Copied to
clipboard

G = C23.725C24  order 128 = 27

442nd central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.725C24, C24.107C23, C22.4982+ (1+4), C23⋊2D4⋊50C2, C23.Q8⋊95C2, C23.105(C4○D4), (C23×C4).181C22, (C22×C4).236C23, C23.11D4⋊131C2, C23.10D4⋊112C2, C23.23D4⋊109C2, (C22×D4).300C22, C23.84C23⋊17C2, C2.48(C22.54C24), C2.114(C22.32C24), C2.C42.428C22, (C2×C4⋊C4).534C22, C22.573(C2×C4○D4), (C2×C22⋊C4).343C22, SmallGroup(128,1557)

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

Derived series
C1 — C23 — C23.725C24
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C23⋊2D4 — C23.725C24
Lower central
C1 — C23 — C23.725C24
Upper central
C1 — C23 — C23.725C24
Jennings
C1 — C23 — C23.725C24

Subgroups: 660 in 261 conjugacy classes, 84 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×5], C4 [×11], C22, C22 [×6], C22 [×31], C2×C4 [×37], D4 [×24], C23, C23 [×2], C23 [×27], C22⋊C4 [×15], C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×9], C22×C4 [×3], C2×D4 [×21], C24, C24 [×3], C2.C42, C2.C42 [×9], C2×C22⋊C4 [×15], C2×C4⋊C4 [×3], C23×C4, C22×D4 [×6], C23.23D4 [×3], C23⋊2D4, C23⋊2D4 [×3], C23.10D4 [×3], C23.Q8, C23.11D4 [×3], C23.84C23, C23.725C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ (1+4) [×6], C22.32C24 [×3], C22.54C24 [×4], C23.725C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=ca=ac, ab=ba, ede=ad=da, ae=ea, gfg=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, geg=abe >

Smallest permutation representation
►On 64 points
Generators in S64
(1 30)(2 31)(3 32)(4 29)(5 60)(6 57)(7 58)(8 59)(9 51)(10 52)(11 49)(12 50)(13 47)(14 48)(15 45)(16 46)(17 44)(18 41)(19 42)(20 43)(21 40)(22 37)(23 38)(24 39)(25 36)(26 33)(27 34)(28 35)(53 63)(54 64)(55 61)(56 62)

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

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

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

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

Matrix representation ►G ⊆ GL12(𝔽5)

400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000004000
000000000400
000000000040
000000000004
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000004000
000000000400
000000000040
000000000004
,
400000000000
040000000000
004000000000
000400000000
000010000000
000001000000
000000100000
000000010000
000000004000
000000000400
000000000040
000000000004
,
300200000000
033000000000
042000000000
100200000000
000000300000
000000020000
000030000000
000002000000
000000000020
000000003231
000000003000
000000000233
,
033000000000
200300000000
000200000000
003000000000
000003000000
000020000000
000000030000
000000200000
000000000010
000000004143
000000001000
000000004044
,
010000000000
100000000000
000400000000
004000000000
000000100000
000000010000
000010000000
000001000000
000000000300
000000002000
000000003231
000000002022
,
400000000000
010000000000
001000000000
000400000000
000001000000
000010000000
000000040000
000000400000
000000000100
000000001000
000000004143
000000001401

G:=sub<GL(12,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[3,0,0,1,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,0,0,2,3,0,3,0,0,0,0,0,0,0,0,0,1,0,3],[0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,3,0,0,3,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,4,0,4,0,0,0,0,0,0,0,0,0,3,0,4],[0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,2,0,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,0,0,1,2],[4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,4,1,0,0,0,0,0,0,0,0,1,0,1,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,3,1] >;
 

Character table of C23.725C24

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 11111111448884444888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-11-11-1-111-11-1-111-11    linear of order 2
ρ311111111-1-11-1-11-1-11-11-111-11-11    linear of order 2
ρ41111111111-1-111111-1-1-1-1-1-1111    linear of order 2
ρ511111111111-1-111111-111-1-1-1-1-1    linear of order 2
ρ611111111-1-1-1-111-1-11111-11-1-11-1    linear of order 2
ρ711111111-1-11111-1-11-1-1-11-11-11-1    linear of order 2
ρ81111111111-11-11111-11-1-111-1-1-1    linear of order 2
ρ91111111111-111-1-1-1-1-1111-1-11-1-1    linear of order 2
ρ1011111111-1-111-1-111-1-1-11-11-111-1    linear of order 2
ρ1111111111-1-1-1-1-1-111-111-11-1111-1    linear of order 2
ρ1211111111111-11-1-1-1-11-1-1-1111-1-1    linear of order 2
ρ131111111111-1-1-1-1-1-1-1-1-11111-111    linear of order 2
ρ1411111111-1-11-11-111-1-111-1-11-1-11    linear of order 2
ρ1511111111-1-1-111-111-11-1-111-1-1-11    linear of order 2
ρ16111111111111-1-1-1-1-111-1-1-1-1-111    linear of order 2
ρ1722-2-22-22-2-220002i2i2i2i000000000    complex lifted from C4○D4
ρ1822-2-22-22-22-20002i2i2i2i000000000    complex lifted from C4○D4
ρ1922-2-22-22-22-20002i2i2i2i000000000    complex lifted from C4○D4
ρ2022-2-22-22-2-220002i2i2i2i000000000    complex lifted from C4○D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-4-4-444000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-44-444-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2444-4-4-44-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ254-4-444-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ264444-4-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)

In GAP, Magma, Sage, TeX 

C_2^3._{725}C_2^4
 
% in TeX
 
G:=Group("C2^3.725C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1557);
 
// by ID
 
G=gap.SmallGroup(128,1557);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,794,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^2=c*a=a*c,a*b=b*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,g*e*g=a*b*e>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀