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G = C23.401C24  order 128 = 27

118th central stem extension by C23 of C24

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

Aliases: C24.14C23, C23.401C24, C22.1512- (1+4), C22⋊C4⋊26D4, C42⋊9C4⋊23C2, (C22×C4).388D4, C23.615(C2×D4), C2.44(D4⋊6D4), C4⋊2(C22.D4), (C23×C4).98C22, C23.4Q8⋊16C2, C22.13(C4⋊1D4), (C2×C42).521C22, C22.277(C22×D4), (C22×C4).1484C23, C24.3C22⋊50C2, (C22×D4).150C22, C2.C42.541C22, C2.18(C23.38C23), (C2×C4).65(C2×D4), C2.8(C2×C4⋊1D4), (C22×C4⋊C4)⋊25C2, (C4×C22⋊C4)⋊76C2, (C2×C4⋊D4).34C2, (C2×C4).813(C4○D4), (C2×C4⋊C4).858C22, C22.278(C2×C4○D4), (C2×C22.D4)⋊18C2, C2.36(C2×C22.D4), (C2×C22⋊C4).161C22, SmallGroup(128,1233)

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

Derived series
C1 — C23 — C23.401C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.401C24
Lower central
C1 — C23 — C23.401C24
Upper central
C1 — C23 — C23.401C24
Jennings
C1 — C23 — C23.401C24

Subgroups: 676 in 354 conjugacy classes, 124 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×16], C22, C22 [×10], C22 [×26], C2×C4 [×16], C2×C4 [×44], D4 [×12], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×20], C4⋊C4 [×26], C22×C4 [×2], C22×C4 [×14], C22×C4 [×16], C2×D4 [×18], C24, C24 [×2], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×12], C2×C4⋊C4 [×4], C4⋊D4 [×4], C22.D4 [×16], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C4×C22⋊C4, C42⋊9C4 [×2], C24.3C22 [×2], C23.4Q8 [×4], C22×C4⋊C4, C2×C4⋊D4, C2×C22.D4 [×4], C23.401C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22.D4 [×4], C4⋊1D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2- (1+4) [×2], C2×C22.D4, C2×C4⋊1D4, C23.38C23, D4⋊6D4 [×4], C23.401C24

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
020000
200000
002000
000200
000010
000044
,
010000
100000
001400
000400
000012
000044
,
300000
020000
003200
001200
000043
000011
,
100000
010000
001000
000100
000012
000044

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4V4W4X
order12···222222244444···444
size11···122228822224···488

38 irreducible representations

dim111111112224
type++++++++++-
imageC1C2C2C2C2C2C2C2D4D4C4○D42- (1+4)
kernelC23.401C24C4×C22⋊C4C42⋊9C4C24.3C22C23.4Q8C22×C4⋊C4C2×C4⋊D4C2×C22.D4C22⋊C4C22×C4C2×C4C22
# reps112241148482

In GAP, Magma, Sage, TeX 

C_2^3._{401}C_2^4
 
% in TeX
 
G:=Group("C2^3.401C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1233);
 
// by ID
 
G=gap.SmallGroup(128,1233);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c,e^2=g^2=a,f^2=b,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^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=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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