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G = C23.214C24  order 128 = 27

67th central extension by C23 of C24

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

Aliases: C23.214C24, C24.201C23, C22.362- 1+4, C22.522+ 1+4, C22⋊Q8⋊18C4, C23.86(C22×C4), (C2×C42).13C22, (C23×C4).293C22, (C22×C4).479C23, C22.105(C23×C4), C23.34D4.9C2, (C22×Q8).89C22, C23.63C23⋊8C2, C24.C22.3C2, C23.67C23⋊16C2, C2.19(C22.11C24), C2.C42.50C22, C2.8(C22.36C24), C2.6(C22.33C24), C2.11(C23.32C23), (C4×C4⋊C4)⋊27C2, C2.17(C4×C4○D4), C4⋊C4.156(C2×C4), C22⋊C4.9(C2×C4), (C4×C22⋊C4).22C2, (C2×C4).34(C22×C4), (C2×Q8).111(C2×C4), C22.99(C2×C4○D4), (C2×C22⋊Q8).17C2, (C2×C4).516(C4○D4), (C2×C4⋊C4).810C22, (C22×C4).305(C2×C4), (C2×C22⋊C4).30C22, SmallGroup(128,1064)

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

Derived series
C1 — C22 — C23.214C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.214C24
Lower central
C1 — C22 — C23.214C24
Upper central
C1 — C23 — C23.214C24
Jennings
C1 — C23 — C23.214C24

Generators and relations for C23.214C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=c, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, fef=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, eg=ge, fg=gf >

Subgroups: 412 in 242 conjugacy classes, 136 normal (22 characteristic)
C1, C2, C2, C2, 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×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C23.34D4, C23.63C23, C24.C22, C23.67C23, C2×C22⋊Q8, C23.214C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, 2+ 1+4, 2- 1+4, C4×C4○D4, C22.11C24, C23.32C23, C22.33C24, C22.36C24, C23.214C24

Smallest permutation representation of C23.214C24
►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)

(2 52)(4 50)(5 62)(6 39)(7 64)(8 37)(10 22)(12 24)(14 26)(16 28)(17 45)(18 58)(19 47)(20 60)(29 57)(30 46)(31 59)(32 48)(33 63)(34 40)(35 61)(36 38)(42 54)(44 56)

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4AH
order12···2224···44···4
size11···1442···24···4

44 irreducible representations

dim111111111244
type+++++++++-
imageC1C2C2C2C2C2C2C2C4C4○D42+ 1+42- 1+4
kernelC23.214C24C4×C22⋊C4C4×C4⋊C4C23.34D4C23.63C23C24.C22C23.67C23C2×C22⋊Q8C22⋊Q8C2×C4C22C22
# reps1121442116822

Matrix representation of C23.214C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
44000000
00010000
00400000
00000020
00000003
00003000
00000200
,
40000000
04000000
00200000
00020000
00000010
00000001
00001000
00000100
,
12000000
04000000
00100000
00040000
00001000
00000100
00000040
00000004
,
30000000
03000000
00300000
00030000
00000100
00004000
00000001
00000040

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
 

C23.214C24 in GAP, Magma, Sage, TeX 

C_2^3._{214}C_2^4
 
% in TeX
 
G:=Group("C2^3.214C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1064);
 
// by ID
 
G=gap.SmallGroup(128,1064);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,675,136]);
 
// 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=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,f*e*f=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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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