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G = C2×D28order 112 = 24·7

Direct product of C2 and D28

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

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

C1C14 — C2×D28
C1C7C14D14C22×D7 — C2×D28
C7C14 — C2×D28
C1C22C2×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 2A2B2C2D2E2F2G4A4B7A7B7C14A···14I28A···28L
order122222224477714···1428···28
size111114141414222222···22···2

34 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D7D14D14D28
kernelC2×D28D28C2×C28C22×D7C14C2×C4C4C22C2
# reps1412236312

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

1000
0100
00280
00028
,
0100
28000
002224
002526
,
0100
1000
002417
0025
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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