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G = C23.386C24  order 128 = 27

103rd central stem extension by C23 of C24

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

Aliases: C23.386C24, C24.299C23, C22.1402- (1+4), (C2×D4).210D4, C23.41(C2×D4), (C22×C4).383D4, C2.39(D4⋊6D4), C4⋊5(C22.D4), (C22×C4).70C23, C23.7Q8⋊54C2, C23.307(C4○D4), C23.11D4⋊27C2, C23.34D4⋊31C2, C22.75(C4⋊D4), (C23×C4).372C22, (C2×C42).514C22, C22.266(C22×D4), (C22×D4).524C22, C23.81C23⋊24C2, C23.65C23⋊68C2, C2.C42.139C22, C2.34(C22.46C24), C2.16(C23.38C23), (C2×C4×D4).56C2, (C2×C4).58(C2×D4), (C22×C4⋊C4)⋊23C2, C2.28(C2×C4⋊D4), (C2×C4).809(C4○D4), (C2×C4⋊C4).856C22, C22.263(C2×C4○D4), C2.31(C2×C22.D4), (C2×C22⋊C4).152C22, (C2×C22.D4).14C2, SmallGroup(128,1218)

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

Derived series
C1 — C23 — C23.386C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.386C24
Lower central
C1 — C23 — C23.386C24
Upper central
C1 — C23 — C23.386C24
Jennings
C1 — C23 — C23.386C24

Subgroups: 564 in 312 conjugacy classes, 116 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×14], C22 [×3], C22 [×8], C22 [×22], C2×C4 [×10], C2×C4 [×54], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×2], C22⋊C4 [×16], C4⋊C4 [×22], C22×C4 [×5], C22×C4 [×12], C22×C4 [×22], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×10], C2×C4⋊C4 [×4], C4×D4 [×4], C22.D4 [×8], C23×C4 [×2], C23×C4 [×2], C22×D4, C23.7Q8, C23.7Q8 [×2], C23.34D4 [×2], C23.65C23 [×2], C23.11D4 [×2], C23.81C23 [×2], C22×C4⋊C4, C2×C4×D4, C2×C22.D4 [×2], C23.386C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C2×C4○D4 [×3], 2- (1+4) [×2], C2×C4⋊D4, C2×C22.D4, C23.38C23, D4⋊6D4 [×2], C22.46C24 [×2], C23.386C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=g2=a, ab=ba, 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 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
330000
420000
000100
001000
000030
000003
,
200000
130000
001000
000100
000013
000004
,
100000
340000
001000
000400
000010
000014
,
200000
130000
001000
000100
000040
000004

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],[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],[3,4,0,0,0,0,3,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[1,3,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[2,1,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4T4U4V4W4X
order12···222222244444···44444
size11···122224422224···48888

38 irreducible representations

dim11111111122224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42- (1+4)
kernelC23.386C24C23.7Q8C23.34D4C23.65C23C23.11D4C23.81C23C22×C4⋊C4C2×C4×D4C2×C22.D4C22×C4C2×D4C2×C4C23C22
# reps13222211244842

In GAP, Magma, Sage, TeX 

C_2^3._{386}C_2^4
 
% in TeX
 
G:=Group("C2^3.386C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1218);
 
// by ID
 
G=gap.SmallGroup(128,1218);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=g^2=a,a*b=b*a,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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