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## G = C7×C4○D4order 112 = 24·7

### Direct product of C7 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C4○D4, D4C28, Q8C28, D42C14, Q82C14, C14.13C23, C28.21C22, C4(C7×D4), C4(C7×Q8), C28(C7×D4), C28(C7×Q8), (C2×C4)⋊3C14, (C2×C28)⋊7C2, (C7×D4)⋊5C2, (C7×Q8)⋊5C2, C4.5(C2×C14), C22.(C2×C14), C2.3(C22×C14), (C2×C14).2C22, SmallGroup(112,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C4○D4
 Chief series C1 — C2 — C14 — C2×C14 — C7×D4 — C7×C4○D4
 Lower central C1 — C2 — C7×C4○D4
 Upper central C1 — C28 — C7×C4○D4

Generators and relations for C7×C4○D4
G = < a,b,c,d | a7=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Smallest permutation representation of C7×C4○D4
On 56 points
Generators in S56
(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 33 19 24)(2 34 20 25)(3 35 21 26)(4 29 15 27)(5 30 16 28)(6 31 17 22)(7 32 18 23)(8 43 55 41)(9 44 56 42)(10 45 50 36)(11 46 51 37)(12 47 52 38)(13 48 53 39)(14 49 54 40)
(1 24 19 33)(2 25 20 34)(3 26 21 35)(4 27 15 29)(5 28 16 30)(6 22 17 31)(7 23 18 32)(8 43 55 41)(9 44 56 42)(10 45 50 36)(11 46 51 37)(12 47 52 38)(13 48 53 39)(14 49 54 40)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 50)(7 51)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 45)(23 46)(24 47)(25 48)(26 49)(27 43)(28 44)(29 41)(30 42)(31 36)(32 37)(33 38)(34 39)(35 40)

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

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

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

C7×C4○D4 is a maximal subgroup of
D42Dic7  Q8.Dic7  D4⋊D14  D4.8D14  D4.9D14  D48D14  D4.10D14  C28.A4
C7×C4○D4 is a maximal quotient of
D4×C28  Q8×C28

70 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A ··· 7F 14A ··· 14F 14G ··· 14X 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 4 4 4 4 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 2 2 2 1 1 2 2 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C7 C14 C14 C14 C4○D4 C7×C4○D4 kernel C7×C4○D4 C2×C28 C7×D4 C7×Q8 C4○D4 C2×C4 D4 Q8 C7 C1 # reps 1 3 3 1 6 18 18 6 2 12

Matrix representation of C7×C4○D4 in GL2(𝔽29) generated by

 20 0 0 20
,
 12 0 0 12
,
 17 0 17 12
,
 12 5 12 17
G:=sub<GL(2,GF(29))| [20,0,0,20],[12,0,0,12],[17,17,0,12],[12,12,5,17] >;

C7×C4○D4 in GAP, Magma, Sage, TeX

C_7\times C_4\circ D_4
% in TeX

G:=Group("C7xC4oD4");
// GroupNames label

G:=SmallGroup(112,40);
// by ID

G=gap.SmallGroup(112,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-2,581,222]);
// Polycyclic

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

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