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G = C2xD5xD7order 280 = 23·5·7

Direct product of C2, D5 and D7

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

Aliases: C2xD5xD7, C35:C23, C70:C22, D70:5C2, C10:1D14, C14:1D10, D35:C22, (C5xD7):C22, (C7xD5):C22, C5:1(C22xD7), C7:1(C22xD5), (C10xD7):3C2, (D5xC14):3C2, SmallGroup(280,36)

Series: Derived Chief Lower central Upper central

C1C35 — C2xD5xD7
C1C7C35C5xD7D5xD7 — C2xD5xD7
C35 — C2xD5xD7
C1C2

Generators and relations for C2xD5xD7
 G = < a,b,c,d,e | a2=b5=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 476 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C22, C5, C7, C23, D5, D5, C10, C10, D7, D7, C14, C14, D10, D10, C2xC10, D14, D14, C2xC14, C35, C22xD5, C22xD7, C7xD5, C5xD7, D35, C70, D5xD7, C10xD7, D5xC14, D70, C2xD5xD7
Quotients: C1, C2, C22, C23, D5, D7, D10, D14, C22xD5, C22xD7, D5xD7, C2xD5xD7

Smallest permutation representation of C2xD5xD7
On 70 points
Generators in S70
(1 41)(2 42)(3 36)(4 37)(5 38)(6 39)(7 40)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)
(1 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)
(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)

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

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

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

40 conjugacy classes

class 1 2A2B2C2D2E2F2G5A5B7A7B7C10A10B10C10D10E10F14A14B14C14D···14I35A···35F70A···70F
order122222225577710101010101014141414···1435···3570···70
size115577353522222221414141422210···104···44···4

40 irreducible representations

dim1111122222244
type+++++++++++++
imageC1C2C2C2C2D5D7D10D10D14D14D5xD7C2xD5xD7
kernelC2xD5xD7D5xD7C10xD7D5xC14D70D14D10D7C14D5C10C2C1
# reps1411123426366

Matrix representation of C2xD5xD7 in GL4(F71) generated by

70000
07000
0010
0001
,
1000
0100
00121
001361
,
1000
0100
00121
00070
,
0100
705200
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(71))| [70,0,0,0,0,70,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,13,0,0,21,61],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,21,70],[0,70,0,0,1,52,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C2xD5xD7 in GAP, Magma, Sage, TeX

C_2\times D_5\times D_7
% in TeX

G:=Group("C2xD5xD7");
// GroupNames label

G:=SmallGroup(280,36);
// by ID

G=gap.SmallGroup(280,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,328,6004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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