Copied to
clipboard

G = C23.178C24order 128 = 27

31st central extension by C23 of C24

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

Aliases: C23.178C24, C24.534C23, C4215(C2×C4), C42⋊C222C4, (C22×C4).97Q8, C23.89(C2×Q8), C23.599(C2×D4), (C22×C4).769D4, (C2×C42).8C22, C22.69(C23×C4), (C22×C42).24C2, C42(C23.8Q8), C22.75(C22×D4), C22.26(C22×Q8), (C22×C4).451C23, (C23×C4).647C22, C23.118(C22×C4), C23.8Q8.72C2, C2.3(C22.19C24), C43(C23.65C23), C23.65C23173C2, C2.C42.513C22, C2.3(C23.37C23), (C4×C4⋊C4)⋊17C2, C4⋊C439(C2×C4), C4.57(C2×C4⋊C4), C2.8(C4×C4○D4), (C2×C4)⋊12(C4⋊C4), C22.28(C2×C4⋊C4), C2.10(C22×C4⋊C4), (C2×C4).352(C2×Q8), (C2×C4).1556(C2×D4), C22⋊C4.53(C2×C4), C22.70(C2×C4○D4), (C2×C4).514(C4○D4), (C2×C4⋊C4).793C22, (C22×C4).411(C2×C4), (C2×C4).211(C22×C4), (C2×C4)(C23.8Q8), (C2×C42⋊C2).23C2, (C2×C22⋊C4).417C22, (C2×C4)(C23.65C23), SmallGroup(128,1028)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.178C24
C1C2C22C23C22×C4C23×C4C22×C42 — C23.178C24
C1C22 — C23.178C24
C1C22×C4 — C23.178C24
C1C23 — C23.178C24

Generators and relations for C23.178C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=d, g2=c, ab=ba, eae-1=ac=ca, ad=da, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 476 in 328 conjugacy classes, 188 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×20], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×44], C2×C4 [×44], C23, C23 [×6], C23 [×4], C42 [×8], C42 [×12], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×20], C22×C4 [×2], C22×C4 [×24], C22×C4 [×12], C24, C2.C42 [×8], C2×C42 [×8], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×12], C42⋊C2 [×8], C42⋊C2 [×4], C23×C4, C23×C4 [×2], C4×C4⋊C4 [×4], C23.8Q8 [×4], C23.65C23 [×4], C22×C42, C2×C42⋊C2 [×2], C23.178C24
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×8], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×4], C22×C4⋊C4, C4×C4○D4 [×2], C22.19C24 [×2], C23.37C23 [×2], C23.178C24

Smallest permutation representation of C23.178C24
On 64 points
Generators in S64
(1 3)(2 50)(4 52)(5 64)(6 8)(7 62)(9 11)(10 22)(12 24)(13 15)(14 26)(16 28)(17 19)(18 30)(20 32)(21 23)(25 27)(29 31)(33 35)(34 38)(36 40)(37 39)(41 43)(42 54)(44 56)(45 47)(46 58)(48 60)(49 51)(53 55)(57 59)(61 63)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)
(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 47 3 45)(2 20 4 18)(5 16 7 14)(6 41 8 43)(9 19 11 17)(10 48 12 46)(13 37 15 39)(21 31 23 29)(22 60 24 58)(25 33 27 35)(26 64 28 62)(30 50 32 52)(34 56 36 54)(38 44 40 42)(49 59 51 57)(53 63 55 61)
(1 55 51 41)(2 56 52 42)(3 53 49 43)(4 54 50 44)(5 48 62 58)(6 45 63 59)(7 46 64 60)(8 47 61 57)(9 27 23 13)(10 28 24 14)(11 25 21 15)(12 26 22 16)(17 33 31 39)(18 34 32 40)(19 35 29 37)(20 36 30 38)

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB4AC···4AR
order12···222224···44···44···4
size11···122221···12···24···4

56 irreducible representations

dim1111111222
type+++++++-
imageC1C2C2C2C2C2C4D4Q8C4○D4
kernelC23.178C24C4×C4⋊C4C23.8Q8C23.65C23C22×C42C2×C42⋊C2C42⋊C2C22×C4C22×C4C2×C4
# reps144412164416

Matrix representation of C23.178C24 in GL5(𝔽5)

10000
01000
00100
00040
00001
,
10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
01000
00100
00040
00004
,
20000
02100
02300
00001
00040
,
30000
01300
00400
00020
00002
,
40000
01000
00100
00030
00003

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

C23.178C24 in GAP, Magma, Sage, TeX

C_2^3._{178}C_2^4
% in TeX

G:=Group("C2^3.178C2^4");
// GroupNames label

G:=SmallGroup(128,1028);
// by ID

G=gap.SmallGroup(128,1028);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,136]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=d,g^2=c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations

׿
×
𝔽