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G = C23.388C24order 128 = 27

105th central stem extension by C23 of C24

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

Aliases: C24.10C23, C23.388C24, C22.1412- 1+4, C22.1892+ 1+4, C22⋊C4.72D4, C428C425C2, C23⋊Q815C2, C23.428(C2×D4), C2.61(D45D4), C2.40(D46D4), (C23×C4).96C22, C23.308(C4○D4), C23.11D428C2, C23.34D432C2, (C2×C42).516C22, (C22×C4).522C23, C22.268(C22×D4), C24.C2263C2, C23.23D4.26C2, C22.14(C4.4D4), (C22×D4).145C22, (C22×Q8).114C22, C23.67C2348C2, C23.83C2318C2, C2.27(C22.45C24), C2.C42.141C22, C2.20(C22.33C24), C2.46(C23.36C23), C2.35(C22.46C24), (C2×C4).59(C2×D4), (C4×C22⋊C4)⋊72C2, (C2×C22⋊Q8)⋊18C2, C2.14(C2×C4.4D4), (C2×C4).375(C4○D4), (C2×C4⋊C4).258C22, C22.265(C2×C4○D4), (C2×C2.C42)⋊33C2, (C2×C22⋊C4).154C22, (C2×C22.D4).15C2, SmallGroup(128,1220)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.388C24
C1C2C22C23C24C23×C4C2×C2.C42 — C23.388C24
C1C23 — C23.388C24
C1C23 — C23.388C24
C1C23 — C23.388C24

Generators and relations for C23.388C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=b, f2=a, ab=ba, ac=ca, ede-1=gdg=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 >

Subgroups: 532 in 273 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×5], C4 [×17], C22 [×7], C22 [×4], C22 [×19], C2×C4 [×8], C2×C4 [×47], D4 [×4], Q8 [×4], C23, C23 [×6], C23 [×11], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×14], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×16], C2×D4 [×6], C2×Q8 [×5], C24 [×2], C2.C42 [×14], C2×C42 [×2], C2×C22⋊C4 [×9], C2×C4⋊C4 [×5], C22⋊Q8 [×4], C22.D4 [×4], C23×C4 [×3], C22×D4, C22×Q8, C2×C2.C42, C4×C22⋊C4, C23.34D4, C428C4, C23.23D4 [×2], C24.C22 [×2], C23.67C23, C23⋊Q8, C23.11D4, C23.83C23 [×2], C2×C22⋊Q8, C2×C22.D4, C23.388C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4, 2- 1+4, C2×C4.4D4, C23.36C23, C22.33C24, D45D4, D46D4, C22.45C24, C22.46C24, C23.388C24

Smallest permutation representation of C23.388C24
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 46)(6 47)(7 48)(8 45)(9 58)(10 59)(11 60)(12 57)(13 54)(14 55)(15 56)(16 53)(21 36)(22 33)(23 34)(24 35)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(49 64)(50 61)(51 62)(52 63)
(1 57)(2 58)(3 59)(4 60)(5 37)(6 38)(7 39)(8 40)(9 18)(10 19)(11 20)(12 17)(13 31)(14 32)(15 29)(16 30)(21 51)(22 52)(23 49)(24 50)(25 48)(26 45)(27 46)(28 47)(33 63)(34 64)(35 61)(36 62)(41 54)(42 55)(43 56)(44 53)
(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 46 57 27)(2 6 58 38)(3 48 59 25)(4 8 60 40)(5 12 37 17)(7 10 39 19)(9 28 18 47)(11 26 20 45)(13 61 31 35)(14 51 32 21)(15 63 29 33)(16 49 30 23)(22 56 52 43)(24 54 50 41)(34 53 64 44)(36 55 62 42)
(1 47 17 6)(2 25 18 39)(3 45 19 8)(4 27 20 37)(5 60 46 11)(7 58 48 9)(10 40 59 26)(12 38 57 28)(13 64 54 49)(14 35 55 24)(15 62 56 51)(16 33 53 22)(21 29 36 43)(23 31 34 41)(30 63 44 52)(32 61 42 50)
(1 24)(2 36)(3 22)(4 34)(5 13)(6 55)(7 15)(8 53)(9 51)(10 63)(11 49)(12 61)(14 47)(16 45)(17 35)(18 21)(19 33)(20 23)(25 43)(26 30)(27 41)(28 32)(29 39)(31 37)(38 42)(40 44)(46 54)(48 56)(50 57)(52 59)(58 62)(60 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A4B4C4D4E···4V4W4X4Y
order12···22222244444···4444
size11···12222822224···4888

38 irreducible representations

dim111111111111122244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC23.388C24C2×C2.C42C4×C22⋊C4C23.34D4C428C4C23.23D4C24.C22C23.67C23C23⋊Q8C23.11D4C23.83C23C2×C22⋊Q8C2×C22.D4C22⋊C4C2×C4C23C22C22
# reps111112211121148811

Matrix representation of C23.388C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
320000
120000
002000
000200
000010
000024
,
300000
030000
004400
000100
000032
000002
,
140000
040000
002200
001300
000023
000003
,
400000
040000
004000
000400
000041
000001

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

C23.388C24 in GAP, Magma, Sage, TeX

C_2^3._{388}C_2^4
% in TeX

G:=Group("C2^3.388C2^4");
// GroupNames label

G:=SmallGroup(128,1220);
// by ID

G=gap.SmallGroup(128,1220);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,100,675]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=b,f^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=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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