Copied to
clipboard

?

G = C42.27C23  order 128 = 27

27th non-split extension by C42 of C23 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.27C23, C4.162- (1+4), C4.352+ (1+4), C8⋊8D4⋊31C2, C8⋊D4⋊24C2, C4⋊C4.142D4, D4.Q8⋊33C2, Q8.Q8⋊32C2, C4⋊SD16⋊16C2, C4⋊C8.88C22, C22⋊C4.34D4, C23.95(C2×D4), D4.D4⋊16C2, C4⋊C4.199C23, (C2×C4).458C24, (C2×C8).341C23, C4⋊Q8.129C22, C4.Q8.97C22, C2.54(D4○SD16), (C2×D4).199C23, (C4×D4).137C22, C4⋊D4.53C22, C4⋊1D4.73C22, (C4×Q8).133C22, (C2×Q8).186C23, C2.D8.115C22, C22⋊Q8.52C22, (C22×C8).352C22, Q8⋊C4.61C22, (C2×SD16).45C22, C22.718(C22×D4), C42.C2.33C22, D4⋊C4.115C22, C22.35C24⋊8C2, (C22×C4).1113C23, C42.6C22⋊15C2, (C2×M4(2)).96C22, C42⋊C2.176C22, C22.34C24.4C2, C2.77(C22.31C24), (C2×C4).582(C2×D4), SmallGroup(128,1992)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.27C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.34C24 — C42.27C23
Lower central
C1 — C2 — C2×C4 — C42.27C23
Upper central
C1 — C22 — C42⋊C2 — C42.27C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.27C23

Subgroups: 348 in 173 conjugacy classes, 84 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×8], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×10], C2×C8 [×4], C2×C8, M4(2), SD16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×2], C42.C2 [×2], C42⋊2C2 [×2], C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×4], C42.6C22, C4⋊SD16 [×2], D4.D4 [×2], C8⋊8D4 [×2], C8⋊D4 [×2], D4.Q8 [×2], Q8.Q8 [×2], C22.34C24, C22.35C24, C42.27C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, D4○SD16 [×2], C42.27C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=a2d >

Smallest permutation representation
►On 64 points
Generators in S64
(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)(57 58 59 60)(61 62 63 64)

(1 22 25 20)(2 23 26 17)(3 24 27 18)(4 21 28 19)(5 15 61 12)(6 16 62 9)(7 13 63 10)(8 14 64 11)(29 36 42 39)(30 33 43 40)(31 34 44 37)(32 35 41 38)(45 57 50 56)(46 58 51 53)(47 59 52 54)(48 60 49 55)

(1 59 25 54)(2 58 26 53)(3 57 27 56)(4 60 28 55)(5 44 61 31)(6 43 62 30)(7 42 63 29)(8 41 64 32)(9 40 16 33)(10 39 13 36)(11 38 14 35)(12 37 15 34)(17 51 23 46)(18 50 24 45)(19 49 21 48)(20 52 22 47)

(1 32 25 41)(2 42 26 29)(3 30 27 43)(4 44 28 31)(5 58 61 53)(6 54 62 59)(7 60 63 55)(8 56 64 57)(9 52 16 47)(10 48 13 49)(11 50 14 45)(12 46 15 51)(17 36 23 39)(18 40 24 33)(19 34 21 37)(20 38 22 35)

(5 10)(6 11)(7 12)(8 9)(13 61)(14 62)(15 63)(16 64)(17 23)(18 24)(19 21)(20 22)(29 31)(30 32)(33 38)(34 39)(35 40)(36 37)(41 43)(42 44)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)
 
G:=sub<Sym(64)| (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)(57,58,59,60)(61,62,63,64), (1,22,25,20)(2,23,26,17)(3,24,27,18)(4,21,28,19)(5,15,61,12)(6,16,62,9)(7,13,63,10)(8,14,64,11)(29,36,42,39)(30,33,43,40)(31,34,44,37)(32,35,41,38)(45,57,50,56)(46,58,51,53)(47,59,52,54)(48,60,49,55), (1,59,25,54)(2,58,26,53)(3,57,27,56)(4,60,28,55)(5,44,61,31)(6,43,62,30)(7,42,63,29)(8,41,64,32)(9,40,16,33)(10,39,13,36)(11,38,14,35)(12,37,15,34)(17,51,23,46)(18,50,24,45)(19,49,21,48)(20,52,22,47), (1,32,25,41)(2,42,26,29)(3,30,27,43)(4,44,28,31)(5,58,61,53)(6,54,62,59)(7,60,63,55)(8,56,64,57)(9,52,16,47)(10,48,13,49)(11,50,14,45)(12,46,15,51)(17,36,23,39)(18,40,24,33)(19,34,21,37)(20,38,22,35), (5,10)(6,11)(7,12)(8,9)(13,61)(14,62)(15,63)(16,64)(17,23)(18,24)(19,21)(20,22)(29,31)(30,32)(33,38)(34,39)(35,40)(36,37)(41,43)(42,44)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54)>;
 
G:=Group( (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)(57,58,59,60)(61,62,63,64), (1,22,25,20)(2,23,26,17)(3,24,27,18)(4,21,28,19)(5,15,61,12)(6,16,62,9)(7,13,63,10)(8,14,64,11)(29,36,42,39)(30,33,43,40)(31,34,44,37)(32,35,41,38)(45,57,50,56)(46,58,51,53)(47,59,52,54)(48,60,49,55), (1,59,25,54)(2,58,26,53)(3,57,27,56)(4,60,28,55)(5,44,61,31)(6,43,62,30)(7,42,63,29)(8,41,64,32)(9,40,16,33)(10,39,13,36)(11,38,14,35)(12,37,15,34)(17,51,23,46)(18,50,24,45)(19,49,21,48)(20,52,22,47), (1,32,25,41)(2,42,26,29)(3,30,27,43)(4,44,28,31)(5,58,61,53)(6,54,62,59)(7,60,63,55)(8,56,64,57)(9,52,16,47)(10,48,13,49)(11,50,14,45)(12,46,15,51)(17,36,23,39)(18,40,24,33)(19,34,21,37)(20,38,22,35), (5,10)(6,11)(7,12)(8,9)(13,61)(14,62)(15,63)(16,64)(17,23)(18,24)(19,21)(20,22)(29,31)(30,32)(33,38)(34,39)(35,40)(36,37)(41,43)(42,44)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54) );
 
G=PermutationGroup([(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),(57,58,59,60),(61,62,63,64)], [(1,22,25,20),(2,23,26,17),(3,24,27,18),(4,21,28,19),(5,15,61,12),(6,16,62,9),(7,13,63,10),(8,14,64,11),(29,36,42,39),(30,33,43,40),(31,34,44,37),(32,35,41,38),(45,57,50,56),(46,58,51,53),(47,59,52,54),(48,60,49,55)], [(1,59,25,54),(2,58,26,53),(3,57,27,56),(4,60,28,55),(5,44,61,31),(6,43,62,30),(7,42,63,29),(8,41,64,32),(9,40,16,33),(10,39,13,36),(11,38,14,35),(12,37,15,34),(17,51,23,46),(18,50,24,45),(19,49,21,48),(20,52,22,47)], [(1,32,25,41),(2,42,26,29),(3,30,27,43),(4,44,28,31),(5,58,61,53),(6,54,62,59),(7,60,63,55),(8,56,64,57),(9,52,16,47),(10,48,13,49),(11,50,14,45),(12,46,15,51),(17,36,23,39),(18,40,24,33),(19,34,21,37),(20,38,22,35)], [(5,10),(6,11),(7,12),(8,9),(13,61),(14,62),(15,63),(16,64),(17,23),(18,24),(19,21),(20,22),(29,31),(30,32),(33,38),(34,39),(35,40),(36,37),(41,43),(42,44),(45,57),(46,58),(47,59),(48,60),(49,55),(50,56),(51,53),(52,54)])
 

Matrix representation ►G ⊆ GL8(𝔽17)

00100000
00010000
160000000
016000000
00000010
00000001
00001000
00000100
,
115000000
116000000
001150000
001160000
00000100
000016000
00000001
000000160
,
06090000
301300000
090110000
1301400000
000012500
00005500
000000125
00000055
,
49650000
4136110000
651380000
6111340000
000010040
000001004
000013070
000001307
,
10000000
116000000
00100000
001160000
00001000
000001600
00000010
000000016

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,3,0,13,0,0,0,0,6,0,9,0,0,0,0,0,0,13,0,14,0,0,0,0,9,0,11,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5],[4,4,6,6,0,0,0,0,9,13,5,11,0,0,0,0,6,6,13,13,0,0,0,0,5,11,8,4,0,0,0,0,0,0,0,0,10,0,13,0,0,0,0,0,0,10,0,13,0,0,0,0,4,0,7,0,0,0,0,0,0,4,0,7],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;
 

Character table of C42.27C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114882244444888888444488
ρ111111111111111111111111111    trivial
ρ2111111-1111-1-1-1-11-11-1-111111-1-1    linear of order 2
ρ31111-11111-1-111-1-1-11-11-11-11-1-11    linear of order 2
ρ41111-11-111-11-1-11-1111-1-11-11-11-1    linear of order 2
ρ511111-11111-1-1-1-1-11-111-11111-1-1    linear of order 2
ρ611111-1-11111111-1-1-1-1-1-1111111    linear of order 2
ρ71111-1-1111-11-1-111-1-1-1111-11-11-1    linear of order 2
ρ81111-1-1-111-1-111-111-11-111-11-1-11    linear of order 2
ρ9111111111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ10111111-1111-1-1-1-111-1-11-1-1-1-1-111    linear of order 2
ρ111111-11111-1-111-1-11-1-1-11-11-111-1    linear of order 2
ρ121111-11-111-11-1-11-1-1-1111-11-11-11    linear of order 2
ρ1311111-11111-1-1-1-1-1-111-11-1-1-1-111    linear of order 2
ρ1411111-1-11111111-111-111-1-1-1-1-1-1    linear of order 2
ρ151111-1-1111-11-1-11111-1-1-1-11-11-11    linear of order 2
ρ161111-1-1-111-1-111-11-1111-1-11-111-1    linear of order 2
ρ172222-200-2-22-22-22000000000000    orthogonal lifted from D4
ρ182222200-2-2-222-2-2000000000000    orthogonal lifted from D4
ρ192222-200-2-222-22-2000000000000    orthogonal lifted from D4
ρ202222200-2-2-2-2-222000000000000    orthogonal lifted from D4
ρ214-44-4000-4400000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-40004-400000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ234-4-4400000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2444-4-4000000000000000002√-202√-200    complex lifted from D4○SD16
ρ254-4-4400000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2644-4-4000000000000000002√-202√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{27}C_2^3
 
% in TeX
 
G:=Group("C4^2.27C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1992);
 
// by ID
 
G=gap.SmallGroup(128,1992);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,e*d*e=a^2*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀