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G = C23.167C24  order 128 = 27

20th central extension by C23 of C24

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

Aliases: C23.167C24, C24.530C23, (C2×C42)⋊23C4, C4○(C42⋊9C4), C4○2(C42⋊8C4), C42⋊9C4⋊42C2, C42⋊8C4⋊77C2, C4⋊4(C42⋊C2), (C22×C4).95Q8, C23.87(C2×Q8), C42.330(C2×C4), (C22×C4).595D4, C23.360(C2×D4), (C22×C42).21C2, C22.58(C23×C4), C4○3(C23.7Q8), C22.65(C22×D4), C22.20(C22×Q8), C23.209(C22×C4), (C22×C4).445C23, (C23×C4).678C22, C23.7Q8.82C2, (C2×C42).1087C22, C2.1(C22.26C24), C2.C42.465C22, C2.1(C23.37C23), (C4×C4⋊C4)⋊14C2, C4.83(C2×C4⋊C4), (C2×C4)⋊10(C4⋊C4), C2.7(C22×C4⋊C4), C22.27(C2×C4⋊C4), (C2×C4).351(C2×Q8), (C2×C4)○(C42⋊9C4), (C2×C4).1555(C2×D4), C22.59(C2×C4○D4), (C2×C4).634(C4○D4), (C2×C4⋊C4).786C22, (C22×C4).491(C2×C4), (C2×C4).488(C22×C4), C2.12(C2×C42⋊C2), (C2×C42⋊C2).22C2, (C2×C22⋊C4).413C22, SmallGroup(128,1017)

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

Derived series
C1 — C22 — C23.167C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C22×C42 — C23.167C24
Lower central
C1 — C22 — C23.167C24
Upper central
C1 — C22×C4 — C23.167C24
Jennings
C1 — C23 — C23.167C24

Subgroups: 476 in 328 conjugacy classes, 196 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×16], C4 [×12], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×44], C2×C4 [×44], C23, C23 [×6], C23 [×4], C42 [×16], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×24], C22×C4 [×2], C22×C4 [×32], C22×C4 [×8], C24, C2.C42 [×8], C2×C42 [×2], C2×C42 [×14], C2×C22⋊C4 [×4], C2×C4⋊C4 [×12], C42⋊C2 [×8], C23×C4, C23×C4 [×2], C4×C4⋊C4 [×4], C23.7Q8 [×4], C42⋊8C4 [×2], C42⋊9C4 [×2], C22×C42, C2×C42⋊C2 [×2], C23.167C24

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×8], C24, C2×C4⋊C4 [×12], C42⋊C2 [×8], C23×C4, C22×D4, C22×Q8, C2×C4○D4 [×4], C22×C4⋊C4, C2×C42⋊C2 [×2], C22.26C24 [×2], C23.37C23 [×2], C23.167C24

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽5)

10000
04000
00400
00010
00001
,
10000
04000
00400
00040
00004
,
40000
01000
00100
00010
00001
,
20000
00100
01000
00040
00001
,
10000
01000
00400
00010
00001
,
10000
02000
00300
00001
00040
,
40000
02000
00200
00010
00001

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB4AC···4AR
order12···222224···44···44···4
size11···122221···12···24···4

56 irreducible representations

dim11111111222
type++++++++-
imageC1C2C2C2C2C2C2C4D4Q8C4○D4
kernelC23.167C24C4×C4⋊C4C23.7Q8C42⋊8C4C42⋊9C4C22×C42C2×C42⋊C2C2×C42C22×C4C22×C4C2×C4
# reps1442212164416

In GAP, Magma, Sage, TeX 

C_2^3._{167}C_2^4
 
% in TeX
 
G:=Group("C2^3.167C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1017);
 
// by ID
 
G=gap.SmallGroup(128,1017);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,184,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=1,d^2=c,f^2=b,g^2=a,a*b=b*a,a*c=c*a,e*d*e=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,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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