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## G = C24.4Dic5order 320 = 26·5

### 2nd non-split extension by C24 of Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.4Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C20.55D4 — C24.4Dic5
 Lower central C5 — C2×C10 — C24.4Dic5
 Upper central C1 — C2×C4 — C23×C4

Generators and relations for C24.4Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=e5, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 398 in 190 conjugacy classes, 79 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×2], C22, C22 [×6], C22 [×14], C5, C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×10], C23, C23 [×2], C23 [×6], C10, C10 [×2], C10 [×6], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C24, C20 [×4], C20 [×2], C2×C10, C2×C10 [×6], C2×C10 [×14], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C52C8 [×4], C2×C20 [×2], C2×C20 [×6], C2×C20 [×10], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.4C4, C2×C52C8 [×4], C4.Dic5 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C20.55D4 [×4], C2×C4.Dic5 [×2], C23×C20, C24.4Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C2×M4(2) [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C24.4C4, C4.Dic5 [×4], C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C4.Dic5 [×2], C2×C23.D5, C24.4Dic5

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

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 5A 5B 8A ··· 8H 10A ··· 10AD 20A ··· 20AF order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 2 20 ··· 20 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 D4 D5 M4(2) Dic5 D10 Dic5 C5⋊D4 C4.Dic5 kernel C24.4Dic5 C20.55D4 C2×C4.Dic5 C23×C20 C22×C20 C23×C10 C2×C20 C23×C4 C2×C10 C22×C4 C22×C4 C24 C2×C4 C22 # reps 1 4 2 1 6 2 4 2 8 6 6 2 16 32

Matrix representation of C24.4Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 10 40
,
 40 12 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 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
,
 21 26 0 0 0 39 0 0 0 0 25 0 0 0 10 23
,
 16 15 0 0 30 25 0 0 0 0 7 15 0 0 24 34
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,10,0,0,0,40],[40,0,0,0,12,1,0,0,0,0,40,0,0,0,0,40],[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],[21,0,0,0,26,39,0,0,0,0,25,10,0,0,0,23],[16,30,0,0,15,25,0,0,0,0,7,24,0,0,15,34] >;

C24.4Dic5 in GAP, Magma, Sage, TeX

C_2^4._4{\rm Dic}_5
% in TeX

G:=Group("C2^4.4Dic5");
// GroupNames label

G:=SmallGroup(320,834);
// by ID

G=gap.SmallGroup(320,834);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d,f^2=e^5,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,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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