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## G = C2×D28order 112 = 24·7

### Direct product of C2 and D28

Aliases: C2×D28, C42D14, C141D4, C282C22, D141C22, C14.3C23, C22.10D14, C71(C2×D4), (C2×C4)⋊2D7, (C2×C28)⋊3C2, (C22×D7)⋊1C2, C2.4(C22×D7), (C2×C14).10C22, SmallGroup(112,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D28
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C2×D28
 Lower central C7 — C14 — C2×D28
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D28
G = < a,b,c | a2=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 232 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C7, C2×C4, D4 [×4], C23 [×2], D7 [×4], C14, C14 [×2], C2×D4, C28 [×2], D14 [×4], D14 [×4], C2×C14, D28 [×4], C2×C28, C22×D7 [×2], C2×D28
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], D28 [×2], C22×D7, C2×D28

Smallest permutation representation of C2×D28
On 56 points
Generators in S56
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 29)(24 30)(25 31)(26 32)(27 33)(28 34)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)

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

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

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

C2×D28 is a maximal subgroup of
C14.D8  C2.D56  C28.46D4  C284D4  C4.D28  C22⋊D28  D14⋊D4  D28⋊C4  D14.5D4  C4⋊D28  C8⋊D14  C287D4  C28⋊D4  C28.23D4  D4⋊D14  C2×D4×D7  D48D14
C2×D28 is a maximal quotient of
C282Q8  C284D4  C4.D28  C22⋊D28  C22.D28  C4⋊D28  D142Q8  D567C2  C8⋊D14  C8.D14  C287D4

34 conjugacy classes

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

34 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 D4 D7 D14 D14 D28 kernel C2×D28 D28 C2×C28 C22×D7 C14 C2×C4 C4 C22 C2 # reps 1 4 1 2 2 3 6 3 12

Matrix representation of C2×D28 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 0 1 0 0 28 0 0 0 0 0 22 24 0 0 25 26
,
 0 1 0 0 1 0 0 0 0 0 24 17 0 0 2 5
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[0,28,0,0,1,0,0,0,0,0,22,25,0,0,24,26],[0,1,0,0,1,0,0,0,0,0,24,2,0,0,17,5] >;

C2×D28 in GAP, Magma, Sage, TeX

C_2\times D_{28}
% in TeX

G:=Group("C2xD28");
// GroupNames label

G:=SmallGroup(112,29);
// by ID

G=gap.SmallGroup(112,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,182,42,2404]);
// Polycyclic

G:=Group<a,b,c|a^2=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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