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G = C23.361C24  order 128 = 27

78th central stem extension by C23 of C24

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

Aliases: C23.361C24, C24.282C23, C22.1682+ (1+4), C22.1232- (1+4), C4⋊C4⋊23D4, C2.23(D42), C22⋊C4⋊41D4, C23.172(C2×D4), C23⋊2D4.6C2, C2.29(D4⋊6D4), C2.27(Q8⋊5D4), C23.31(C4○D4), (C23×C4).87C22, C23.Q8⋊16C2, C23.8Q8⋊49C2, C23.10D4⋊29C2, C23.23D4⋊44C2, (C2×C42).504C22, (C22×C4).814C23, C22.241(C22×D4), C24.C22⋊46C2, C24.3C22⋊41C2, (C22×D4).135C22, (C22×Q8).109C22, C23.78C23⋊10C2, C2.33(C22.19C24), C2.17(C22.45C24), C2.C42.118C22, C2.19(C22.26C24), C2.21(C22.36C24), (C2×C4×D4)⋊37C2, (C4×C4⋊C4)⋊60C2, (C2×C4).55(C2×D4), (C2×C22⋊Q8)⋊14C2, (C2×C4.4D4)⋊11C2, (C2×C4).365(C4○D4), (C2×C4⋊C4).242C22, C22.238(C2×C4○D4), (C2×C22.D4)⋊15C2, (C2×C22⋊C4).137C22, SmallGroup(128,1193)

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

Derived series
C1 — C23 — C23.361C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.361C24
Lower central
C1 — C23 — C23.361C24
Upper central
C1 — C23 — C23.361C24
Jennings
C1 — C23 — C23.361C24

Subgroups: 644 in 323 conjugacy classes, 108 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×19], C22 [×7], C22 [×27], C2×C4 [×14], C2×C4 [×37], D4 [×16], Q8 [×4], C23, C23 [×4], C23 [×19], C42 [×6], C22⋊C4 [×4], C22⋊C4 [×24], C4⋊C4 [×4], C4⋊C4 [×11], C22×C4 [×12], C22×C4 [×10], C2×D4 [×16], C2×Q8 [×5], C24 [×3], C2.C42 [×6], C2×C42 [×3], C2×C22⋊C4 [×13], C2×C4⋊C4 [×7], C4×D4 [×4], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×4], C23×C4 [×2], C22×D4 [×3], C22×Q8, C4×C4⋊C4, C23.8Q8, C23.23D4, C24.C22 [×3], C24.3C22, C23⋊2D4, C23.10D4, C23.78C23, C23.Q8, C2×C4×D4, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C23.361C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×6], C24, C22×D4 [×2], C2×C4○D4 [×3], 2+ (1+4), 2- (1+4), C22.19C24, C22.26C24, C22.36C24, D42, D4⋊6D4, Q8⋊5D4, C22.45C24, C23.361C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=a, g2=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 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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(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 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
030000
200000
001000
000400
000040
000004
,
030000
200000
000400
001000
000023
000043
,
040000
400000
000400
001000
000010
000024
,
300000
030000
000100
004000
000040
000004

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4H4I···4V4W4X4Y
order12···2222224···44···4444
size11···1444482···24···4888

38 irreducible representations

dim11111111111111222244
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.361C24C4×C4⋊C4C23.8Q8C23.23D4C24.C22C24.3C22C23⋊2D4C23.10D4C23.78C23C23.Q8C2×C4×D4C2×C22⋊Q8C2×C22.D4C2×C4.4D4C22⋊C4C4⋊C4C2×C4C23C22C22
# reps11113111111111448411

In GAP, Magma, Sage, TeX 

C_2^3._{361}C_2^4
 
% in TeX
 
G:=Group("C2^3.361C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1193);
 
// by ID
 
G=gap.SmallGroup(128,1193);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,100,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=a,g^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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