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G = C5×C22.45C24order 320 = 26·5

Direct product of C5 and C22.45C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22.45C24, C10.1632+ 1+4, (C4×D4)⋊19C10, (D4×C20)⋊48C2, (C4×C20)⋊44C22, C4210(C2×C10), C22⋊Q815C10, C422C24C10, C22≀C2.2C10, C4.4D412C10, C24.20(C2×C10), (C22×C20)⋊6C22, (Q8×C10)⋊30C22, C42⋊C214C10, (C2×C10).371C24, (C2×C20).678C23, (D4×C10).323C22, C22.D410C10, C23.18(C22×C10), (C23×C10).20C22, C22.45(C23×C10), C2.15(C5×2+ 1+4), (C22×C10).266C23, C4⋊C417(C2×C10), (C2×Q8)⋊5(C2×C10), (C5×C4⋊C4)⋊74C22, C22⋊C46(C2×C10), (C22×C4)⋊4(C2×C10), C2.24(C10×C4○D4), (C5×C22⋊Q8)⋊42C2, C22.9(C5×C4○D4), (C10×C22⋊C4)⋊35C2, (C2×C22⋊C4)⋊15C10, (C5×C22≀C2).4C2, (C2×D4).69(C2×C10), C10.243(C2×C4○D4), (C5×C4.4D4)⋊32C2, (C5×C422C2)⋊15C2, (C5×C42⋊C2)⋊35C2, (C5×C22⋊C4)⋊41C22, (C2×C4).61(C22×C10), (C2×C10).118(C4○D4), (C5×C22.D4)⋊29C2, SmallGroup(320,1553)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C22.45C24
C1C2C22C2×C10C2×C20C5×C4⋊C4C5×C22.D4 — C5×C22.45C24
C1C22 — C5×C22.45C24
C1C2×C10 — C5×C22.45C24

Generators and relations for C5×C22.45C24
 G = < a,b,c,d,e,f,g | a5=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 394 in 248 conjugacy classes, 150 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C5, C2×C4, C2×C4 [×10], C2×C4 [×7], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C10, C10 [×2], C10 [×6], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, C20 [×11], C2×C10, C2×C10 [×4], C2×C10 [×14], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C422C2 [×2], C2×C20, C2×C20 [×10], C2×C20 [×7], C5×D4 [×5], C5×Q8, C22×C10 [×2], C22×C10 [×2], C22×C10 [×5], C22.45C24, C4×C20, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C22⋊C4 [×12], C5×C4⋊C4 [×8], C22×C20, C22×C20 [×4], D4×C10, D4×C10 [×2], Q8×C10, C23×C10, C10×C22⋊C4 [×2], C5×C42⋊C2 [×2], D4×C20 [×2], C5×C22≀C2, C5×C22⋊Q8 [×2], C5×C22.D4, C5×C22.D4 [×2], C5×C4.4D4, C5×C422C2 [×2], C5×C22.45C24
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×4], C24, C2×C10 [×35], C2×C4○D4 [×2], 2+ 1+4, C22×C10 [×15], C22.45C24, C5×C4○D4 [×4], C23×C10, C10×C4○D4 [×2], C5×2+ 1+4, C5×C22.45C24

Smallest permutation representation of C5×C22.45C24
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 35)(2 31)(3 32)(4 33)(5 34)(6 80)(7 76)(8 77)(9 78)(10 79)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 54 80 40)(7 55 76 36)(8 51 77 37)(9 52 78 38)(10 53 79 39)(11 42 17 48)(12 43 18 49)(13 44 19 50)(14 45 20 46)(15 41 16 47)(21 67 27 61)(22 68 28 62)(23 69 29 63)(24 70 30 64)(25 66 26 65)
(1 41 21 36)(2 42 22 37)(3 43 23 38)(4 44 24 39)(5 45 25 40)(6 60 20 65)(7 56 16 61)(8 57 17 62)(9 58 18 63)(10 59 19 64)(11 68 77 71)(12 69 78 72)(13 70 79 73)(14 66 80 74)(15 67 76 75)(26 54 34 46)(27 55 35 47)(28 51 31 48)(29 52 32 49)(30 53 33 50)
(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 14)(7 15)(8 11)(9 12)(10 13)(16 76)(17 77)(18 78)(19 79)(20 80)(26 34)(27 35)(28 31)(29 32)(30 33)(36 47)(37 48)(38 49)(39 50)(40 46)(41 55)(42 51)(43 52)(44 53)(45 54)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)

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

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

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

125 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4O5A5B5C5D10A···10L10M···10AB10AC···10AJ20A···20AF20AG···20BH
order12222222224···44···4555510···1010···1010···1020···2020···20
size11112222442···24···411111···12···24···42···24···4

125 irreducible representations

dim1111111111111111112244
type++++++++++
imageC1C2C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10C10C4○D4C5×C4○D42+ 1+4C5×2+ 1+4
kernelC5×C22.45C24C10×C22⋊C4C5×C42⋊C2D4×C20C5×C22≀C2C5×C22⋊Q8C5×C22.D4C5×C4.4D4C5×C422C2C22.45C24C2×C22⋊C4C42⋊C2C4×D4C22≀C2C22⋊Q8C22.D4C4.4D4C422C2C2×C10C22C10C2
# reps122212312488848124883214

Matrix representation of C5×C22.45C24 in GL4(𝔽41) generated by

37000
03700
00160
00016
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
53900
123600
00320
00409
,
9000
0900
0092
00132
,
1000
54000
0010
0001
,
40000
04000
0010
003240
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,39,36,0,0,0,0,32,40,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,9,1,0,0,2,32],[1,5,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,32,0,0,0,40] >;

C5×C22.45C24 in GAP, Magma, Sage, TeX

C_5\times C_2^2._{45}C_2^4
% in TeX

G:=Group("C5xC2^2.45C2^4");
// GroupNames label

G:=SmallGroup(320,1553);
// by ID

G=gap.SmallGroup(320,1553);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1128,3446,1242]);
// Polycyclic

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

׿
×
𝔽