Copied to
clipboard

G = C23.2D28  order 448 = 26·7

2nd non-split extension by C23 of D28 acting via D28/C7=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.2D28, (C2×D28)⋊3C4, C23⋊C4⋊2D7, (C4×Dic7)⋊2C4, (C2×D4).4D14, C7⋊1(C42⋊C4), C28⋊D4.1C2, C23⋊Dic7⋊1C2, (D4×C14).4C22, (C22×C14).11D4, C23.4(C7⋊D4), C14.10(C23⋊C4), C22.11(D14⋊C4), C2.11(C23.1D14), (C2×C4).2(C4×D7), (C7×C23⋊C4)⋊2C2, (C2×C28).2(C2×C4), (C2×C14).4(C22⋊C4), SmallGroup(448,31)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C28 — C23.2D28
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — D4×C14 — C28⋊D4 — C23.2D28
Lower central
C7 — C14 — C2×C14 — C2×C28 — C23.2D28
Upper central
C1 — C2 — C22 — C2×D4 — C23⋊C4

Generators and relations for C23.2D28
 G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=a, dad-1=ab=ba, ac=ca, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=ad-1 >

Subgroups: 668 in 86 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C2×D4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C23⋊C4, C23⋊C4, C4⋊1D4, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C42⋊C4, C4×Dic7, C23.D7, C7×C22⋊C4, C2×D28, C2×C7⋊D4, D4×C14, C23⋊Dic7, C7×C23⋊C4, C28⋊D4, C23.2D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C42⋊C4, D14⋊C4, C23.1D14, C23.2D28

Smallest permutation representation of C23.2D28
►On 56 points
Generators in S56
(1 44)(2 45)(5 48)(6 49)(9 52)(10 53)(13 56)(14 29)(17 32)(18 33)(21 36)(22 37)(25 40)(26 41)

(2 45)(4 47)(6 49)(8 51)(10 53)(12 55)(14 29)(16 31)(18 33)(20 35)(22 37)(24 39)(26 41)(28 43)

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

46 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G7A7B7C14A14B14C14D···14L14M14N14O28A···28O
order122222444444477714141414···1414141428···28
size1124456488282856562222224···48888···8

46 irreducible representations

dim1111112222224448
type+++++++++++
imageC1C2C2C2C4C4D4D7D14C4×D7D28C7⋊D4C23⋊C4C42⋊C4C23.1D14C23.2D28
kernelC23.2D28C23⋊Dic7C7×C23⋊C4C28⋊D4C4×Dic7C2×D28C22×C14C23⋊C4C2×D4C2×C4C23C23C14C7C2C1
# reps1111222336661263

Matrix representation of C23.2D28 ►in GL8(𝔽29)

2816740000
010220000
001130000
000280000
000028000
000016100
000017010
00001302228
,
280000000
028000000
002800000
000280000
00001000
00000100
0000120280
000000028
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
200000000
220700000
19000000
0132490000
000010240
00000001
0000128280
00000100
,
2028000000
242222250000
14000000
11285160000
0000102411
000000028
0000021288
0000132800

G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,7,0,1,0,0,0,0,0,4,22,13,28,0,0,0,0,0,0,0,0,28,16,17,13,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,22,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[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],[20,22,1,0,0,0,0,0,0,0,9,13,0,0,0,0,0,7,0,24,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,0,0,8,1,0,0,0,0,24,0,28,0,0,0,0,0,0,1,0,0],[20,24,1,11,0,0,0,0,28,22,4,28,0,0,0,0,0,22,0,5,0,0,0,0,0,25,0,16,0,0,0,0,0,0,0,0,1,0,0,13,0,0,0,0,0,0,21,28,0,0,0,0,24,0,28,0,0,0,0,0,11,28,8,0] >;
 

C23.2D28 in GAP, Magma, Sage, TeX 

C_2^3._2D_{28}
 
% in TeX
 
G:=Group("C2^3.2D28");
 
// GroupNames label
 
G:=SmallGroup(448,31);
 
// by ID
 
G=gap.SmallGroup(448,31);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,1123,794,297,136,851,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=a,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁