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G = C23.379C24  order 128 = 27

96th central stem extension by C23 of C24

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

Aliases: C23.379C24, C24.296C23, C22.1362- 1+4, C22.1822+ 1+4, C4⋊C4⋊46D4, C2.38(D4⋊6D4), C2.19(Q8⋊6D4), (C2×C42).38C22, C23.8Q8⋊58C2, C23.4Q8⋊12C2, C23.143(C4○D4), (C22×C4).825C23, (C23×C4).367C22, C22.259(C22×D4), C24.C22⋊58C2, C23.10D4.11C2, C23.23D4.23C2, (C22×D4).143C22, C23.63C23⋊57C2, C23.81C23⋊22C2, C23.65C23⋊66C2, C2.51(C22.19C24), C24.3C22.37C2, C2.23(C22.45C24), C2.C42.135C22, C2.27(C22.36C24), C2.30(C22.46C24), C2.45(C23.36C23), (C4×C4⋊C4)⋊66C2, (C2×C4).346(C2×D4), (C2×C42⋊2C2)⋊4C2, (C2×C42⋊C2)⋊27C2, (C2×C4).120(C4○D4), (C2×C4⋊C4).854C22, C22.256(C2×C4○D4), (C2×C22⋊C4).147C22, SmallGroup(128,1211)

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

Derived series
C1 — C23 — C23.379C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.379C24
Lower central
C1 — C23 — C23.379C24
Upper central
C1 — C23 — C23.379C24
Jennings
C1 — C23 — C23.379C24

Generators and relations for C23.379C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=c, f2=b, 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, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, gfg-1=cdf >

Subgroups: 484 in 259 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊2C2, C23×C4, C22×D4, C4×C4⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C24.3C22, C23.10D4, C23.81C23, C23.4Q8, C2×C42⋊C2, C2×C42⋊2C2, C23.379C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.19C24, C23.36C23, C22.36C24, D4⋊6D4, Q8⋊6D4, C22.45C24, C22.46C24, C23.379C24

Smallest permutation representation of C23.379C24
►On 64 points
Generators in S64
(1 16)(2 45)(3 14)(4 47)(5 44)(6 9)(7 42)(8 11)(10 19)(12 17)(13 40)(15 38)(18 41)(20 43)(21 30)(22 59)(23 32)(24 57)(25 62)(26 34)(27 64)(28 36)(29 52)(31 50)(33 53)(35 55)(37 46)(39 48)(49 58)(51 60)(54 63)(56 61)

(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J4A···4H4I···4X4Y4Z4AA
order12···22224···44···4444
size11···14482···24···4888

38 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.379C24C4×C4⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C24.3C22C23.10D4C23.81C23C23.4Q8C2×C42⋊C2C2×C42⋊2C2C4⋊C4C2×C4C23C22C22
# reps1111131111211412411

Matrix representation of C23.379C24 ►in GL6(𝔽5)

400000
110000
000200
003000
000004
000040
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
310000
020000
003000
000300
000001
000010
,
240000
030000
000100
001000
000040
000001
,
120000
440000
000100
004000
000001
000010

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

C23.379C24 in GAP, Magma, Sage, TeX 

C_2^3._{379}C_2^4
 
% in TeX
 
G:=Group("C2^3.379C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1211);
 
// by ID
 
G=gap.SmallGroup(128,1211);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,100,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=c,f^2=b,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,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*f*g^-1=c*d*f>;
 
// generators/relations
 

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