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G = C20.76C24order 320 = 26·5

23rd non-split extension by C20 of C24 acting via C24/C23=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.76C24, C4○D4.45D10, (D4×C10).26C4, C56(Q8○M4(2)), D4.Dic59C2, (Q8×C10).23C4, C4○D4.2Dic5, (C2×Q8).8Dic5, D4.9(C2×Dic5), C4.75(C23×D5), C10.70(C23×C4), C52C8.34C23, (C2×D4).10Dic5, Q8.10(C2×Dic5), (C2×C20).554C23, C20.157(C22×C4), (C22×C4).280D10, C4.Dic535C22, C4.20(C22×Dic5), C2.11(C23×Dic5), C23.11(C2×Dic5), C22.2(C22×Dic5), (C22×C20).289C22, (C2×C4○D4).9D5, (C5×C4○D4).8C4, (C5×D4).40(C2×C4), (C5×Q8).43(C2×C4), (C2×C20).309(C2×C4), (C2×C52C8)⋊21C22, (C10×C4○D4).10C2, (C2×C4.Dic5)⋊29C2, (C2×C4).31(C2×Dic5), (C5×C4○D4).49C22, (C2×C4).635(C22×D5), (C22×C10).149(C2×C4), (C2×C10).130(C22×C4), SmallGroup(320,1491)

Series: Derived Chief Lower central Upper central

C1C10 — C20.76C24
C1C5C10C20C52C8C2×C52C8D4.Dic5 — C20.76C24
C5C10 — C20.76C24
C1C4C2×C4○D4

Generators and relations for C20.76C24
 G = < a,b,c,d,e | a20=c2=d2=e2=1, b2=a5, bab-1=a9, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=a10b, dcd=a10c, ce=ec, de=ed >

Subgroups: 494 in 258 conjugacy classes, 187 normal (17 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×3], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×7], C2×C8 [×12], M4(2) [×16], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4, C52C8 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], Q8○M4(2), C2×C52C8 [×12], C4.Dic5 [×16], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4.Dic5 [×6], D4.Dic5 [×8], C10×C4○D4, C20.76C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, C2×Dic5 [×28], C22×D5 [×7], Q8○M4(2), C22×Dic5 [×14], C23×D5, C23×Dic5, C20.76C24

Smallest permutation representation of C20.76C24
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 55 6 60 11 45 16 50)(2 44 7 49 12 54 17 59)(3 53 8 58 13 43 18 48)(4 42 9 47 14 52 19 57)(5 51 10 56 15 41 20 46)(21 70 26 75 31 80 36 65)(22 79 27 64 32 69 37 74)(23 68 28 73 33 78 38 63)(24 77 29 62 34 67 39 72)(25 66 30 71 35 76 40 61)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 76)(42 77)(43 78)(44 79)(45 80)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)
(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)

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

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

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

74 conjugacy classes

class 1 2A2B···2H4A4B4C···4I5A5B8A···8P10A···10F10G···10R20A···20H20I···20T
order122···2444···4558···810···1010···1020···2020···20
size112···2112···22210···102···24···42···24···4

74 irreducible representations

dim111111122222244
type++++++---+
imageC1C2C2C2C4C4C4D5D10Dic5Dic5Dic5D10Q8○M4(2)C20.76C24
kernelC20.76C24C2×C4.Dic5D4.Dic5C10×C4○D4D4×C10Q8×C10C5×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C5C1
# reps168162826628828

Matrix representation of C20.76C24 in GL6(𝔽41)

2300000
38250000
009000
000900
000090
000009
,
5240000
16360000
00015123
00017026
0094001
0005024
,
4000000
0400000
00092632
003202440
000010
00003640
,
4000000
0400000
00012927
00101132
0000920
00003732
,
100000
010000
0010035
0001026
0000400
0000040

G:=sub<GL(6,GF(41))| [23,38,0,0,0,0,0,25,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[5,16,0,0,0,0,24,36,0,0,0,0,0,0,0,0,9,0,0,0,15,17,40,5,0,0,1,0,0,0,0,0,23,26,1,24],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,26,24,1,36,0,0,32,40,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,29,11,9,37,0,0,27,32,20,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,35,26,0,40] >;

C20.76C24 in GAP, Magma, Sage, TeX

C_{20}._{76}C_2^4
% in TeX

G:=Group("C20.76C2^4");
// GroupNames label

G:=SmallGroup(320,1491);
// by ID

G=gap.SmallGroup(320,1491);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^20=c^2=d^2=e^2=1,b^2=a^5,b*a*b^-1=a^9,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=a^10*b,d*c*d=a^10*c,c*e=e*c,d*e=e*d>;
// generators/relations

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