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G = C23.738C24  order 128 = 27

455th central stem extension by C23 of C24

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

Aliases: C23.738C24, C24.117C23, C22.5112+ (1+4), C22.3922- (1+4), C23⋊Q8.34C2, (C2×C42).744C22, (C22×C4).249C23, C23.4Q8.34C2, C23.Q8.46C2, C23.11D4.65C2, (C22×Q8).243C22, C23.84C23⋊19C2, C23.78C23⋊70C2, C24.C22.84C2, C23.65C23⋊166C2, C23.81C23⋊143C2, C23.63C23⋊201C2, C23.83C23⋊142C2, C2.56(C22.54C24), C2.C42.441C22, C2.72(C22.57C24), C2.129(C22.36C24), C2.129(C22.33C24), (C2×C4).260(C4○D4), (C2×C4⋊C4).547C22, C22.586(C2×C4○D4), (C2×C22⋊C4).355C22, SmallGroup(128,1570)

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

Derived series
C1 — C23 — C23.738C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.83C23 — C23.738C24
Lower central
C1 — C23 — C23.738C24
Upper central
C1 — C23 — C23.738C24
Jennings
C1 — C23 — C23.738C24

Subgroups: 372 in 183 conjugacy classes, 84 normal (82 characteristic)
C1, C2 [×7], C2, C4 [×15], C22 [×7], C22 [×7], C2×C4 [×2], C2×C4 [×41], Q8 [×4], C23, C23 [×7], C42, C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×14], C2×Q8 [×3], C24, C2.C42 [×16], C2×C42, C2×C22⋊C4 [×7], C2×C4⋊C4 [×10], C22×Q8, C23.63C23, C24.C22, C23.65C23, C23⋊Q8, C23.78C23 [×2], C23.Q8, C23.11D4 [×3], C23.81C23, C23.4Q8, C23.83C23 [×2], C23.84C23, C23.738C24

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×2], C24, C2×C4○D4, 2+ (1+4) [×3], 2- (1+4) [×3], C22.33C24, C22.36C24 [×2], C22.54C24, C22.57C24 [×3], C23.738C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=c, e2=g2=a, f2=b, ab=ba, ac=ca, ede-1=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 20)(2 17)(3 18)(4 19)(5 46)(6 47)(7 48)(8 45)(9 56)(10 53)(11 54)(12 55)(13 60)(14 57)(15 58)(16 59)(21 33)(22 34)(23 35)(24 36)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(49 64)(50 61)(51 62)(52 63)

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

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

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

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

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
000000004000
000000000400
000000000040
000000000004
,
400000000000
040000000000
001000000000
000100000000
000010400000
000001040000
000020400000
000002040000
000000000402
000000001020
000000000004
000000000010
,
001000000000
000400000000
400000000000
010000000000
000020300000
000002030000
000000300000
000000030000
000000000300
000000002000
000000000002
000000000030
,
001000000000
000100000000
400000000000
040000000000
000020000000
000003000000
000040300000
000001020000
000000002040
000000000304
000000000030
000000000002
,
010000000000
400000000000
000400000000
001000000000
000001000000
000040000000
000000010000
000000400000
000000000100
000000001000
000000000401
000000001010

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

Character table of C23.738C24

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 11111111844444488888888888
ρ111111111111111111111111111    trivial
ρ2111111111-1-11-11-1-111-1-11-11-11-1    linear of order 2
ρ311111111-11-1-1-1-11111-1-111-11-1-1    linear of order 2
ρ411111111-1-11-11-1-1-111111-1-1-1-11    linear of order 2
ρ5111111111-1-11-11-1-1-11-11-11-11-11    linear of order 2
ρ61111111111111111-111-1-1-1-1-1-1-1    linear of order 2
ρ711111111-1-11-11-1-1-1-111-1-11111-1    linear of order 2
ρ811111111-11-1-1-1-111-11-11-1-11-111    linear of order 2
ρ911111111-1111111-1-1-1-11111-1-1-1    linear of order 2
ρ1011111111-1-1-11-11-11-1-11-11-111-11    linear of order 2
ρ111111111111-1-1-1-11-1-1-11-111-1-111    linear of order 2
ρ12111111111-11-11-1-11-1-1-111-1-111-1    linear of order 2
ρ1311111111-1-1-11-11-111-111-11-1-11-1    linear of order 2
ρ1411111111-1111111-11-1-1-1-1-1-1111    linear of order 2
ρ15111111111-11-11-1-111-1-1-1-111-1-11    linear of order 2
ρ161111111111-1-1-1-11-11-111-1-111-1-1    linear of order 2
ρ172-22-22-22-202i-22i22i2i00000000000    complex lifted from C4○D4
ρ182-22-22-22-202i-22i22i2i00000000000    complex lifted from C4○D4
ρ192-22-22-22-202i22i-22i2i00000000000    complex lifted from C4○D4
ρ202-22-22-22-202i22i-22i2i00000000000    complex lifted from C4○D4
ρ214-4-4-444-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2244-4-4-4-444000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ24444-4-44-4-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-444-4-4-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_2^3._{738}C_2^4
 
% in TeX
 
G:=Group("C2^3.738C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1570);
 
// by ID
 
G=gap.SmallGroup(128,1570);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,184,794,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=g^2=a,f^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=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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