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G = (D4×C10)⋊21C4order 320 = 26·5

5th semidirect product of D4×C10 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (D4×C10)⋊21C4, (Q8×C10)⋊19C4, C4○D44Dic5, (C2×Q8)⋊7Dic5, (C2×D4)⋊9Dic5, C4○D4.37D10, (C2×C20).198D4, C20.453(C2×D4), Q8.7(C2×Dic5), D4.7(C2×Dic5), (C4×Dic5)⋊7C22, D42Dic510C2, C20.85(C22⋊C4), C20.145(C22×C4), (C2×C20).482C23, C57(C42⋊C22), (C22×C10).114D4, (C22×C4).162D10, C23.31(C5⋊D4), C4.Dic524C22, C4.23(C23.D5), C4.16(C22×Dic5), C22.6(C23.D5), C23.21D1020C2, (C22×C20).208C22, (C5×C4○D4)⋊10C4, (C2×C4○D4).5D5, (C10×C4○D4).5C2, (C5×D4).38(C2×C4), (C2×C10).40(C2×D4), C4.144(C2×C5⋊D4), (C5×Q8).40(C2×C4), (C2×C20).299(C2×C4), (C2×C4).90(C5⋊D4), (C2×C4.Dic5)⋊22C2, (C2×C4).29(C2×Dic5), C22.12(C2×C5⋊D4), C2.21(C2×C23.D5), C10.126(C2×C22⋊C4), (C5×C4○D4).42C22, (C2×C4).567(C22×D5), (C2×C10).91(C22⋊C4), SmallGroup(320,863)

Series: Derived Chief Lower central Upper central

C1C20 — (D4×C10)⋊21C4
C1C5C10C20C2×C20C4.Dic5C2×C4.Dic5 — (D4×C10)⋊21C4
C5C10C20 — (D4×C10)⋊21C4
C1C4C22×C4C2×C4○D4

Generators and relations for (D4×C10)⋊21C4
 G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=bc >

Subgroups: 398 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×4], C22 [×3], C22 [×5], C5, C8 [×2], C2×C4 [×6], C2×C4 [×7], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×5], C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic5 [×2], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×5], C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C52C8 [×2], C2×Dic5 [×2], C2×C20 [×6], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C42⋊C22, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×Dic5 [×2], C4⋊Dic5, C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], D42Dic5 [×4], C2×C4.Dic5, C23.21D10, C10×C4○D4, (D4×C10)⋊21C4
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, C42⋊C22, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C23.D5, (D4×C10)⋊21C4

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F10G···10R20A···20H20I···20T
order12222224444444444455888810···1010···1020···2020···20
size112224411222442020202022202020202···24···42···24···4

62 irreducible representations

dim11111111222222222244
type+++++++++---+
imageC1C2C2C2C2C4C4C4D4D4D5D10Dic5Dic5Dic5D10C5⋊D4C5⋊D4C42⋊C22(D4×C10)⋊21C4
kernel(D4×C10)⋊21C4D42Dic5C2×C4.Dic5C23.21D10C10×C4○D4D4×C10Q8×C10C5×C4○D4C2×C20C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C2×C4C23C5C1
# reps141112243122224412428

Matrix representation of (D4×C10)⋊21C4 in GL4(𝔽41) generated by

351500
32000
003515
00320
,
32000
03200
0090
0009
,
003937
00322
2400
93900
,
6600
13500
002828
003213
G:=sub<GL(4,GF(41))| [35,3,0,0,15,20,0,0,0,0,35,3,0,0,15,20],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,9],[0,0,2,9,0,0,4,39,39,32,0,0,37,2,0,0],[6,1,0,0,6,35,0,0,0,0,28,32,0,0,28,13] >;

(D4×C10)⋊21C4 in GAP, Magma, Sage, TeX

(D_4\times C_{10})\rtimes_{21}C_4
% in TeX

G:=Group("(D4xC10):21C4");
// GroupNames label

G:=SmallGroup(320,863);
// by ID

G=gap.SmallGroup(320,863);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,1684,438,102,12550]);
// Polycyclic

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

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