Copied to
clipboard

G = C24.208C23order 128 = 27

48th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.208C23, C23.228C24, C22.642+ 1+4, C22.462- 1+4, C42⋊C227C4, C428C416C2, C424C414C2, C42.184(C2×C4), C23.88(C22×C4), (C2×C42).17C22, C4.28(C42⋊C2), (C23×C4).302C22, (C22×C4).757C23, C22.119(C23×C4), C23.7Q8.31C2, C24.C22.5C2, C23.65C2320C2, C2.C42.520C22, C2.3(C22.46C24), C2.1(C22.49C24), C2.25(C23.33C23), (C4×C4⋊C4)⋊34C2, C2.26(C4×C4○D4), C4⋊C4.238(C2×C4), C22⋊C4.59(C2×C4), (C4×C22⋊C4).27C2, (C2×C4).791(C4○D4), (C2×C4⋊C4).186C22, (C22×C4).311(C2×C4), (C2×C4).233(C22×C4), C2.29(C2×C42⋊C2), C22.113(C2×C4○D4), (C2×C42⋊C2).32C2, (C2×C22⋊C4).33C22, SmallGroup(128,1078)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.208C23
C1C2C22C23C22×C4C23×C4C2×C42⋊C2 — C24.208C23
C1C22 — C24.208C23
C1C23 — C24.208C23
C1C23 — C24.208C23

Generators and relations for C24.208C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=c, g2=b, faf-1=ab=ba, eae-1=ac=ca, ad=da, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 396 in 252 conjugacy classes, 144 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×20], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×20], C2×C4 [×40], C23, C23 [×2], C23 [×6], C42 [×8], C42 [×12], C22⋊C4 [×8], C22⋊C4 [×6], C4⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C22×C4 [×4], C24, C2.C42 [×2], C2.C42 [×10], C2×C42 [×8], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4 [×6], C42⋊C2 [×8], C23×C4, C424C4 [×2], C4×C22⋊C4, C4×C4⋊C4 [×2], C23.7Q8, C428C4 [×2], C24.C22 [×4], C23.65C23 [×2], C2×C42⋊C2, C24.208C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×8], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C42⋊C2, C4×C4○D4, C23.33C23, C22.46C24 [×2], C22.49C24 [×2], C24.208C23

Smallest permutation representation of C24.208C23
On 64 points
Generators in S64
(2 52)(4 50)(5 36)(6 39)(7 34)(8 37)(10 24)(12 22)(14 28)(16 26)(17 45)(18 60)(19 47)(20 58)(29 57)(30 48)(31 59)(32 46)(33 63)(35 61)(38 62)(40 64)(42 56)(44 54)
(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 59 51 45)(2 32 52 18)(3 57 49 47)(4 30 50 20)(5 16 62 26)(6 41 63 55)(7 14 64 28)(8 43 61 53)(9 31 23 17)(10 60 24 46)(11 29 21 19)(12 58 22 48)(13 33 27 39)(15 35 25 37)(34 56 40 42)(36 54 38 44)
(1 41 9 13)(2 56 10 28)(3 43 11 15)(4 54 12 26)(5 30 38 58)(6 17 39 45)(7 32 40 60)(8 19 37 47)(14 52 42 24)(16 50 44 22)(18 34 46 64)(20 36 48 62)(21 25 49 53)(23 27 51 55)(29 35 57 61)(31 33 59 63)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I4A···4X4Y···4AN
order12···2224···44···4
size11···1442···24···4

50 irreducible representations

dim1111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4C4○D42+ 1+42- 1+4
kernelC24.208C23C424C4C4×C22⋊C4C4×C4⋊C4C23.7Q8C428C4C24.C22C23.65C23C2×C42⋊C2C42⋊C2C2×C4C22C22
# reps121212421161611

Matrix representation of C24.208C23 in GL6(𝔽5)

100000
240000
001000
003400
000010
000004
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
230000
030000
002200
001300
000040
000001
,
300000
030000
002000
000200
000001
000010
,
100000
240000
001000
003400
000030
000003

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

C24.208C23 in GAP, Magma, Sage, TeX

C_2^4._{208}C_2^3
% in TeX

G:=Group("C2^4.208C2^3");
// GroupNames label

G:=SmallGroup(128,1078);
// by ID

G=gap.SmallGroup(128,1078);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,100,346,80]);
// Polycyclic

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

׿
×
𝔽