Copied to
clipboard

G = C23.381C24  order 128 = 27

98th central stem extension by C23 of C24

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

Aliases: C23.381C24, C24.573C23, C22.1842+ (1+4), C22.1372- (1+4), C4⋊C4⋊47D4, C2.60(D4⋊5D4), C2.32(Q8⋊5D4), (C2×C42).39C22, (C22×C4).67C23, C23.Q8⋊22C2, C23.8Q8⋊60C2, C23.7Q8⋊52C2, C22⋊3(C42⋊2C2), C23.237(C4○D4), C23.11D4⋊23C2, (C23×C4).368C22, C22.261(C22×D4), C24.C22⋊60C2, C23.23D4.24C2, (C22×D4).144C22, C23.83C23⋊15C2, C23.63C23⋊58C2, C2.53(C22.19C24), C2.C42.136C22, C2.29(C22.47C24), C2.19(C22.33C24), C2.31(C22.46C24), (C22×C4⋊C4)⋊22C2, (C4×C22⋊C4)⋊14C2, (C2×C4).905(C2×D4), (C2×C42⋊2C2)⋊6C2, (C2×C4).373(C4○D4), (C2×C4⋊C4).855C22, C2.11(C2×C42⋊2C2), C22.258(C2×C4○D4), (C2×C22⋊C4).149C22, SmallGroup(128,1213)

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

Derived series
C1 — C23 — C23.381C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.381C24
Lower central
C1 — C23 — C23.381C24
Upper central
C1 — C23 — C23.381C24
Jennings
C1 — C23 — C23.381C24

Subgroups: 516 in 273 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×4], C22 [×19], C2×C4 [×8], C2×C4 [×51], D4 [×4], C23, C23 [×6], C23 [×11], C42 [×3], C22⋊C4 [×16], C4⋊C4 [×4], C4⋊C4 [×13], C22×C4 [×13], C22×C4 [×16], C2×D4 [×6], C24 [×2], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×9], C2×C4⋊C4 [×8], C2×C4⋊C4 [×4], C42⋊2C2 [×4], C23×C4 [×3], C22×D4, C4×C22⋊C4, C23.7Q8, C23.8Q8, C23.23D4 [×3], C23.63C23 [×2], C24.C22, C23.Q8, C23.11D4, C23.83C23 [×2], C22×C4⋊C4, C2×C42⋊2C2, C23.381C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C42⋊2C2 [×4], C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C22.19C24, C2×C42⋊2C2, C22.33C24, D4⋊5D4, Q8⋊5D4, C22.46C24, C22.47C24, C23.381C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=f2=a, e2=g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, 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, eg=ge, fg=gf >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
020000
200000
001000
000100
000044
000001
,
010000
400000
004300
000100
000030
000003
,
300000
030000
004000
001100
000033
000042
,
010000
400000
004000
000400
000030
000003

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,3,2],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4V4W4X4Y
order12···22222244444···4444
size11···12222822224···4888

38 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.381C24C4×C22⋊C4C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.Q8C23.11D4C23.83C23C22×C4⋊C4C2×C42⋊2C2C4⋊C4C2×C4C23C22C22
# reps111132111211441211

In GAP, Magma, Sage, TeX 

C_2^3._{381}C_2^4
 
% in TeX
 
G:=Group("C2^3.381C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1213);
 
// by ID
 
G=gap.SmallGroup(128,1213);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,344,758,723,100,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=f^2=a,e^2=g^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀