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G = C23×C10order 80 = 24·5

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

direct product, abelian, monomial, 2-elementary

Aliases: C23×C10, SmallGroup(80,52)

Series: Derived Chief Lower central Upper central

C1 — C23×C10
C1C5C10C2×C10C22×C10 — C23×C10
C1 — C23×C10
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···2O5A5B5C5D10A···10BH
order12···2555510···10
size11···111111···1

80 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC23×C10C22×C10C24C23
# reps115460

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

10000
0100
0010
00010
,
1000
0100
0010
00010
,
10000
01000
00100
00010
,
10000
0800
0090
0007
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

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