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## G = C22×Dic5order 80 = 24·5

### Direct product of C22 and Dic5

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

 Derived series C1 — C5 — C22×Dic5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22×Dic5
 Lower central C5 — C22×Dic5
 Upper central C1 — C23

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 ··· 2G 4A ··· 4H 5A 5B 10A ··· 10N order 1 2 ··· 2 4 ··· 4 5 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2

32 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D5 Dic5 D10 kernel C22×Dic5 C2×Dic5 C22×C10 C2×C10 C23 C22 C22 # reps 1 6 1 8 2 8 6

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

 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
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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