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G = C23.377C24  order 128 = 27

94th central stem extension by C23 of C24

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

Aliases: C23.377C24, C24.294C23, C22.1812+ 1+4, C22.1342- 1+4, C4⋊C4⋊45D4, C23⋊Q8⋊14C2, C2.58(D4⋊5D4), C2.31(Q8⋊5D4), C23.38(C4○D4), (C2×C42).36C22, C23.8Q8⋊57C2, C23.11D4⋊21C2, (C22×C4).823C23, (C23×C4).365C22, C22.257(C22×D4), C24.C22⋊57C2, C23.23D4.22C2, C23.10D4.10C2, (C22×D4).142C22, (C22×Q8).113C22, C23.83C23⋊14C2, C23.67C23⋊47C2, C23.63C23⋊55C2, C2.49(C22.19C24), C2.22(C22.45C24), C2.C42.133C22, C2.26(C22.36C24), C2.43(C23.36C23), C2.28(C22.46C24), (C4×C22⋊C4)⋊71C2, (C2×C4).345(C2×D4), (C2×C22⋊Q8)⋊17C2, (C2×C42⋊C2)⋊26C2, (C2×C4).372(C4○D4), (C2×C4⋊C4).853C22, C22.254(C2×C4○D4), (C2×C22⋊C4).146C22, SmallGroup(128,1209)

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

Derived series
C1 — C23 — C23.377C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.377C24
Lower central
C1 — C23 — C23.377C24
Upper central
C1 — C23 — C23.377C24
Jennings
C1 — C23 — C23.377C24

Generators and relations for C23.377C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=a, e2=b, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 516 in 270 conjugacy classes, 100 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.67C23, C23⋊Q8, C23.10D4, C23.11D4, C23.83C23, C2×C42⋊C2, C2×C22⋊Q8, C23.377C24
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⋊5D4, Q8⋊5D4, C22.45C24, C22.46C24, C23.377C24

Smallest permutation representation of C23.377C24
►On 64 points
Generators in S64
(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 33)(2 34)(3 35)(4 36)(5 43)(6 44)(7 41)(8 42)(9 13)(10 14)(11 15)(12 16)(17 58)(18 59)(19 60)(20 57)(21 54)(22 55)(23 56)(24 53)(25 52)(26 49)(27 50)(28 51)(29 40)(30 37)(31 38)(32 39)(45 62)(46 63)(47 64)(48 61)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.377C24C4×C22⋊C4C23.8Q8C23.23D4C23.63C23C24.C22C23.67C23C23⋊Q8C23.10D4C23.11D4C23.83C23C2×C42⋊C2C2×C22⋊Q8C4⋊C4C2×C4C23C22C22
# reps111222111111148811

Matrix representation of C23.377C24 ►in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
300000
220000
004000
000400
000040
000001
,
240000
330000
001000
000400
000020
000002
,
400000
040000
000100
001000
000001
000010
,
430000
110000
004000
000400
000020
000002

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

C23.377C24 in GAP, Magma, Sage, TeX 

C_2^3._{377}C_2^4
 
% in TeX
 
G:=Group("C2^3.377C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1209);
 
// by ID
 
G=gap.SmallGroup(128,1209);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,675,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=a,e^2=b,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,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,f*e*f=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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