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G = C24⋊3D10  order 320 = 26·5

2nd semidirect product of C24 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24⋊3D10, C10.252+ (1+4), (C2×D4)⋊5D10, C22≀C2⋊2D5, C22⋊C4⋊5D10, (C22×D5)⋊6D4, C23⋊D10⋊3C2, C20⋊2D4⋊12C2, D10.14(C2×D4), (D4×C10)⋊6C22, C24⋊2D5⋊6C2, C5⋊2(C23⋊3D4), D10⋊D4⋊12C2, C22.40(D4×D5), (C2×D20)⋊18C22, (C2×C20).27C23, C4⋊Dic5⋊25C22, (C23×C10)⋊9C22, C10.55(C22×D4), (C23×D5)⋊6C22, (C2×C10).133C24, C10.D4⋊8C22, C22.D20⋊9C2, (C22×C10).8C23, D10.12D4⋊12C2, C23.D5⋊14C22, C2.27(D4⋊6D10), D10⋊C4⋊10C22, C23.18D10⋊4C2, (C2×Dic5).60C23, C22.154(C23×D5), C23.107(C22×D5), (C22×Dic5)⋊12C22, (C22×D5).192C23, (C2×D4×D5)⋊6C2, C2.28(C2×D4×D5), (C2×C4×D5)⋊6C22, (D5×C22⋊C4)⋊2C2, (C5×C22≀C2)⋊4C2, (C2×C10).53(C2×D4), (C22×C5⋊D4)⋊7C2, (C2×C5⋊D4)⋊38C22, (C5×C22⋊C4)⋊4C22, (C2×C4).27(C22×D5), SmallGroup(320,1261)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C24⋊3D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C24⋊3D10
Lower central
C5 — C2×C10 — C24⋊3D10

Subgroups: 1502 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×2], C22 [×34], C5, C2×C4, C2×C4 [×2], C2×C4 [×11], D4 [×20], C23 [×2], C23 [×2], C23 [×17], D5 [×5], C10, C10 [×2], C10 [×5], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×17], C24, C24 [×2], Dic5 [×5], C20 [×3], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×15], C2×C22⋊C4, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×14], C2×C20, C2×C20 [×2], C5×D4 [×4], C22×D5 [×3], C22×D5 [×4], C22×D5 [×6], C22×C10 [×2], C22×C10 [×2], C22×C10 [×4], C23⋊3D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5, C23.D5 [×4], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, D4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C5⋊D4 [×6], C2×C5⋊D4 [×4], D4×C10, D4×C10 [×2], C23×D5 [×2], C23×C10, D5×C22⋊C4, D10.12D4 [×2], D10⋊D4 [×2], C22.D20, C23.18D10, C23⋊D10 [×2], C20⋊2D4 [×2], C24⋊2D5, C5×C22≀C2, C2×D4×D5, C22×C5⋊D4, C24⋊3D10

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], C22×D5 [×7], C23⋊3D4, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10 [×2], C24⋊3D10

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

400000000
040000000
004000000
000400000
000040000
000004000
00000010
00000001
,
00100000
00010000
10000000
01000000
0000233500
000061800
000000186
0000003523
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
400000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
91311250000
281316250000
301632280000
251613280000
0000004035
000000635
0000403500
000063500
,
901100000
283216300000
3003200000
25111390000
000000400
00000061
000040000
00006100

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,23,6,0,0,0,0,0,0,35,18,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[9,28,30,25,0,0,0,0,13,13,16,16,0,0,0,0,11,16,32,13,0,0,0,0,25,25,28,28,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0],[9,28,30,25,0,0,0,0,0,32,0,11,0,0,0,0,11,16,32,13,0,0,0,0,0,30,0,9,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,0,1,0,0] >;
 

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D···4H5A5B10A···10F10G···10R10S10T20A···20F
order122222222222224444···45510···1010···10101020···20
size111122444101010102044420···20222···24···4888···8

50 irreducible representations

dim11111111111122222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D5D10D10D102+ (1+4)D4×D5D4⋊6D10
kernelC24⋊3D10D5×C22⋊C4D10.12D4D10⋊D4C22.D20C23.18D10C23⋊D10C20⋊2D4C24⋊2D5C5×C22≀C2C2×D4×D5C22×C5⋊D4C22×D5C22≀C2C22⋊C4C2×D4C24C10C22C2
# reps11221122111142662248

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_3D_{10}
 
% in TeX
 
G:=Group("C2^4:3D10");
 
// GroupNames label
 
G:=SmallGroup(320,1261);
 
// by ID
 
G=gap.SmallGroup(320,1261);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,297,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=f^2=1,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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