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## G = C2×D5×D7order 280 = 23·5·7

### Direct product of C2, D5 and D7

Aliases: C2×D5×D7, C35⋊C23, C70⋊C22, D705C2, C101D14, C141D10, D35⋊C22, (C5×D7)⋊C22, (C7×D5)⋊C22, C51(C22×D7), C71(C22×D5), (C10×D7)⋊3C2, (D5×C14)⋊3C2, SmallGroup(280,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C2×D5×D7
 Chief series C1 — C7 — C35 — C5×D7 — D5×D7 — C2×D5×D7
 Lower central C35 — C2×D5×D7
 Upper central C1 — C2

Generators and relations for C2×D5×D7
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, C2×C10, D14, D14, C2×C14, C35, C22×D5, C22×D7, C7×D5, C5×D7, D35, C70, D5×D7, C10×D7, D5×C14, D70, C2×D5×D7
Quotients: C1, C2, C22, C23, D5, D7, D10, D14, C22×D5, C22×D7, D5×D7, C2×D5×D7

Smallest permutation representation of C2×D5×D7
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 2A 2B 2C 2D 2E 2F 2G 5A 5B 7A 7B 7C 10A 10B 10C 10D 10E 10F 14A 14B 14C 14D ··· 14I 35A ··· 35F 70A ··· 70F order 1 2 2 2 2 2 2 2 5 5 7 7 7 10 10 10 10 10 10 14 14 14 14 ··· 14 35 ··· 35 70 ··· 70 size 1 1 5 5 7 7 35 35 2 2 2 2 2 2 2 14 14 14 14 2 2 2 10 ··· 10 4 ··· 4 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D5 D7 D10 D10 D14 D14 D5×D7 C2×D5×D7 kernel C2×D5×D7 D5×D7 C10×D7 D5×C14 D70 D14 D10 D7 C14 D5 C10 C2 C1 # reps 1 4 1 1 1 2 3 4 2 6 3 6 6

Matrix representation of C2×D5×D7 in GL4(𝔽71) generated by

 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 21 0 0 13 61
,
 1 0 0 0 0 1 0 0 0 0 1 21 0 0 0 70
,
 0 1 0 0 70 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
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] >;

C2×D5×D7 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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