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## G = C5×C23.C23order 320 = 26·5

### Direct product of C5 and C23.C23

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23.C23
 Chief series C1 — C2 — C22 — C23 — C22×C10 — C5×C22⋊C4 — C5×C23⋊C4 — C5×C23.C23
 Lower central C1 — C2 — C22 — C5×C23.C23
 Upper central C1 — C20 — C22×C20 — C5×C23.C23

Generators and relations for C5×C23.C23
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=1, e2=c, g2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 274 in 158 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C23⋊C4, C42⋊C2, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C23.C23, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C23⋊C4, C5×C42⋊C2, C10×C4○D4, C5×C23.C23
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C23.C23, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C23.C23

Smallest permutation representation of C5×C23.C23
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 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 28)(22 29)(23 30)(24 26)(25 27)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)
(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 70)(57 66)(58 67)(59 68)(60 69)(61 73)(62 74)(63 75)(64 71)(65 72)
(6 16 76 11)(7 17 77 12)(8 18 78 13)(9 19 79 14)(10 20 80 15)(21 33)(22 34)(23 35)(24 31)(25 32)(41 53)(42 54)(43 55)(44 51)(45 52)(56 75 70 63)(57 71 66 64)(58 72 67 65)(59 73 68 61)(60 74 69 62)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 50)(7 46)(8 47)(9 48)(10 49)(11 55)(12 51)(13 52)(14 53)(15 54)(16 43)(17 44)(18 45)(19 41)(20 42)(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 50 30 36)(2 46 26 37)(3 47 27 38)(4 48 28 39)(5 49 29 40)(6 70 76 56)(7 66 77 57)(8 67 78 58)(9 68 79 59)(10 69 80 60)(11 75 16 63)(12 71 17 64)(13 72 18 65)(14 73 19 61)(15 74 20 62)(21 53 33 41)(22 54 34 42)(23 55 35 43)(24 51 31 44)(25 52 32 45)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F ··· 4O 5A 5B 5C 5D 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10X 20A ··· 20H 20I ··· 20T 20U ··· 20BH order 1 2 2 2 2 2 2 4 4 4 4 4 4 ··· 4 5 5 5 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 2 2 2 4 4 1 1 2 2 2 4 ··· 4 1 1 1 1 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C20 C20 C20 D4 C5×D4 C23.C23 C5×C23.C23 kernel C5×C23.C23 C5×C23⋊C4 C5×C42⋊C2 C10×C4○D4 C22×C20 D4×C10 Q8×C10 C23.C23 C23⋊C4 C42⋊C2 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C2×C20 C2×C4 C5 C1 # reps 1 4 2 1 4 2 2 4 16 8 4 16 8 8 4 16 2 8

Matrix representation of C5×C23.C23 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 39 40 33 33 0 0 0 1 0 0 1 0
,
 1 0 8 8 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 1 0 8 0 40 0 0 0 0 0 40 0 0 1 0
,
 8 0 11 11 0 0 0 1 39 40 33 33 0 1 0 0
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,39,0,0,0,40,0,0,0,33,0,1,0,33,1,0],[1,0,0,0,0,1,0,0,8,0,40,0,8,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,1,40,0,0,0,0,0,1,8,0,40,0],[8,0,39,0,0,0,40,1,11,0,33,0,11,1,33,0],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32] >;

C5×C23.C23 in GAP, Magma, Sage, TeX

C_5\times C_2^3.C_2^3
% in TeX

G:=Group("C5xC2^3.C2^3");
// GroupNames label

G:=SmallGroup(320,911);
// by ID

G=gap.SmallGroup(320,911);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,856,7004,5052]);
// Polycyclic

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

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