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## G = C24.Dic5order 320 = 26·5

### 1st non-split extension by C24 of Dic5 acting via Dic5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C20.55D4 — C24.Dic5
 Lower central C5 — C10 — C2×C10 — C24.Dic5
 Upper central C1 — C22 — C22×C4 — C2×C22⋊C4

Generators and relations for C24.Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=e5, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 278 in 98 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, C23, C10, C10, C22⋊C4, C2×C8, C22×C4, C24, C20, C2×C10, C2×C10, C22⋊C8, C2×C22⋊C4, C52C8, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23⋊C8, C2×C52C8, C5×C22⋊C4, C22×C20, C23×C10, C20.55D4, C10×C22⋊C4, C24.Dic5
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), Dic5, D10, C22⋊C8, C23⋊C4, C4.D4, C52C8, C2×Dic5, C5⋊D4, C23⋊C8, C2×C52C8, C4.Dic5, C23.D5, C20.55D4, C20.D4, C23⋊Dic5, C24.Dic5

Smallest permutation representation of C24.Dic5
On 80 points
Generators in S80
(1 11)(2 70)(3 13)(4 72)(5 15)(6 74)(7 17)(8 76)(9 19)(10 78)(12 80)(14 62)(16 64)(18 66)(20 68)(21 31)(22 49)(23 33)(24 51)(25 35)(26 53)(27 37)(28 55)(29 39)(30 57)(32 59)(34 41)(36 43)(38 45)(40 47)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(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 28 6 33 11 38 16 23)(2 37 7 22 12 27 17 32)(3 26 8 31 13 36 18 21)(4 35 9 40 14 25 19 30)(5 24 10 29 15 34 20 39)(41 68 46 73 51 78 56 63)(42 77 47 62 52 67 57 72)(43 66 48 71 53 76 58 61)(44 75 49 80 54 65 59 70)(45 64 50 69 55 74 60 79)

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + - + + image C1 C2 C2 C4 C4 C8 D4 D5 M4(2) Dic5 D10 Dic5 C5⋊D4 C5⋊2C8 C4.Dic5 C23⋊C4 C4.D4 C20.D4 C23⋊Dic5 kernel C24.Dic5 C20.55D4 C10×C22⋊C4 C22×C20 C23×C10 C22×C10 C2×C20 C2×C22⋊C4 C2×C10 C22×C4 C22×C4 C24 C2×C4 C23 C22 C10 C10 C2 C2 # reps 1 2 1 2 2 8 2 2 2 2 2 2 8 8 8 1 1 4 4

Matrix representation of C24.Dic5 in GL6(𝔽41)

 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 25 40 0 0 0 0 25 39 1 0 0 0 4 16 25 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 16 2 40 0 0 0 29 14 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 40 5 0 0 0 0 0 1 0 0 0 0 30 7 40 5 0 0 12 11 0 1
,
 0 1 0 0 0 0 9 0 0 0 0 0 0 0 10 32 9 0 0 0 13 19 0 9 0 0 20 25 31 9 0 0 31 14 28 22

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,25,25,4,0,0,0,40,39,16,0,0,0,0,1,25,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,16,29,0,0,0,1,2,14,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,40,0,30,12,0,0,5,1,7,11,0,0,0,0,40,0,0,0,0,0,5,1],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,10,13,20,31,0,0,32,19,25,14,0,0,9,0,31,28,0,0,0,9,9,22] >;

C24.Dic5 in GAP, Magma, Sage, TeX

C_2^4.{\rm Dic}_5
% in TeX

G:=Group("C2^4.Dic5");
// GroupNames label

G:=SmallGroup(320,83);
// by ID

G=gap.SmallGroup(320,83);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,387,100,1123,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=c,f^2=e^5,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations

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