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G = C23⋊D28  order 448 = 26·7

The semidirect product of C23 and D28 acting via D28/C7=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊D28, (C2×C4)⋊D28, C7⋊1C2≀C22, (C2×C28)⋊1D4, C23⋊C4⋊3D7, C22⋊C4⋊1D14, (C2×Dic7)⋊1D4, (C22×D7)⋊1D4, (C22×C14)⋊2D4, C23⋊D14⋊1C2, C22⋊D28⋊1C2, D4⋊6D14⋊1C2, (C2×D4).11D14, C22.8(C2×D28), C22.24(D4×D7), C14.13C22≀C2, (D4×C14).8C22, (C23×D7)⋊1C22, C23.D7⋊1C22, C23.2(C22×D7), C23.1D14⋊1C2, (C22×C14).2C23, C2.16(C22⋊D28), (C7×C23⋊C4)⋊4C2, (C2×C14).17(C2×D4), (C7×C22⋊C4)⋊1C22, (C2×C7⋊D4).2C22, SmallGroup(448,275)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — C23⋊D28
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C2×C7⋊D4 — D4⋊6D14 — C23⋊D28
Lower central
C7 — C14 — C22×C14 — C23⋊D28
Upper central
C1 — C2 — C23 — C23⋊C4

Generators and relations for C23⋊D28
 G = < a,b,c,d,e | a2=b2=c2=d28=e2=1, ab=ba, ac=ca, dad-1=eae=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1500 in 198 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C23⋊C4, C23⋊C4, C22≀C2, 2+ 1+4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C2≀C22, D14⋊C4, C23.D7, C7×C22⋊C4, C2×D28, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, C2×C7⋊D4, D4×C14, C23×D7, C23.1D14, C7×C23⋊C4, C22⋊D28, C23⋊D14, D4⋊6D14, C23⋊D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, C2≀C22, C2×D28, D4×D7, C22⋊D28, C23⋊D28

Smallest permutation representation of C23⋊D28
►On 56 points
Generators in S56
(1 8)(2 38)(3 53)(4 25)(5 12)(6 42)(7 29)(9 16)(10 46)(11 33)(13 20)(14 50)(15 37)(17 24)(18 54)(19 41)(21 28)(22 30)(23 45)(26 34)(27 49)(31 52)(32 39)(35 56)(36 43)(40 47)(44 51)(48 55)

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

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

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

49 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F7A7B7C14A14B14C14D···14L14M14N14O28A···28O
order122222222244444477714141414···1414141428···28
size112224282828284882828562222224···48888···8

49 irreducible representations

dim111111222222222448
type++++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D4D7D14D14D28D28C2≀C22D4×D7C23⋊D28
kernelC23⋊D28C23.1D14C7×C23⋊C4C22⋊D28C23⋊D14D4⋊6D14C2×Dic7C2×C28C22×D7C22×C14C23⋊C4C22⋊C4C2×D4C2×C4C23C7C22C1
# reps121211212136366263

Matrix representation of C23⋊D28 ►in GL8(𝔽29)

2801800000
0280180000
00100000
00010000
0000280280
000018282611
00000010
0000270281
,
280000000
028000000
002800000
000280000
0000111600
000071800
000020127
00001216028
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
51326270000
38410000
72424160000
101726210000
0000280280
0000161818
0000121610
00001216028
,
2113000000
138000000
12248160000
241716210000
0000181300
0000221100
0000121610
0000171301

G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,18,0,1,0,0,0,0,0,0,18,0,1,0,0,0,0,0,0,0,0,28,18,0,27,0,0,0,0,0,28,0,0,0,0,0,0,28,26,1,28,0,0,0,0,0,11,0,1],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,11,7,2,12,0,0,0,0,16,18,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[5,3,7,10,0,0,0,0,13,8,24,17,0,0,0,0,26,4,24,26,0,0,0,0,27,1,16,21,0,0,0,0,0,0,0,0,28,16,12,12,0,0,0,0,0,1,16,16,0,0,0,0,28,8,1,0,0,0,0,0,0,18,0,28],[21,13,12,24,0,0,0,0,13,8,24,17,0,0,0,0,0,0,8,16,0,0,0,0,0,0,16,21,0,0,0,0,0,0,0,0,18,22,12,17,0,0,0,0,13,11,16,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C23⋊D28 in GAP, Magma, Sage, TeX 

C_2^3\rtimes D_{28}
 
% in TeX
 
G:=Group("C2^3:D28");
 
// GroupNames label
 
G:=SmallGroup(448,275);
 
// by ID
 
G=gap.SmallGroup(448,275);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,570,1684,438,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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