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## G = C2×D4×D7order 224 = 25·7

### Direct product of C2, D4 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D4×D7
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — C2×D4×D7
 Lower central C7 — C14 — C2×D4×D7
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×D4×D7
G = < a,b,c,d,e | a2=b4=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: 1054 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, D4, C23, C23, D7, D7, C14, C14, C14, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22×D4, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C2×D4×D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4×D7, C23×D7, C2×D4×D7

Smallest permutation representation of C2×D4×D7
On 56 points
Generators in S56
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
(1 34 13 41)(2 35 14 42)(3 29 8 36)(4 30 9 37)(5 31 10 38)(6 32 11 39)(7 33 12 40)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 56)
(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 7A 7B 7C 14A ··· 14I 14J ··· 14U 28A ··· 28F order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 7 7 7 7 14 14 14 14 2 2 14 14 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D7 D14 D14 D14 D4×D7 kernel C2×D4×D7 C2×C4×D7 C2×D28 D4×D7 C2×C7⋊D4 D4×C14 C23×D7 D14 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 1 8 2 1 2 4 3 3 12 6 6

Matrix representation of C2×D4×D7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 1 13 0 0 11 28
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 11 28
,
 19 1 0 0 9 28 0 0 0 0 1 0 0 0 0 1
,
 22 4 0 0 17 7 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,11,0,0,13,28],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,28],[19,9,0,0,1,28,0,0,0,0,1,0,0,0,0,1],[22,17,0,0,4,7,0,0,0,0,1,0,0,0,0,1] >;

C2×D4×D7 in GAP, Magma, Sage, TeX

C_2\times D_4\times D_7
% in TeX

G:=Group("C2xD4xD7");
// GroupNames label

G:=SmallGroup(224,178);
// by ID

G=gap.SmallGroup(224,178);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,159,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=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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