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G = C23.737C24order 128 = 27

454th central stem extension by C23 of C24

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

Aliases: C24.116C23, C23.737C24, C22.3912- 1+4, C22.5102+ 1+4, C23.4Q870C2, (C22×C4).248C23, (C2×C42).743C22, C23.Q8100C2, C23.11D4138C2, C23.10D4.77C2, (C22×D4).307C22, C24.C22181C2, C23.83C23141C2, C23.81C23142C2, C23.65C23165C2, C2.55(C22.54C24), C2.C42.440C22, C2.71(C22.34C24), C2.61(C22.56C24), C2.71(C22.57C24), C2.128(C22.36C24), C2.128(C22.33C24), (C2×C4).259(C4○D4), (C2×C4⋊C4).546C22, C22.585(C2×C4○D4), (C2×C22⋊C4).354C22, SmallGroup(128,1569)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.737C24
C1C2C22C23C22×C4C2×C4⋊C4C23.4Q8 — C23.737C24
C1C23 — C23.737C24
C1C23 — C23.737C24
C1C23 — C23.737C24

Generators and relations for C23.737C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=ca=ac, f2=b, 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 >

Subgroups: 436 in 200 conjugacy classes, 84 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×14], C22 [×7], C22 [×14], C2×C4 [×2], C2×C4 [×38], D4 [×4], C23, C23 [×14], C42, C22⋊C4 [×14], C4⋊C4 [×12], C22×C4 [×13], C2×D4 [×3], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×11], C22×D4, C24.C22 [×2], C23.65C23, C23.10D4 [×3], C23.Q8 [×2], C23.11D4 [×2], C23.81C23 [×2], C23.4Q8 [×2], C23.83C23, C23.737C24
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ 1+4 [×4], 2- 1+4 [×2], C22.33C24, C22.34C24, C22.36C24, C22.54C24, C22.56C24 [×2], C22.57C24, C23.737C24

Character table of C23.737C24

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111884444448888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-11-1-1-1-111-11-11-11-111    linear of order 2
ρ31111111111-11-11-1-1-1-1-1-111-1-111    linear of order 2
ρ411111111-1-1-1-11-11-1-11-111-1-1111    linear of order 2
ρ51111111111-1-11-11-11-1-111-11-1-1-1    linear of order 2
ρ611111111-1-1-11-11-1-111-1-11111-1-1    linear of order 2
ρ711111111111-1-1-1-11-111-11-1-11-1-1    linear of order 2
ρ811111111-1-1111111-1-11111-1-1-1-1    linear of order 2
ρ9111111111-1-11-11-1-1-1-111-1-111-11    linear of order 2
ρ1011111111-11-1-11-11-1-111-1-111-1-11    linear of order 2
ρ11111111111-111111111-1-1-1-1-1-1-11    linear of order 2
ρ1211111111-111-1-1-1-111-1-11-11-11-11    linear of order 2
ρ13111111111-11-1-1-1-11-11-11-111-11-1    linear of order 2
ρ1411111111-11111111-1-1-1-1-1-1111-1    linear of order 2
ρ15111111111-1-1-11-11-11-11-1-11-111-1    linear of order 2
ρ1611111111-11-11-11-1-11111-1-1-1-11-1    linear of order 2
ρ172-22-22-22-200-2i-2-2i22i2i0000000000    complex lifted from C4○D4
ρ182-22-22-22-200-2i22i-2-2i2i0000000000    complex lifted from C4○D4
ρ192-22-22-22-2002i2-2i-22i-2i0000000000    complex lifted from C4○D4
ρ202-22-22-22-2002i-22i2-2i-2i0000000000    complex lifted from C4○D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ224-444-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ23444-4-44-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ244-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2544-444-4-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C23.737C24
On 64 points
Generators in S64
(1 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)
(1 26)(2 27)(3 28)(4 25)(5 23)(6 24)(7 21)(8 22)(9 54)(10 55)(11 56)(12 53)(13 60)(14 57)(15 58)(16 59)(17 62)(18 63)(19 64)(20 61)(29 52)(30 49)(31 50)(32 51)(33 48)(34 45)(35 46)(36 47)(37 41)(38 42)(39 43)(40 44)
(1 12)(2 9)(3 10)(4 11)(5 39)(6 40)(7 37)(8 38)(13 50)(14 51)(15 52)(16 49)(17 48)(18 45)(19 46)(20 47)(21 41)(22 42)(23 43)(24 44)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(33 62)(34 63)(35 64)(36 61)
(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)
(2 11)(4 9)(5 21)(6 44)(7 23)(8 42)(13 29)(14 57)(15 31)(16 59)(17 19)(18 45)(20 47)(22 38)(24 40)(25 54)(27 56)(30 49)(32 51)(33 35)(34 63)(36 61)(37 43)(39 41)(46 48)(50 58)(52 60)(62 64)
(1 36 26 47)(2 48 27 33)(3 34 28 45)(4 46 25 35)(5 58 23 15)(6 16 24 59)(7 60 21 13)(8 14 22 57)(9 17 54 62)(10 63 55 18)(11 19 56 64)(12 61 53 20)(29 43 52 39)(30 40 49 44)(31 41 50 37)(32 38 51 42)
(1 15 10 50)(2 16 11 51)(3 13 12 52)(4 14 9 49)(5 36 37 63)(6 33 38 64)(7 34 39 61)(8 35 40 62)(17 22 46 44)(18 23 47 41)(19 24 48 42)(20 21 45 43)(25 57 54 30)(26 58 55 31)(27 59 56 32)(28 60 53 29)

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

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

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

Matrix representation of C23.737C24 in 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
000000004000
000000000400
000000000040
000000000004
,
001000000000
000100000000
400000000000
040000000000
000000100000
000000010000
000010000000
000001000000
000000000241
000000000143
000000004200
000000003024
,
100000000000
010000000000
004000000000
000400000000
000010000000
000004000000
000000400000
000000010000
000000001000
000000001400
000000000040
000000003431
,
010000000000
400000000000
000400000000
001000000000
000001000000
000040000000
000000040000
000000100000
000000000010
000000000103
000000004000
000000000104
,
200000000000
030000000000
002000000000
000300000000
000030000000
000002000000
000000300000
000000020000
000000003400
000000003200
000000000432
000000004012

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],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4],[0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,2,1,2,0,0,0,0,0,0,0,0,0,4,4,0,2,0,0,0,0,0,0,0,0,1,3,0,4],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,3,0,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,3,0,4],[2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,3,3,0,4,0,0,0,0,0,0,0,0,4,2,4,0,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,2,2] >;

C23.737C24 in GAP, Magma, Sage, TeX

C_2^3._{737}C_2^4
% in TeX

G:=Group("C2^3.737C2^4");
// GroupNames label

G:=SmallGroup(128,1569);
// by ID

G=gap.SmallGroup(128,1569);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,758,723,184,794,185,80]);
// 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=b,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

Export

Character table of C23.737C24 in TeX

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