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G = C24.38D10order 320 = 26·5

38th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.38D10, C10.872+ 1+4, (D4×C10)⋊24C4, D46(C2×Dic5), (C2×D4)⋊11Dic5, (D4×Dic5)⋊37C2, C233(C2×Dic5), (C2×D4).251D10, C10.66(C23×C4), C4⋊Dic576C22, (C22×D4).12D5, C2.5(D46D10), C2.7(C23×Dic5), C20.153(C22×C4), (C2×C10).293C24, (C2×C20).541C23, C55(C22.11C24), (C4×Dic5)⋊40C22, (C22×C4).270D10, C23.D559C22, C4.17(C22×Dic5), C22.45(C23×D5), (D4×C10).270C22, (C23×C10).75C22, C23.204(C22×D5), C23.21D1032C2, C22.1(C22×Dic5), (C22×C10).229C23, (C22×C20).274C22, (C2×Dic5).293C23, (C22×Dic5)⋊31C22, (D4×C2×C10).9C2, (C2×C20)⋊28(C2×C4), (C5×D4)⋊30(C2×C4), (C2×C4)⋊4(C2×Dic5), (C22×C10)⋊19(C2×C4), (C2×C23.D5)⋊26C2, (C2×C4).624(C22×D5), (C2×C10).128(C22×C4), SmallGroup(320,1470)

Series: Derived Chief Lower central Upper central

C1C10 — C24.38D10
C1C5C10C2×C10C2×Dic5C22×Dic5D4×Dic5 — C24.38D10
C5C10 — C24.38D10
C1C22C22×D4

Generators and relations for C24.38D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 910 in 338 conjugacy classes, 191 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×18], C5, C2×C4 [×6], C2×C4 [×16], D4 [×16], C23, C23 [×12], C23 [×4], C10, C10 [×2], C10 [×10], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×8], C2×D4 [×12], C24 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×18], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4 [×8], C22×D4, C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×6], C5×D4 [×16], C22×C10, C22×C10 [×12], C22×C10 [×4], C22.11C24, C4×Dic5 [×4], C4⋊Dic5 [×4], C23.D5 [×12], C22×Dic5 [×8], C22×C20, D4×C10 [×12], C23×C10 [×2], C23.21D10 [×2], D4×Dic5 [×8], C2×C23.D5 [×4], D4×C2×C10, C24.38D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2+ 1+4 [×2], C2×Dic5 [×28], C22×D5 [×7], C22.11C24, C22×Dic5 [×14], C23×D5, D46D10 [×2], C23×Dic5, C24.38D10

Smallest permutation representation of C24.38D10
On 80 points
Generators in S80
(1 43)(2 49)(3 45)(4 41)(5 47)(6 46)(7 42)(8 48)(9 44)(10 50)(11 56)(12 52)(13 58)(14 54)(15 60)(16 64)(17 70)(18 66)(19 62)(20 68)(21 65)(22 61)(23 67)(24 63)(25 69)(26 71)(27 77)(28 73)(29 79)(30 75)(31 57)(32 53)(33 59)(34 55)(35 51)(36 74)(37 80)(38 76)(39 72)(40 78)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 35)(7 31)(8 32)(9 33)(10 34)(16 26)(17 27)(18 28)(19 29)(20 30)(21 39)(22 40)(23 36)(24 37)(25 38)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 78)(62 79)(63 80)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 35)(7 31)(8 32)(9 33)(10 34)(16 38)(17 39)(18 40)(19 36)(20 37)(21 27)(22 28)(23 29)(24 30)(25 26)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 35)(12 31)(13 32)(14 33)(15 34)(16 25)(17 21)(18 22)(19 23)(20 24)(26 38)(27 39)(28 40)(29 36)(30 37)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 36 13 19)(2 40 14 18)(3 39 15 17)(4 38 11 16)(5 37 12 20)(6 26 35 25)(7 30 31 24)(8 29 32 23)(9 28 33 22)(10 27 34 21)(41 71 56 69)(42 80 57 68)(43 79 58 67)(44 78 59 66)(45 77 60 65)(46 76 51 64)(47 75 52 63)(48 74 53 62)(49 73 54 61)(50 72 55 70)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D···2M4A4B4C4D4E···4T5A5B10A···10N10O···10AD20A···20H
order12222···244444···45510···1010···1020···20
size11112···2222210···10222···24···44···4

74 irreducible representations

dim1111112222244
type+++++++-+++
imageC1C2C2C2C2C4D5D10Dic5D10D102+ 1+4D46D10
kernelC24.38D10C23.21D10D4×Dic5C2×C23.D5D4×C2×C10D4×C10C22×D4C22×C4C2×D4C2×D4C24C10C2
# reps128411622168428

Matrix representation of C24.38D10 in GL6(𝔽41)

100000
010000
000100
001000
000001
000010
,
4000000
0400000
001000
000100
001714400
001417040
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
2300000
0250000
001000
0004000
0002710
00140040
,
0210000
3900000
002523260
001816015
002001623
000211825

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,17,14,0,0,0,1,14,17,0,0,0,0,40,0,0,0,0,0,0,40],[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],[23,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,14,0,0,0,40,27,0,0,0,0,0,1,0,0,0,0,0,0,40],[0,39,0,0,0,0,21,0,0,0,0,0,0,0,25,18,20,0,0,0,23,16,0,21,0,0,26,0,16,18,0,0,0,15,23,25] >;

C24.38D10 in GAP, Magma, Sage, TeX

C_2^4._{38}D_{10}
% in TeX

G:=Group("C2^4.38D10");
// GroupNames label

G:=SmallGroup(320,1470);
// by ID

G=gap.SmallGroup(320,1470);
# by ID

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

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

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