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## G = C2×C4×D7order 112 = 24·7

### Direct product of C2×C4 and D7

Aliases: C2×C4×D7, C283C22, C14.2C23, C22.9D14, Dic73C22, D14.4C22, C4(C4×D7), (C2×C28)⋊5C2, C141(C2×C4), C71(C22×C4), (C2×C4)Dic7, C4(C2×Dic7), (C2×Dic7)⋊5C2, C2.1(C22×D7), (C2×C14).9C22, (C22×D7).2C2, (C2×C4)(C2×Dic7), SmallGroup(112,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C4×D7
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×C4×D7
 Lower central C7 — C2×C4×D7
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D7
G = < a,b,c,d | a2=b4=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 168 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C7, C2×C4, C2×C4 [×5], C23, D7 [×4], C14, C14 [×2], C22×C4, Dic7 [×2], C28 [×2], D14 [×6], C2×C14, C4×D7 [×4], C2×Dic7, C2×C28, C22×D7, C2×C4×D7
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D7, C22×C4, D14 [×3], C4×D7 [×2], C22×D7, C2×C4×D7

Smallest permutation representation of C2×C4×D7
On 56 points
Generators in S56
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)
(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)
(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 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)

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

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

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

C2×C4×D7 is a maximal subgroup of
D14⋊C8  C42⋊D7  Dic74D4  D14.D4  D14⋊D4  C4⋊C47D7  D28⋊C4  D14.5D4  C4⋊D28  D14⋊Q8  D142Q8  C282D4  D143Q8
C2×C4×D7 is a maximal quotient of
C42⋊D7  C23.11D14  Dic74D4  Dic73Q8  C4⋊C47D7  D28⋊C4  D28.2C4  D28.C4

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 14A ··· 14I 28A ··· 28L order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 7 7 7 7 1 1 1 1 7 7 7 7 2 2 2 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 D7 D14 D14 C4×D7 kernel C2×C4×D7 C4×D7 C2×Dic7 C2×C28 C22×D7 D14 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 8 3 6 3 12

Matrix representation of C2×C4×D7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 28 1 0 0 20 8
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 20 1
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,20,0,0,1,8],[28,0,0,0,0,28,0,0,0,0,28,20,0,0,0,1] >;

C2×C4×D7 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_7
% in TeX

G:=Group("C2xC4xD7");
// GroupNames label

G:=SmallGroup(112,28);
// by ID

G=gap.SmallGroup(112,28);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,42,2404]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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