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G = C23.3D28  order 448 = 26·7

3rd non-split extension by C23 of D28 acting via D28/C7=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.3D28, C7⋊2C2≀C4, (C2×D4).5D14, C4.D4⋊5D7, (C2×C28).13D4, (C23×D7)⋊2C4, C23.D7⋊3C4, C23.3(C4×D7), C23⋊Dic7⋊7C2, C23⋊D14.4C2, (C22×C14).12D4, C14.11(C23⋊C4), C22.12(D14⋊C4), (D4×C14).170C22, C2.12(C23.1D14), (C2×C4).1(C7⋊D4), (C7×C4.D4)⋊11C2, (C22×C14).3(C2×C4), (C2×C14).5(C22⋊C4), SmallGroup(448,32)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — C23.3D28
Chief series
C1 — C7 — C14 — C2×C14 — C2×C28 — D4×C14 — C23⋊D14 — C23.3D28
Lower central
C7 — C14 — C2×C14 — C22×C14 — C23.3D28
Upper central
C1 — C2 — C22 — C2×D4 — C4.D4

Generators and relations for C23.3D28
 G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=a, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd27 >

Subgroups: 780 in 94 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, M4(2), C2×D4, C2×D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C23⋊C4, C4.D4, C22≀C2, C56, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C2≀C4, D14⋊C4, C23.D7, C23.D7, C7×M4(2), C2×C7⋊D4, D4×C14, C23×D7, C23⋊Dic7, C7×C4.D4, C23⋊D14, C23.3D28
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D14, C23⋊C4, C4×D7, D28, C7⋊D4, C2≀C4, D14⋊C4, C23.1D14, C23.3D28

Smallest permutation representation of C23.3D28
►On 56 points
Generators in S56
(1 29)(2 30)(5 33)(6 34)(9 37)(10 38)(13 41)(14 42)(17 45)(18 46)(21 49)(22 50)(25 53)(26 54)

(1 29)(3 31)(5 33)(7 35)(9 37)(11 39)(13 41)(15 43)(17 45)(19 47)(21 49)(23 51)(25 53)(27 55)

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

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

46 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D7A7B7C8A8B14A14B14C14D14E14F14G···14L28A···28F56A···56L
order122222244447778814141414141414···1428···2856···56
size1124428284565656222882224448···84···48···8

46 irreducible representations

dim11111122222224448
type++++++++++++
imageC1C2C2C2C4C4D4D4D7D14C7⋊D4C4×D7D28C23⋊C4C2≀C4C23.1D14C23.3D28
kernelC23.3D28C23⋊Dic7C7×C4.D4C23⋊D14C23.D7C23×D7C2×C28C22×C14C4.D4C2×D4C2×C4C23C23C14C7C2C1
# reps11112211336661263

Matrix representation of C23.3D28 ►in GL8(𝔽113)

112361770000
0101120000
001770000
0001120000
0000112000
000036100
00000010
0000144112112
,
1120000000
0112000000
0011200000
0001120000
0000112000
0000011200
00000010
000006901
,
10000000
01000000
00100000
00010000
0000112000
0000011200
0000001120
0000000112
,
97496800000
59041610000
104870750000
09028160000
00000010
00000112036
00001126900
00001001
,
97496800000
28913520000
104870750000
5698570000
0000044112111
00000112036
00001126900
00001001

G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,77,112,77,112,0,0,0,0,0,0,0,0,112,36,0,1,0,0,0,0,0,1,0,44,0,0,0,0,0,0,1,112,0,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,69,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[97,59,104,0,0,0,0,0,49,0,87,90,0,0,0,0,68,41,0,28,0,0,0,0,0,61,75,16,0,0,0,0,0,0,0,0,0,0,112,1,0,0,0,0,0,112,69,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,1],[97,28,104,56,0,0,0,0,49,9,87,9,0,0,0,0,68,13,0,85,0,0,0,0,0,52,75,7,0,0,0,0,0,0,0,0,0,0,112,1,0,0,0,0,44,112,69,0,0,0,0,0,112,0,0,0,0,0,0,0,111,36,0,1] >;
 

C23.3D28 in GAP, Magma, Sage, TeX 

C_2^3._3D_{28}
 
% in TeX
 
G:=Group("C2^3.3D28");
 
// GroupNames label
 
G:=SmallGroup(448,32);
 
// by ID
 
G=gap.SmallGroup(448,32);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,422,184,346,297,851,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,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*c*d^27>;
 
// generators/relations
 

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