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G = C23.732C24  order 128 = 27

449th central stem extension by C23 of C24

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

Aliases: C24.114C23, C23.732C24, C22.5052+ (1+4), C22.3862- (1+4), C23⋊Q8⋊65C2, (C22×C4).243C23, (C2×C42).740C22, C23.11D4⋊137C2, C23.10D4.76C2, (C22×D4).306C22, (C22×Q8).240C22, C24.C22⋊180C2, C23.78C23⋊68C2, C2.19(C24⋊C22), C23.67C23⋊107C2, C23.83C23⋊138C2, C2.121(C22.32C24), C2.C42.435C22, C2.59(C22.56C24), C2.66(C22.57C24), C2.127(C22.36C24), (C2×C4).256(C4○D4), (C2×C4⋊C4).541C22, C22.580(C2×C4○D4), (C2×C22⋊C4).350C22, SmallGroup(128,1564)

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

Derived series
C1 — C23 — C23.732C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C23⋊Q8 — C23.732C24
Lower central
C1 — C23 — C23.732C24
Upper central
C1 — C23 — C23.732C24
Jennings
C1 — C23 — C23.732C24

Subgroups: 468 in 208 conjugacy classes, 84 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×14], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×2], C2×C4 [×38], D4 [×4], Q8 [×8], C23, C23 [×14], C42, C22⋊C4 [×14], C4⋊C4 [×5], C22×C4 [×3], C22×C4 [×10], C2×D4 [×3], C2×Q8 [×7], C24 [×2], C2.C42 [×2], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×4], C22×D4, C22×Q8 [×2], C24.C22 [×2], C23.67C23, C23⋊Q8 [×4], C23.10D4, C23.10D4 [×2], C23.78C23, C23.11D4 [×2], C23.83C23 [×2], C23.732C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ (1+4) [×4], 2- (1+4) [×2], C22.32C24, C22.36C24 [×2], C24⋊C22, C22.56C24 [×2], C22.57C24, C23.732C24

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL12(𝔽5)

400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000001000
000000000100
000000000010
000000000001
,
400000000000
040000000000
004000000000
000400000000
000040000000
000004000000
000000400000
000000040000
000000004000
000000000400
000000000040
000000000004
,
100000000000
010000000000
001000000000
000100000000
000040000000
000004000000
000000400000
000000040000
000000001000
000000000100
000000000010
000000000001
,
200000000000
020000000000
003000000000
000300000000
000000010000
000000100000
000001000000
000010000000
000000002010
000000000304
000000002030
000000000302
,
000200000000
002000000000
030000000000
300000000000
000040000000
000004000000
000000100000
000000010000
000000002010
000000000201
000000002030
000000000203
,
001000000000
000100000000
400000000000
040000000000
000000100000
000000010000
000040000000
000004000000
000000000100
000000001000
000000000001
000000000010
,
010000000000
400000000000
000400000000
001000000000
000001000000
000040000000
000000040000
000000100000
000000000100
000000001000
000000000104
000000001040

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

Character table of C23.732C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111884444448888888888
ρ111111111111111111111111111    trivial
ρ211111111-11-1-11-11-11-1-111-11-11-1    linear of order 2
ρ311111111-111-1-1-1-11-111-11-1-111-1    linear of order 2
ρ41111111111-11-11-1-1-1-1-1-111-1-111    linear of order 2
ρ511111111-1-1-11-11-1-111-1-11111-1-1    linear of order 2
ρ6111111111-11-1-1-1-111-11-11-11-1-11    linear of order 2
ρ7111111111-1-1-11-11-1-11-111-1-11-11    linear of order 2
ρ811111111-1-1111111-1-11111-1-1-1-1    linear of order 2
ρ911111111-1-1-1-11-11-11-11-1-11-1111    linear of order 2
ρ10111111111-111111111-1-1-1-1-1-11-1    linear of order 2
ρ11111111111-1-11-11-1-1-1-111-1-1111-1    linear of order 2
ρ1211111111-1-11-1-1-1-11-11-11-111-111    linear of order 2
ρ1311111111111-1-1-1-111-1-11-11-11-1-1    linear of order 2
ρ1411111111-11-11-11-1-11111-1-1-1-1-11    linear of order 2
ρ1511111111-11111111-1-1-1-1-1-111-11    linear of order 2
ρ161111111111-1-11-11-1-111-1-111-1-1-1    linear of order 2
ρ172-22-22-22-2002i-22i22i2i0000000000    complex lifted from C4○D4
ρ182-22-22-22-2002i22i-22i2i0000000000    complex lifted from C4○D4
ρ192-22-22-22-2002i22i-22i2i0000000000    complex lifted from C4○D4
ρ202-22-22-22-2002i-22i22i2i0000000000    complex lifted from C4○D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ244-4-4-444-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ25444-4-44-4-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_2^3._{732}C_2^4
 
% in TeX
 
G:=Group("C2^3.732C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1564);
 
// by ID
 
G=gap.SmallGroup(128,1564);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,100,794,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c*a=a*c,f^2=g^2=a,a*b=b*a,e*d*e=a*d=d*a,a*e=e*a,g*f*g^-1=a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,g*e*g^-1=a*b*e>;
 
// generators/relations
 

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