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G = C2×C23⋊D10order 320 = 26·5

Direct product of C2 and C23⋊D10

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

Aliases: C2×C23⋊D10, C246D10, C102C22≀C2, D1012(C2×D4), (C2×D4)⋊35D10, (C2×C20)⋊9C23, (D5×C24)⋊3C2, (C22×D4)⋊5D5, (C22×D5)⋊15D4, (C22×C4)⋊26D10, (C22×C10)⋊11D4, C236(C5⋊D4), C233(C22×D5), (D4×C10)⋊55C22, (C22×C10)⋊5C23, (C2×Dic5)⋊3C23, C22.146(D4×D5), (C2×C10).294C24, (C23×C10)⋊12C22, (C22×C20)⋊43C22, C10.141(C22×D4), (C23×D5)⋊21C22, C23.D560C22, D10⋊C471C22, C22.307(C23×D5), (C22×Dic5)⋊32C22, (C22×D5).248C23, C53(C2×C22≀C2), (D4×C2×C10)⋊15C2, (C2×C10)⋊7(C2×D4), C2.101(C2×D4×D5), (C2×C4)⋊4(C22×D5), C223(C2×C5⋊D4), (C2×C5⋊D4)⋊43C22, (C22×C5⋊D4)⋊12C2, (C2×C23.D5)⋊27C2, (C2×D10⋊C4)⋊41C2, C2.14(C22×C5⋊D4), SmallGroup(320,1471)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C23⋊D10
C1C5C10C2×C10C22×D5C23×D5D5×C24 — C2×C23⋊D10
C5C2×C10 — C2×C23⋊D10
C1C23C22×D4

Generators and relations for C2×C23⋊D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2782 in 662 conjugacy classes, 143 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×14], C4 [×6], C22, C22 [×10], C22 [×86], C5, C2×C4 [×2], C2×C4 [×10], D4 [×24], C23, C23 [×8], C23 [×86], D5 [×8], C10, C10 [×6], C10 [×6], C22⋊C4 [×12], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24 [×18], Dic5 [×4], C20 [×2], D10 [×8], D10 [×56], C2×C10, C2×C10 [×10], C2×C10 [×22], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4, C22×D4 [×2], C25, C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×D5 [×12], C22×D5 [×64], C22×C10, C22×C10 [×8], C22×C10 [×10], C2×C22≀C2, D10⋊C4 [×8], C23.D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×8], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5 [×6], C23×D5 [×12], C23×C10 [×2], C2×D10⋊C4 [×2], C23⋊D10 [×8], C2×C23.D5, C22×C5⋊D4 [×2], D4×C2×C10, D5×C24, C2×C23⋊D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C22≀C2 [×4], C22×D4 [×3], C5⋊D4 [×4], C22×D5 [×7], C2×C22≀C2, D4×D5 [×4], C2×C5⋊D4 [×6], C23×D5, C23⋊D10 [×4], C2×D4×D5 [×2], C22×C5⋊D4, C2×C23⋊D10

Smallest permutation representation of C2×C23⋊D10
On 80 points
Generators in S80
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 51)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 31)(18 32)(19 33)(20 34)(21 67)(22 68)(23 69)(24 70)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)
(1 80)(2 17)(3 72)(4 19)(5 74)(6 11)(7 76)(8 13)(9 78)(10 15)(12 25)(14 27)(16 29)(18 21)(20 23)(22 73)(24 75)(26 77)(28 79)(30 71)(31 53)(32 67)(33 55)(34 69)(35 57)(36 61)(37 59)(38 63)(39 51)(40 65)(41 66)(42 54)(43 68)(44 56)(45 70)(46 58)(47 62)(48 60)(49 64)(50 52)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 26)(22 27)(23 28)(24 29)(25 30)(31 46)(32 47)(33 48)(34 49)(35 50)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 29)(2 30)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 71)(18 72)(19 73)(20 74)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 61)(59 62)(60 63)
(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 23)(2 22)(3 21)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 74)(12 73)(13 72)(14 71)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 50)(40 49)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N···2U4A4B4C4D4E4F5A5B10A···10N10O···10AD20A···20H
order12···22222222···24444445510···1010···1020···20
size11···122224410···104420202020222···24···44···4

68 irreducible representations

dim111111122222224
type++++++++++++++
imageC1C2C2C2C2C2C2D4D4D5D10D10D10C5⋊D4D4×D5
kernelC2×C23⋊D10C2×D10⋊C4C23⋊D10C2×C23.D5C22×C5⋊D4D4×C2×C10D5×C24C22×D5C22×C10C22×D4C22×C4C2×D4C24C23C22
# reps1281211842284168

Matrix representation of C2×C23⋊D10 in GL6(𝔽41)

100000
010000
0040000
0004000
000010
000001
,
23360000
40180000
00234000
00361800
0000040
0000400
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
001000
000100
0000400
0000040
,
3560000
3400000
000700
00353500
0000400
000001
,
40360000
010000
001000
0054000
0000400
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,40,0,0,0,0,36,18,0,0,0,0,0,0,23,36,0,0,0,0,40,18,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,34,0,0,0,0,6,0,0,0,0,0,0,0,0,35,0,0,0,0,7,35,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,36,1,0,0,0,0,0,0,1,5,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C2×C23⋊D10 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes D_{10}
% in TeX

G:=Group("C2xC2^3:D10");
// GroupNames label

G:=SmallGroup(320,1471);
// by ID

G=gap.SmallGroup(320,1471);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,12550]);
// Polycyclic

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

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