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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22.45C24
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C4⋊C4 — C5×C22.D4 — C5×C22.45C24
 Lower central C1 — C22 — C5×C22.45C24
 Upper central C1 — C2×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, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C422C2, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C22.45C24, C4×C20, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C23×C10, C10×C22⋊C4, C5×C42⋊C2, D4×C20, C5×C22≀C2, C5×C22⋊Q8, C5×C22.D4, C5×C22.D4, C5×C4.4D4, C5×C422C2, C5×C22.45C24
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, 2+ 1+4, C22×C10, C22.45C24, C5×C4○D4, C23×C10, C10×C4○D4, 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 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(11 46 76 36)(12 47 77 37)(13 48 78 38)(14 49 79 39)(15 50 80 40)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)
(1 41 21 36)(2 42 22 37)(3 43 23 38)(4 44 24 39)(5 45 25 40)(6 71 76 66)(7 72 77 67)(8 73 78 68)(9 74 79 69)(10 75 80 70)(11 56 16 61)(12 57 17 62)(13 58 18 63)(14 59 19 64)(15 60 20 65)(26 51 31 46)(27 52 32 47)(28 53 33 48)(29 54 34 49)(30 55 35 50)
(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(26 31)(27 32)(28 33)(29 34)(30 35)(36 51)(37 52)(38 53)(39 54)(40 55)(41 46)(42 47)(43 48)(44 49)(45 50)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)

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

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

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

125 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I ··· 4O 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AB 10AC ··· 10AJ 20A ··· 20AF 20AG ··· 20BH order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 4 4 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

125 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 C10 C4○D4 C5×C4○D4 2+ 1+4 C5×2+ 1+4 kernel C5×C22.45C24 C10×C22⋊C4 C5×C42⋊C2 D4×C20 C5×C22≀C2 C5×C22⋊Q8 C5×C22.D4 C5×C4.4D4 C5×C42⋊2C2 C22.45C24 C2×C22⋊C4 C42⋊C2 C4×D4 C22≀C2 C22⋊Q8 C22.D4 C4.4D4 C42⋊2C2 C2×C10 C22 C10 C2 # reps 1 2 2 2 1 2 3 1 2 4 8 8 8 4 8 12 4 8 8 32 1 4

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

 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 39 0 0 12 36 0 0 0 0 32 0 0 0 40 9
,
 9 0 0 0 0 9 0 0 0 0 9 2 0 0 1 32
,
 1 0 0 0 5 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 32 40
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

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