Copied to
clipboard

G = C23⋊C45D5order 320 = 26·5

The semidirect product of C23⋊C4 and D5 acting through Inn(C23⋊C4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊C45D5, C23.5(C4×D5), (C2×Dic10)⋊6C4, C23⋊Dic53C2, C22.23(D4×D5), (C2×D4).119D10, C22⋊C4.43D10, (C22×Dic5)⋊4C4, (D4×C10).7C22, C23.1(C22×D5), (C2×Dic5).188D4, C23.1D103C2, (C22×C10).1C23, C23.D5.1C22, C54(C23.C23), C23.11D1023C2, Dic5.20(C22⋊C4), (C22×Dic5).25C22, (C2×C4×D5)⋊1C4, (C2×C5⋊D4)⋊1C4, (C2×C4).5(C4×D5), (C5×C23⋊C4)⋊3C2, C22.12(C2×C4×D5), (C2×C20).19(C2×C4), (C2×C10).16(C2×D4), C2.11(D5×C22⋊C4), (C2×D42D5).1C2, C10.51(C2×C22⋊C4), (C22×C10).5(C2×C4), (C2×Dic5).1(C2×C4), (C22×D5).1(C2×C4), (C2×C5⋊D4).1C22, (C2×C10).107(C22×C4), (C5×C22⋊C4).82C22, SmallGroup(320,367)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C23⋊C45D5
C1C5C10C2×C10C22×C10C22×Dic5C2×D42D5 — C23⋊C45D5
C5C10C2×C10 — C23⋊C45D5
C1C2C23C23⋊C4

Generators and relations for C23⋊C45D5
 G = < a,b,c,d,e,f | a2=b2=c2=d4=e5=f2=1, ab=ba, faf=ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=bcd, fef=e-1 >

Subgroups: 622 in 158 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2 [×5], C4 [×10], C22, C22 [×2], C22 [×5], C5, C2×C4, C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10, C10 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], Dic5 [×3], C20 [×3], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×3], C23⋊C4, C23⋊C4 [×3], C42⋊C2 [×2], C2×C4○D4, Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C22×D5, C22×C10 [×2], C23.C23, C4×Dic5 [×2], C10.D4 [×2], C23.D5 [×2], C5×C22⋊C4 [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C23.1D10 [×2], C23⋊Dic5, C5×C23⋊C4, C23.11D10 [×2], C2×D42D5, C23⋊C45D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], C22×D5, C23.C23, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, C23⋊C45D5

Smallest permutation representation of C23⋊C45D5
On 80 points
Generators in S80
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 66 56 71)(42 67 57 72)(43 68 58 73)(44 69 59 74)(45 70 60 75)(46 61 51 76)(47 62 52 77)(48 63 53 78)(49 64 54 79)(50 65 55 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 33)(2 32)(3 31)(4 35)(5 34)(6 36)(7 40)(8 39)(9 38)(10 37)(11 26)(12 30)(13 29)(14 28)(15 27)(16 21)(17 25)(18 24)(19 23)(20 22)(41 76)(42 80)(43 79)(44 78)(45 77)(46 71)(47 75)(48 74)(49 73)(50 72)(51 61)(52 65)(53 64)(54 63)(55 62)(56 66)(57 70)(58 69)(59 68)(60 67)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A···4E4F4G4H4I4J4K···4O5A5B10A10B10C···10H10I10J20A···20J
order12222224···4444444···455101010···10101020···20
size112224204···45510101020···2022224···4888···8

44 irreducible representations

dim1111111111222222448
type+++++++++++-
imageC1C2C2C2C2C2C4C4C4C4D4D5D10D10C4×D5C4×D5C23.C23D4×D5C23⋊C45D5
kernelC23⋊C45D5C23.1D10C23⋊Dic5C5×C23⋊C4C23.11D10C2×D42D5C2×Dic10C2×C4×D5C22×Dic5C2×C5⋊D4C2×Dic5C23⋊C4C22⋊C4C2×D4C2×C4C23C5C22C1
# reps1211212222424244242

Matrix representation of C23⋊C45D5 in GL8(𝔽41)

201133400000
10188330000
8023300000
35831210000
00000090
000000032
000032000
00000900
,
400000000
040000000
004000000
000400000
00000100
00001000
000000040
000000400
,
10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
302928240000
0113280000
151840120000
15150110000
00000010
00000001
00000100
00001000
,
61000000
400000000
230010000
0184060000
00001000
00000100
00000010
00000001
,
11617130000
0113280000
37142910000
37143000000
00000001
000000400
000004000
00001000

G:=sub<GL(8,GF(41))| [20,10,8,35,0,0,0,0,11,18,0,8,0,0,0,0,33,8,23,31,0,0,0,0,40,33,30,21,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[30,0,15,15,0,0,0,0,29,1,18,15,0,0,0,0,28,13,40,0,0,0,0,0,24,28,12,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[6,40,23,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,0,37,37,0,0,0,0,6,1,14,14,0,0,0,0,17,13,29,30,0,0,0,0,13,28,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0] >;

C23⋊C45D5 in GAP, Magma, Sage, TeX

C_2^3\rtimes C_4\rtimes_5D_5
% in TeX

G:=Group("C2^3:C4:5D5");
// GroupNames label

G:=SmallGroup(320,367);
// by ID

G=gap.SmallGroup(320,367);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,58,570,438,12550]);
// Polycyclic

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

׿
×
𝔽