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G = C24.558C23  order 128 = 27

39th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.558C23, C23.242C24, C22.762+ 1+4, C22.562- 1+4, C2.4(D4×Q8), C4.73(C4×D4), C22⋊2(C4×Q8), C4⋊C4.394D4, C22⋊Q8⋊19C4, C22⋊C4⋊22Q8, C2.8(D4⋊5D4), C2.5(D4⋊3Q8), C23.115(C2×Q8), (C23×C4).55C22, C23.8Q8.7C2, C23.294(C4○D4), C22.43(C22×Q8), (C2×C42).436C22, (C22×C4).764C23, C23.133(C22×C4), C22.133(C23×C4), C22.113(C22×D4), (C22×Q8).404C22, C23.63C23⋊18C2, C23.67C23⋊20C2, C23.65C23⋊25C2, C2.C42.522C22, C2.7(C22.46C24), C2.33(C23.33C23), (C2×C4×Q8)⋊8C2, (C4×C4⋊C4)⋊42C2, C4⋊C4⋊15(C2×C4), C2.36(C2×C4×D4), C2.18(C2×C4×Q8), C22⋊C4○6(C4⋊C4), (C2×Q8)⋊25(C2×C4), (C2×C4).888(C2×D4), (C2×C4).252(C2×Q8), (C4×C22⋊C4).29C2, (C22×C4⋊C4).29C2, C22⋊C4.32(C2×C4), (C2×C4).41(C22×C4), (C2×C22⋊Q8).18C2, (C2×C4).798(C4○D4), (C2×C4⋊C4).826C22, (C22×C4).314(C2×C4), C22.127(C2×C4○D4), (C2×C22⋊C4).556C22, C22⋊C4○2(C2×C4⋊C4), SmallGroup(128,1092)

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

Derived series
C1 — C22 — C24.558C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C4⋊C4 — C24.558C23
Lower central
C1 — C22 — C24.558C23
Upper central
C1 — C23 — C24.558C23
Jennings
C1 — C23 — C24.558C23

Generators and relations for C24.558C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=c, g2=a, ab=ba, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 492 in 310 conjugacy classes, 164 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22⋊Q8, C23×C4, C23×C4, C22×Q8, C4×C22⋊C4, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C23.63C23, C23.65C23, C23.65C23, C23.67C23, C22×C4⋊C4, C2×C4×Q8, C2×C22⋊Q8, C24.558C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C4×D4, C4×Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C4×D4, C2×C4×Q8, C23.33C23, D4⋊5D4, D4×Q8, C22.46C24, D4⋊3Q8, C24.558C23

Smallest permutation representation of C24.558C23
►On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(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 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim11111111111222244
type++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C4Q8D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.558C23C4×C22⋊C4C4×C4⋊C4C23.8Q8C23.63C23C23.65C23C23.67C23C22×C4⋊C4C2×C4×Q8C2×C22⋊Q8C22⋊Q8C22⋊C4C4⋊C4C2×C4C23C22C22
# reps131223111116444411

Matrix representation of C24.558C23 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
010000
400000
000100
001000
000044
000021
,
300000
030000
004000
000400
000030
000042
,
400000
010000
001000
000400
000040
000004
,
100000
010000
004000
000400
000044
000021

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

C24.558C23 in GAP, Magma, Sage, TeX 

C_2^4._{558}C_2^3
 
% in TeX
 
G:=Group("C2^4.558C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1092);
 
// by ID
 
G=gap.SmallGroup(128,1092);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=c,g^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,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*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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