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## G = C5⋊(C23⋊C8)  order 320 = 26·5

### The semidirect product of C5 and C23⋊C8 acting via C23⋊C8/C22×C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C5⋊(C23⋊C8)
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.2F5 — C5⋊(C23⋊C8)
 Lower central C5 — C10 — C2×C10 — C5⋊(C23⋊C8)
 Upper central C1 — C22 — C23 — C22×C4

Generators and relations for C5⋊(C23⋊C8)
G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, eae-1=a3, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >

Subgroups: 546 in 98 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C2×C8, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22⋊C8, C2×C22⋊C4, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C23⋊C8, D10⋊C4, C2×C5⋊C8, C22×Dic5, C22×C20, C23×D5, C23.2F5, C2×D10⋊C4, C5⋊(C23⋊C8)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.D4, C2×F5, C23⋊C8, D5⋊C8, C4.F5, C22⋊F5, D10.D4, D10⋊C8, C23.F5, C5⋊(C23⋊C8)

Smallest permutation representation of C5⋊(C23⋊C8)
On 80 points
Generators in S80
(1 75 39 44 68)(2 45 76 69 40)(3 70 46 33 77)(4 34 71 78 47)(5 79 35 48 72)(6 41 80 65 36)(7 66 42 37 73)(8 38 67 74 43)(9 51 26 62 22)(10 63 52 23 27)(11 24 64 28 53)(12 29 17 54 57)(13 55 30 58 18)(14 59 56 19 31)(15 20 60 32 49)(16 25 21 50 61)
(1 20)(2 6)(4 19)(5 24)(8 23)(9 62)(10 74)(11 79)(12 61)(13 58)(14 78)(15 75)(16 57)(17 21)(25 54)(26 51)(27 43)(28 48)(29 50)(30 55)(31 47)(32 44)(33 46)(34 56)(35 53)(36 45)(37 42)(38 52)(39 49)(40 41)(59 71)(60 68)(63 67)(64 72)(65 76)(66 73)(69 80)(70 77)
(1 5)(2 17)(3 7)(4 19)(6 21)(8 23)(9 13)(10 67)(11 15)(12 69)(14 71)(16 65)(18 22)(20 24)(25 36)(26 30)(27 38)(28 32)(29 40)(31 34)(33 37)(35 39)(41 50)(42 46)(43 52)(44 48)(45 54)(47 56)(49 53)(51 55)(57 76)(58 62)(59 78)(60 64)(61 80)(63 74)(66 70)(68 72)(73 77)(75 79)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)
(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)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5 8A ··· 8H 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 20 20 4 4 10 10 10 10 4 20 ··· 20 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C4 C4 C8 D4 M4(2) F5 C23⋊C4 C4.D4 C2×F5 D5⋊C8 C4.F5 C22⋊F5 D10.D4 C23.F5 kernel C5⋊(C23⋊C8) C23.2F5 C2×D10⋊C4 C22×C20 C23×D5 C22×D5 C2×Dic5 C2×C10 C22×C4 C10 C10 C23 C22 C22 C22 C2 C2 # reps 1 2 1 2 2 8 2 2 1 1 1 1 2 2 2 4 4

Matrix representation of C5⋊(C23⋊C8) in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 40 6 0 0 0 0 35 35
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 1 0 0 0 0 0 6 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 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 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 18 35 0 0 0 0 20 23 0 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,20,0,0,0,0,35,23,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5⋊(C23⋊C8) in GAP, Magma, Sage, TeX

C_5\rtimes (C_2^3\rtimes C_8)
% in TeX

G:=Group("C5:(C2^3:C8)");
// GroupNames label

G:=SmallGroup(320,253);
// by ID

G=gap.SmallGroup(320,253);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,268,1123,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^3,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations

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