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G = C2×C4×D7order 112 = 24·7

Direct product of C2×C4 and D7

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

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

C1C7 — C2×C4×D7
C1C7C14D14C22×D7 — C2×C4×D7
C7 — C2×C4×D7
C1C2×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 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H7A7B7C14A···14I28A···28L
order122222224444444477714···1428···28
size11117777111177772222···22···2

40 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C4D7D14D14C4×D7
kernelC2×C4×D7C4×D7C2×Dic7C2×C28C22×D7D14C2×C4C4C22C2
# reps14111836312

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

28000
02800
0010
0001
,
12000
02800
0010
0001
,
1000
0100
00281
00208
,
28000
02800
00280
00201
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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