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G = C24.264C23  order 128 = 27

104th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.264C23, C23.332C24, C22.1042- 1+4, C4.29C22≀C2, (C2×D4).284D4, (C2×Q8).221D4, C23.161(C2×D4), (C22×C4).376D4, C2.20(D4⋊6D4), C2.16(Q8⋊5D4), C23.8Q8⋊36C2, C23.10D4⋊21C2, (C23×C4).345C22, (C22×C4).799C23, (C2×C42).478C22, C22.212(C22×D4), C24.3C22⋊36C2, (C22×D4).507C22, (C22×Q8).422C22, C23.67C23⋊36C2, C2.C42.93C22, C2.16(C22.26C24), C2.12(C23.38C23), (C2×C4⋊Q8)⋊5C2, (C2×C4)⋊4(C4○D4), (C2×C22⋊Q8)⋊8C2, (C4×C22⋊C4)⋊56C2, (C2×C4).317(C2×D4), (C2×C4.4D4)⋊7C2, C2.20(C2×C22≀C2), (C2×C4⋊C4).218C22, (C22×C4○D4).11C2, C22.211(C2×C4○D4), (C2×C22⋊C4).122C22, SmallGroup(128,1164)

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

Derived series
C1 — C23 — C24.264C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.264C23
Lower central
C1 — C23 — C24.264C23
Upper central
C1 — C23 — C24.264C23
Jennings
C1 — C23 — C24.264C23

Generators and relations for C24.264C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=c, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 772 in 418 conjugacy classes, 120 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C4.4D4, C4⋊Q8, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C4×C22⋊C4, C23.8Q8, C24.3C22, C23.67C23, C23.10D4, C2×C22⋊Q8, C2×C4.4D4, C2×C4⋊Q8, C22×C4○D4, C24.264C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22≀C2, C22×D4, C2×C4○D4, 2- 1+4, C2×C22≀C2, C22.26C24, C23.38C23, D4⋊6D4, Q8⋊5D4, C24.264C23

Smallest permutation representation of C24.264C23
►On 64 points
Generators in S64
(1 11)(2 10)(3 9)(4 12)(5 51)(6 50)(7 49)(8 52)(13 35)(14 34)(15 33)(16 36)(17 23)(18 22)(19 21)(20 24)(25 44)(26 43)(27 42)(28 41)(29 48)(30 47)(31 46)(32 45)(37 62)(38 61)(39 64)(40 63)(53 57)(54 60)(55 59)(56 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 13)(2 14)(3 15)(4 16)(5 43)(6 44)(7 41)(8 42)(9 33)(10 34)(11 35)(12 36)(17 58)(18 59)(19 60)(20 57)(21 54)(22 55)(23 56)(24 53)(25 50)(26 51)(27 52)(28 49)(29 38)(30 39)(31 40)(32 37)(45 62)(46 63)(47 64)(48 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4T4U4V4W4X
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim111111111122224
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D42- 1+4
kernelC24.264C23C4×C22⋊C4C23.8Q8C24.3C22C23.67C23C23.10D4C2×C22⋊Q8C2×C4.4D4C2×C4⋊Q8C22×C4○D4C22×C4C2×D4C2×Q8C2×C4C22
# reps114114111144482

Matrix representation of C24.264C23 ►in GL6(𝔽5)

040000
400000
004000
000100
000010
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
200000
030000
001000
000100
000040
000001
,
010000
400000
000100
001000
000001
000040
,
010000
400000
004000
000400
000010
000001

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

C24.264C23 in GAP, Magma, Sage, TeX 

C_2^4._{264}C_2^3
 
% in TeX
 
G:=Group("C2^4.264C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1164);
 
// by ID
 
G=gap.SmallGroup(128,1164);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,g*e*g^-1=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*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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