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G = C23.550C24  order 128 = 27

267th central stem extension by C23 of C24

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

Aliases: C24.47C23, C23.550C24, C22.2412- 1+4, C22.3252+ 1+4, (C2×C42).82C22, C23⋊Q8.17C2, (C22×C4).160C23, C23.Q8.21C2, C23.11D4.29C2, (C22×Q8).162C22, C23.83C23⋊68C2, C23.67C23⋊74C2, C23.78C23⋊31C2, C2.48(C22.32C24), C24.C22.44C2, C23.63C23⋊117C2, C2.C42.268C22, C2.47(C22.33C24), C2.57(C22.36C24), C2.32(C22.35C24), C2.102(C23.36C23), (C4×C4⋊C4)⋊112C2, (C2×C4).175(C4○D4), (C2×C4⋊C4).894C22, C22.422(C2×C4○D4), (C2×C22⋊C4).233C22, SmallGroup(128,1382)

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

Derived series
C1 — C23 — C23.550C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C24.C22 — C23.550C24
Lower central
C1 — C23 — C23.550C24
Upper central
C1 — C23 — C23.550C24
Jennings
C1 — C23 — C23.550C24

Generators and relations for C23.550C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=a, g2=b, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg-1=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 372 in 193 conjugacy classes, 88 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×Q8, C4×C4⋊C4, C23.63C23, C24.C22, C23.67C23, C23⋊Q8, C23.78C23, C23.Q8, C23.11D4, C23.83C23, C23.550C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, 2- 1+4, C23.36C23, C22.32C24, C22.33C24, C22.35C24, C22.36C24, C23.550C24

Smallest permutation representation of C23.550C24
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(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 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 39)(34 40)(35 37)(36 38)(41 55)(42 56)(43 53)(44 54)(45 59)(46 60)(47 57)(48 58)

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

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

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

32 conjugacy classes

class 1 2A···2G2H4A4B4C4D4E···4P4Q···4W
order12···2244444···44···4
size11···1822224···48···8

32 irreducible representations

dim1111111111244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4○D42+ 1+42- 1+4
kernelC23.550C24C4×C4⋊C4C23.63C23C24.C22C23.67C23C23⋊Q8C23.78C23C23.Q8C23.11D4C23.83C23C2×C4C22C22
# reps11231111231222

Matrix representation of C23.550C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
30000000
03000000
00010000
00100000
00000032
00000012
00003200
00001200
,
01000000
10000000
00100000
00010000
00000010
00000001
00004000
00000400
,
10000000
04000000
00100000
00040000
00001000
00002400
00000010
00000024
,
10000000
01000000
00200000
00020000
00004100
00003100
00000041
00000031

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,1,1] >;
 

C23.550C24 in GAP, Magma, Sage, TeX 

C_2^3._{550}C_2^4
 
% in TeX
 
G:=Group("C2^3.550C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1382);
 
// by ID
 
G=gap.SmallGroup(128,1382);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,185,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=c,e^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,g*f*g^-1=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,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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