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G = C23.387C24  order 128 = 27

104th central stem extension by C23 of C24

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

Aliases: C23.387C24, C24.300C23, C22.1882+ (1+4), (C2×D4).30Q8, C23.16(C2×Q8), C23.614(C2×D4), (C22×C4).384D4, C2.18(D4⋊3Q8), (C22×C4).71C23, C23.4Q8⋊14C2, C23.8Q8⋊63C2, C23.7Q8⋊55C2, C23.329(C4○D4), C22.84(C22×Q8), (C2×C42).515C22, (C23×C4).373C22, C22.267(C22×D4), C22.52(C22⋊Q8), C23.23D4.25C2, (C22×D4).525C22, C23.83C23⋊17C2, C23.65C23⋊69C2, C2.16(C22.29C24), C4.125(C22.D4), C2.C42.140C22, C2.31(C22.47C24), (C2×C4×D4).57C2, (C2×C4).37(C2×Q8), (C22×C4⋊C4)⋊24C2, C2.28(C2×C22⋊Q8), (C2×C4).1394(C2×D4), (C2×C4).810(C4○D4), (C2×C4⋊C4).257C22, C22.264(C2×C4○D4), C2.32(C2×C22.D4), (C2×C22⋊C4).153C22, SmallGroup(128,1219)

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

Derived series
C1 — C23 — C23.387C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.387C24
Lower central
C1 — C23 — C23.387C24
Upper central
C1 — C23 — C23.387C24
Jennings
C1 — C23 — C23.387C24

Subgroups: 548 in 292 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 [×12], C4⋊C4 [×18], C22×C4 [×5], C22×C4 [×12], C22×C4 [×18], 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], C23×C4 [×2], C23×C4 [×2], C22×D4, C23.7Q8, C23.7Q8 [×2], C23.8Q8 [×2], C23.23D4 [×2], C23.65C23 [×2], C23.4Q8 [×2], C23.83C23 [×2], C22×C4⋊C4, C2×C4×D4, C23.387C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22.D4 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C22⋊Q8, C2×C22.D4, C22.29C24, C22.47C24 [×2], D4⋊3Q8 [×2], C23.387C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=ca=ac, e2=g2=a, ab=ba, 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 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)

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

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

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

(2 27)(4 25)(5 39)(6 44)(7 37)(8 42)(9 54)(11 56)(14 57)(16 59)(17 33)(18 45)(19 35)(20 47)(21 41)(22 38)(23 43)(24 40)(30 49)(32 51)(34 63)(36 61)(46 64)(48 62)

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
020000
200000
000100
001000
000030
000003
,
010000
400000
001000
000100
000001
000010
,
100000
010000
001000
000400
000010
000004
,
010000
400000
004000
000400
000010
000001

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

38 conjugacy classes

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

38 irreducible representations

dim11111111122224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D4C4○D42+ (1+4)
kernelC23.387C24C23.7Q8C23.8Q8C23.23D4C23.65C23C23.4Q8C23.83C23C22×C4⋊C4C2×C4×D4C22×C4C2×D4C2×C4C23C22
# reps13222221144842

In GAP, Magma, Sage, TeX 

C_2^3._{387}C_2^4
 
% in TeX
 
G:=Group("C2^3.387C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1219);
 
// by ID
 
G=gap.SmallGroup(128,1219);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c*a=a*c,e^2=g^2=a,a*b=b*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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