Copied to
clipboard

## G = C23×C10order 80 = 24·5

### Abelian group of type [2,2,2,10]

Aliases: C23×C10, SmallGroup(80,52)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C23×C10
G = < a,b,c,d | a2=b2=c2=d10=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 134, all normal (4 characteristic)
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], C22×C10 [×15], C23×C10
Quotients: C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C24, C2×C10 [×35], C22×C10 [×15], C23×C10

Smallest permutation representation of C23×C10
Regular action on 80 points
Generators in S80
(1 47)(2 48)(3 49)(4 50)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 27)(2 28)(3 29)(4 30)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)(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 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)

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

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

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

C23×C10 is a maximal subgroup of   C242D5  C24⋊C25

80 conjugacy classes

 class 1 2A ··· 2O 5A 5B 5C 5D 10A ··· 10BH order 1 2 ··· 2 5 5 5 5 10 ··· 10 size 1 1 ··· 1 1 1 1 1 1 ··· 1

80 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C5 C10 kernel C23×C10 C22×C10 C24 C23 # reps 1 15 4 60

Matrix representation of C23×C10 in GL4(𝔽11) generated by

 10 0 0 0 0 1 0 0 0 0 1 0 0 0 0 10
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 10
,
 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 10
,
 10 0 0 0 0 8 0 0 0 0 9 0 0 0 0 7
G:=sub<GL(4,GF(11))| [10,0,0,0,0,1,0,0,0,0,1,0,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,10],[10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[10,0,0,0,0,8,0,0,0,0,9,0,0,0,0,7] >;

C23×C10 in GAP, Magma, Sage, TeX

C_2^3\times C_{10}
% in TeX

G:=Group("C2^3xC10");
// GroupNames label

G:=SmallGroup(80,52);
// by ID

G=gap.SmallGroup(80,52);
# by ID

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

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

׿
×
𝔽