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G = C24.Dic5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.1Dic5, C54(C23⋊C8), C23⋊(C52C8), (C22×C10)⋊2C8, (C2×C20).227D4, (C23×C10).3C4, (C22×C20).1C4, (C22×C4).3D10, C20.55D421C2, (C22×C4).1Dic5, C10.37(C23⋊C4), C10.28(C22⋊C8), C2.1(C20.D4), (C2×C10).40M4(2), C2.1(C23⋊Dic5), C23.20(C2×Dic5), C10.13(C4.D4), C2.3(C20.55D4), C22.4(C4.Dic5), (C22×C20).324C22, C22.23(C23.D5), (C2×C10).57(C2×C8), C22.2(C2×C52C8), (C2×C22⋊C4).1D5, (C2×C4).159(C5⋊D4), (C10×C22⋊C4).21C2, (C22×C10).191(C2×C4), (C2×C10).149(C22⋊C4), SmallGroup(320,83)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.Dic5
C1C5C10C2×C10C2×C20C22×C20C20.55D4 — C24.Dic5
C5C10C2×C10 — C24.Dic5
C1C22C22×C4C2×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 [×3], C2 [×4], C4 [×3], C22 [×3], C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×3], C23, C23 [×2], C23 [×4], C10 [×3], C10 [×4], C22⋊C4 [×2], C2×C8 [×2], C22×C4 [×2], C24, C20 [×3], C2×C10 [×3], C2×C10 [×10], C22⋊C8 [×2], C2×C22⋊C4, C52C8 [×2], C2×C20 [×2], C2×C20 [×3], C22×C10, C22×C10 [×2], C22×C10 [×4], C23⋊C8, C2×C52C8 [×2], C5×C22⋊C4 [×2], C22×C20 [×2], C23×C10, C20.55D4 [×2], C10×C22⋊C4, C24.Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C22⋊C8, C23⋊C4, C4.D4, C52C8 [×2], C2×Dic5, C5⋊D4 [×2], 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 62)(3 13)(4 64)(5 15)(6 66)(7 17)(8 68)(9 19)(10 70)(12 72)(14 74)(16 76)(18 78)(20 80)(21 47)(22 32)(23 49)(24 34)(25 51)(26 36)(27 53)(28 38)(29 55)(30 40)(31 57)(33 59)(35 41)(37 43)(39 45)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 45 6 50 11 55 16 60)(2 54 7 59 12 44 17 49)(3 43 8 48 13 53 18 58)(4 52 9 57 14 42 19 47)(5 41 10 46 15 51 20 56)(21 64 26 69 31 74 36 79)(22 73 27 78 32 63 37 68)(23 62 28 67 33 72 38 77)(24 71 29 76 34 61 39 66)(25 80 30 65 35 70 40 75)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F5A5B8A···8H10A···10N10O···10V20A···20P
order12222222444444558···810···1010···1020···20
size111122442222442220···202···24···44···4

62 irreducible representations

dim1111112222222224444
type+++++-+-++
imageC1C2C2C4C4C8D4D5M4(2)Dic5D10Dic5C5⋊D4C52C8C4.Dic5C23⋊C4C4.D4C20.D4C23⋊Dic5
kernelC24.Dic5C20.55D4C10×C22⋊C4C22×C20C23×C10C22×C10C2×C20C2×C22⋊C4C2×C10C22×C4C22×C4C24C2×C4C23C22C10C10C2C2
# reps1212282222228881144

Matrix representation of C24.Dic5 in GL6(𝔽41)

4000000
010000
001000
00254000
00253910
004162540
,
4000000
0400000
001000
000100
00162400
002914040
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
500000
080000
0040500
000100
00307405
00121101
,
010000
900000
00103290
00131909
002025319
0031142822

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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