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G = C22×Dic5order 80 = 24·5

Direct product of C22 and Dic5

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

Aliases: C22×Dic5, C23.2D5, C10.9C23, C22.11D10, (C2×C10)⋊5C4, C103(C2×C4), C53(C22×C4), C2.2(C22×D5), (C22×C10).3C2, (C2×C10).12C22, SmallGroup(80,43)

Series: Derived Chief Lower central Upper central

C1C5 — C22×Dic5
C1C5C10Dic5C2×Dic5 — C22×Dic5
C5 — C22×Dic5
C1C23

Generators and relations for C22×Dic5
 G = < a,b,c,d | a2=b2=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 98 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×7], C5, C2×C4 [×6], C23, C10, C10 [×6], C22×C4, Dic5 [×4], C2×C10 [×7], C2×Dic5 [×6], C22×C10, C22×Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, Dic5 [×4], D10 [×3], C2×Dic5 [×6], C22×D5, C22×Dic5

Smallest permutation representation of C22×Dic5
Regular action on 80 points
Generators in S80
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 21)(9 22)(10 23)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 11)(9 12)(10 13)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(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 56 6 51)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 74 26 79)(22 73 27 78)(23 72 28 77)(24 71 29 76)(25 80 30 75)(31 64 36 69)(32 63 37 68)(33 62 38 67)(34 61 39 66)(35 70 40 65)

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

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

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

C22×Dic5 is a maximal subgroup of
C10.10C42  C23.2F5  C23.11D10  Dic5.14D4  Dic54D4  C22.D20  C23.18D10  Dic5⋊D4  D5×C22×C4
C22×Dic5 is a maximal quotient of
C23.21D10  D4.Dic5

32 conjugacy classes

class 1 2A···2G4A···4H5A5B10A···10N
order12···24···45510···10
size11···15···5222···2

32 irreducible representations

dim1111222
type++++-+
imageC1C2C2C4D5Dic5D10
kernelC22×Dic5C2×Dic5C22×C10C2×C10C23C22C22
# reps1618286

Matrix representation of C22×Dic5 in GL4(𝔽41) generated by

1000
0100
00400
00040
,
1000
04000
00400
00040
,
40000
0100
00100
00037
,
9000
04000
00040
00400
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,10,0,0,0,0,37],[9,0,0,0,0,40,0,0,0,0,0,40,0,0,40,0] >;

C22×Dic5 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_5
% in TeX

G:=Group("C2^2xDic5");
// GroupNames label

G:=SmallGroup(80,43);
// by ID

G=gap.SmallGroup(80,43);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,40,1604]);
// Polycyclic

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

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