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G = C2×C23.D5order 160 = 25·5

Direct product of C2 and C23.D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23.D5, C24.D5, C232Dic5, C23.32D10, (C22×C10)⋊7C4, (C2×C10).44D4, C10.62(C2×D4), C103(C22⋊C4), (C23×C10).2C2, C222(C2×Dic5), C10.41(C22×C4), (C2×C10).60C23, (C22×Dic5)⋊7C2, (C2×Dic5)⋊7C22, C2.9(C22×Dic5), C22.25(C5⋊D4), C22.27(C22×D5), (C22×C10).41C22, C54(C2×C22⋊C4), C2.4(C2×C5⋊D4), (C2×C10)⋊11(C2×C4), SmallGroup(160,173)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C23.D5
C1C5C10C2×C10C2×Dic5C22×Dic5 — C2×C23.D5
C5C10 — C2×C23.D5
C1C23C24

Generators and relations for C2×C23.D5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 296 in 132 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C22, C22 [×10], C22 [×12], C5, C2×C4 [×8], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22×C4 [×2], C24, Dic5 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×Dic5 [×4], C2×Dic5 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.D5 [×4], C22×Dic5 [×2], C23×C10, C2×C23.D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], C2×C23.D5

Smallest permutation representation of C2×C23.D5
On 80 points
Generators in S80
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 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 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(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 53 11 43)(2 52 12 42)(3 51 13 41)(4 55 14 45)(5 54 15 44)(6 58 16 48)(7 57 17 47)(8 56 18 46)(9 60 19 50)(10 59 20 49)(21 73 31 63)(22 72 32 62)(23 71 33 61)(24 75 34 65)(25 74 35 64)(26 78 36 68)(27 77 37 67)(28 76 38 66)(29 80 39 70)(30 79 40 69)

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

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

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

C2×C23.D5 is a maximal subgroup of
C24.D10  C24.2D10  C24.F5  C22⋊C4×Dic5  C24.44D10  C23.42D20  C24.3D10  C24.4D10  C24.46D10  C23⋊Dic10  C24.6D10  C24.7D10  C24.47D10  C24.8D10  C24.9D10  C23.14D20  C24.48D10  C24.12D10  C24.13D10  C24.14D10  C24.16D10  C24.62D10  C24.63D10  C24.64D10  C24.65D10  C24.18D10  C24.19D10  C24.20D10  C24.21D10  C25.2D5  C232Dic10  C2×D5×C22⋊C4  C24.24D10  C24.31D10  C24.32D10  C24.33D10  C24.35D10  C2×C4×C5⋊D4  C2×D4×Dic5  C24.38D10  C248D10  C24.42D10
C2×C23.D5 is a maximal quotient of
C24.4Dic5  C24.63D10  C24.64D10  (D4×C10)⋊18C4  C24.18D10  C24.19D10  (Q8×C10)⋊16C4  (Q8×C10)⋊17C4  C4○D4⋊Dic5  C20.(C2×D4)  (D4×C10).24C4  (D4×C10)⋊21C4  (D4×C10).29C4  (D4×C10)⋊22C4  C25.2D5

52 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H5A5B10A···10AD
order12···222224···45510···10
size11···1222210···10222···2

52 irreducible representations

dim1111122222
type++++++-+
imageC1C2C2C2C4D4D5Dic5D10C5⋊D4
kernelC2×C23.D5C23.D5C22×Dic5C23×C10C22×C10C2×C10C24C23C23C22
# reps14218428616

Matrix representation of C2×C23.D5 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
40000
04000
0010
002040
,
40000
0100
00400
00040
,
1000
0100
00400
00040
,
1000
0100
00100
001737
,
32000
0100
00520
001136
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,20,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,10,17,0,0,0,37],[32,0,0,0,0,1,0,0,0,0,5,11,0,0,20,36] >;

C2×C23.D5 in GAP, Magma, Sage, TeX

C_2\times C_2^3.D_5
% in TeX

G:=Group("C2xC2^3.D5");
// GroupNames label

G:=SmallGroup(160,173);
// by ID

G=gap.SmallGroup(160,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,4613]);
// Polycyclic

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