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## G = C2×C7⋊D4order 112 = 24·7

### Direct product of C2 and C7⋊D4

Aliases: C2×C7⋊D4, C23⋊D7, C142D4, C222D14, D143C22, C14.10C23, Dic72C22, C73(C2×D4), (C2×C14)⋊3C22, (C22×C14)⋊2C2, (C2×Dic7)⋊4C2, (C22×D7)⋊3C2, C2.10(C22×D7), SmallGroup(112,36)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C7⋊D4
G = < a,b,c,d | a2=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 184 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C23, D7, C14, C14, C14, C2×D4, Dic7, D14, D14, C2×C14, C2×C14, C2×C14, C2×Dic7, C7⋊D4, C22×D7, C22×C14, C2×C7⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C7⋊D4, C22×D7, C2×C7⋊D4

Smallest permutation representation of C2×C7⋊D4
On 56 points
Generators in S56
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 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 15 8 22)(2 21 9 28)(3 20 10 27)(4 19 11 26)(5 18 12 25)(6 17 13 24)(7 16 14 23)(29 43 36 50)(30 49 37 56)(31 48 38 55)(32 47 39 54)(33 46 40 53)(34 45 41 52)(35 44 42 51)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 22)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 50)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)

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

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

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

C2×C7⋊D4 is a maximal subgroup of
C23.1D14  Dic74D4  C22⋊D28  D14.D4  D14⋊D4  Dic7.D4  C22.D28  C23.23D14  C287D4  C23⋊D14  C282D4  Dic7⋊D4  C28⋊D4  C24⋊D7  C2×D4×D7  D46D14
C2×C7⋊D4 is a maximal quotient of
C28.48D4  C23.23D14  C287D4  D4.D14  C23.18D14  C28.17D4  C23⋊D14  C282D4  Dic7⋊D4  C28⋊D4  C28.C23  Dic7⋊Q8  D143Q8  C28.23D4  D4⋊D14  D4.8D14  D4.9D14  C24⋊D7

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A 7B 7C 14A ··· 14U order 1 2 2 2 2 2 2 2 4 4 7 7 7 14 ··· 14 size 1 1 1 1 2 2 14 14 14 14 2 2 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D7 D14 C7⋊D4 kernel C2×C7⋊D4 C2×Dic7 C7⋊D4 C22×D7 C22×C14 C14 C23 C22 C2 # reps 1 1 4 1 1 2 3 9 12

Matrix representation of C2×C7⋊D4 in GL3(𝔽29) generated by

 28 0 0 0 1 0 0 0 1
,
 1 0 0 0 3 28 0 1 0
,
 28 0 0 0 5 16 0 2 24
,
 28 0 0 0 1 0 0 3 28
G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,1],[1,0,0,0,3,1,0,28,0],[28,0,0,0,5,2,0,16,24],[28,0,0,0,1,3,0,0,28] >;

C2×C7⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes D_4
% in TeX

G:=Group("C2xC7:D4");
// GroupNames label

G:=SmallGroup(112,36);
// by ID

G=gap.SmallGroup(112,36);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,182,2404]);
// Polycyclic

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

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