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G = C23.456C24  order 128 = 27

173rd central stem extension by C23 of C24

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

Aliases: C23.456C24, C24.329C23, C22.1852- 1+4, C22.2412+ 1+4, C2.25(D4×Q8), C4⋊C4.236D4, C22⋊C4⋊18Q8, C23.22(C2×Q8), C2.76(D4⋊5D4), C2.33(D4⋊3Q8), C23.4Q8.8C2, (C23×C4).400C22, (C2×C42).561C22, C22.307(C22×D4), C23.8Q8.30C2, C23.Q8.13C2, C22.103(C22×Q8), (C22×C4).1260C23, (C22×Q8).136C22, C23.67C23⋊62C2, C23.65C23⋊87C2, C23.63C23⋊85C2, C23.78C23⋊17C2, C23.81C23⋊40C2, C24.C22.31C2, C2.C42.193C22, C2.36(C22.26C24), C2.25(C22.33C24), C2.62(C22.46C24), C2.27(C23.37C23), (C4×C4⋊C4)⋊93C2, (C2×C4).80(C2×D4), (C2×C4).255(C2×Q8), (C4×C22⋊C4).62C2, (C2×C42.C2)⋊11C2, (C2×C22⋊Q8).35C2, (C2×C4).389(C4○D4), (C2×C4⋊C4).308C22, C22.332(C2×C4○D4), (C2×C22⋊C4).507C22, SmallGroup(128,1288)

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

Derived series
C1 — C23 — C23.456C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.456C24
Lower central
C1 — C23 — C23.456C24
Upper central
C1 — C23 — C23.456C24
Jennings
C1 — C23 — C23.456C24

Generators and relations for C23.456C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=b, f2=db=bd, g2=c, ab=ba, ac=ca, faf-1=ad=da, eae-1=abc, ag=ga, bc=cb, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 436 in 246 conjugacy classes, 108 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C42.C2, C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23.67C23, C23.78C23, C23.Q8, C23.81C23, C23.4Q8, C2×C22⋊Q8, C2×C42.C2, C23.456C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.26C24, C23.37C23, C22.33C24, D4⋊5D4, D4×Q8, C22.46C24, D4⋊3Q8, C23.456C24

Smallest permutation representation of C23.456C24
►On 64 points
Generators in S64
(2 12)(4 10)(5 22)(6 49)(7 24)(8 51)(14 44)(16 42)(17 36)(18 64)(19 34)(20 62)(21 40)(23 38)(26 54)(28 56)(30 58)(32 60)(33 48)(35 46)(37 52)(39 50)(45 61)(47 63)

(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim1111111111111122244
type++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ 1+42- 1+4
kernelC23.456C24C4×C22⋊C4C4×C4⋊C4C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23.78C23C23.Q8C23.81C23C23.4Q8C2×C22⋊Q8C2×C42.C2C22⋊C4C4⋊C4C2×C4C22C22
# reps11111221111111441211

Matrix representation of C23.456C24 ►in GL6(𝔽5)

100000
040000
001000
001400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
010000
400000
001300
000400
000002
000030
,
200000
030000
004000
000400
000004
000010
,
100000
040000
002000
000200
000020
000002

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

C23.456C24 in GAP, Magma, Sage, TeX 

C_2^3._{456}C_2^4
 
% in TeX
 
G:=Group("C2^3.456C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1288);
 
// by ID
 
G=gap.SmallGroup(128,1288);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,568,758,723,100,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b,f^2=d*b=b*d,g^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,e*a*e^-1=a*b*c,a*g=g*a,b*c=c*b,f*e*f^-1=g*e*g^-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,f*g=g*f>;
 
// generators/relations
 

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