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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, C22.D20⋊9C2, C10.D4⋊8C22, (C22×C10).8C23, D10.12D4⋊12C2, C2.27(D4⋊6D10), C23.D5⋊14C22, D10⋊C4⋊10C22, C23.18D10⋊4C2, (C2×Dic5).60C23, C23.107(C22×D5), C22.154(C23×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
Upper central
C1 — C22 — C22≀C2

Generators and relations for C24⋊3D10
 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 >

Subgroups: 1502 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C23⋊3D4, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C2×C5⋊D4, D4×C10, D4×C10, C23×D5, C23×C10, D5×C22⋊C4, D10.12D4, D10⋊D4, C22.D20, C23.18D10, C23⋊D10, C20⋊2D4, C24⋊2D5, C5×C22≀C2, C2×D4×D5, C22×C5⋊D4, C24⋊3D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C23⋊3D4, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, C24⋊3D10

Smallest permutation representation of C24⋊3D10
►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 75)(12 49)(13 77)(14 41)(15 79)(16 43)(17 71)(18 45)(19 73)(20 47)(21 64)(22 32)(23 66)(24 34)(25 68)(26 36)(27 70)(28 38)(29 62)(30 40)(42 51)(44 53)(46 55)(48 57)(50 59)(52 80)(54 72)(56 74)(58 76)(60 78)

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

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

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

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

Matrix representation of C24⋊3D10 ►in 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] >;
 

C24⋊3D10 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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