C10×C4○D4, (D4×C10).29C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, M4(2).8C22, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C23.D5, (D4×C10).29C4

Smallest permutation representation of (D4×C10).29C4
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 20 37 14)(2 16 38 15)(3 17 39 11)(4 18 40 12)(5 19 36 13)(6 21 31 30)(7 22 32 26)(8 23 33 27)(9 24 34 28)(10 25 35 29)(41 52 46 57)(42 53 47 58)(43 54 48 59)(44 55 49 60)(45 56 50 51)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 22)(2 23)(3 24)(4 25)(5 21)(6 19)(7 20)(8 16)(9 17)(10 18)(11 34)(12 35)(13 31)(14 32)(15 33)(26 37)(27 38)(28 39)(29 40)(30 36)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 71)(48 72)(49 73)(50 74)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 77 7 54 37 72 32 59)(2 71 8 58 38 76 33 53)(3 75 9 52 39 80 34 57)(4 79 10 56 40 74 35 51)(5 73 6 60 36 78 31 55)(11 70 24 46 17 65 28 41)(12 64 25 50 18 69 29 45)(13 68 21 44 19 63 30 49)(14 62 22 48 20 67 26 43)(15 66 23 42 16 61 27 47)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A···8H10A···10F10G···10R20A···20H20I···20T
order12222224444444558···810···1010···1020···2020···20
size112224411222442220···202···24···42···24···4

62 irreducible representations

dim11111112222222244
type+++++++-+-++
imageC1C2C2C2C2C4C4D4D5Dic5D10Dic5D10D10C5⋊D4M4(2).8C22(D4×C10).29C4
kernel(D4×C10).29C4C20.D4C20.10D4C2×C4.Dic5C10×C4○D4C22×C20D4×C10C2×C20C2×C4○D4C22×C4C22×C4C2×D4C2×D4C2×Q8C2×C4C5C1
# reps122214442424221628

Matrix representation of (D4×C10).29C4 in GL6(𝔽41)

3100000
040000
00400537
00040413
000010
000001
,
4000000
010000
0032005
000950
0000320
000009
,
4000000
0400000
000323640
0090136
000009
0000320
,
010000
4000000
003736404
0013373731
00039413
0020364

G:=sub<GL(6,GF(41))| [31,0,0,0,0,0,0,4,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,5,4,1,0,0,0,37,13,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,5,32,0,0,0,5,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,36,1,0,32,0,0,40,36,9,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,37,13,0,2,0,0,36,37,39,0,0,0,40,37,4,36,0,0,4,31,13,4] >;

(D4×C10).29C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})._{29}C_4
% in TeX

G:=Group("(D4xC10).29C4");
// GroupNames label

G:=SmallGroup(320,864);
// by ID

G=gap.SmallGroup(320,864);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,1684,12550]);
// Polycyclic

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

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