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G = C23.350C24  order 128 = 27

67th central stem extension by C23 of C24

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

Aliases: C23.350C24, C24.273C23, C22.1582+ 1+4, (C2×D4)⋊13Q8, C23⋊Q8⋊8C2, C23.12(C2×Q8), C22⋊C4.128D4, C23.427(C2×D4), C2.39(D4⋊5D4), C2.13(D4⋊3Q8), (C23×C4).80C22, C23.7Q8⋊45C2, C23.Q8⋊10C2, C23.8Q8⋊42C2, C23.234(C4○D4), C22.76(C22×Q8), (C2×C42).493C22, (C22×C4).514C23, C22.230(C22×D4), C22.17(C22⋊Q8), C23.83C23⋊8C2, C23.23D4.19C2, (C22×D4).511C22, (C22×Q8).106C22, C23.67C23⋊43C2, C23.63C23⋊37C2, C23.81C23⋊14C2, C2.13(C22.32C24), C2.C42.107C22, C2.27(C23.36C23), (C2×C4×D4).49C2, (C2×C4).31(C2×Q8), (C2×C4).330(C2×D4), (C2×C22⋊Q8)⋊12C2, C2.23(C2×C22⋊Q8), (C2×C4).364(C4○D4), (C2×C4⋊C4).232C22, C22.227(C2×C4○D4), (C2×C2.C42)⋊32C2, (C2×C22⋊C4).129C22, SmallGroup(128,1182)

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

Derived series
C1 — C23 — C23.350C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C23.350C24
Lower central
C1 — C23 — C23.350C24
Upper central
C1 — C23 — C23.350C24
Jennings
C1 — C23 — C23.350C24

Generators and relations for C23.350C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=b, e2=ba=ab, 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: 564 in 292 conjugacy classes, 112 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C2.C42, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C23.67C23, C23⋊Q8, C23.Q8, C23.81C23, C23.83C23, C2×C4×D4, C2×C22⋊Q8, C23.350C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C23.36C23, C22.32C24, D4⋊5D4, D4⋊3Q8, C23.350C24

Smallest permutation representation of C23.350C24
►On 64 points
Generators in S64
(1 13)(2 14)(3 15)(4 16)(5 40)(6 37)(7 38)(8 39)(9 43)(10 44)(11 41)(12 42)(17 46)(18 47)(19 48)(20 45)(21 50)(22 51)(23 52)(24 49)(25 58)(26 59)(27 60)(28 57)(29 54)(30 55)(31 56)(32 53)(33 64)(34 61)(35 62)(36 63)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4T4U4V4W4X
order12···222222244444···44444
size11···122224422224···48888

38 irreducible representations

dim111111111111122224
type++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4Q8C4○D4C4○D42+ 1+4
kernelC23.350C24C2×C2.C42C23.7Q8C23.8Q8C23.23D4C23.63C23C23.67C23C23⋊Q8C23.Q8C23.81C23C23.83C23C2×C4×D4C2×C22⋊Q8C22⋊C4C2×D4C2×C4C23C22
# reps111321111111144842

Matrix representation of C23.350C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
400000
040000
004300
001100
000002
000030
,
400000
010000
001200
004400
000002
000020
,
010000
100000
002000
003300
000040
000004
,
100000
010000
004000
000400
000001
000010

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

C23.350C24 in GAP, Magma, Sage, TeX 

C_2^3._{350}C_2^4
 
% in TeX
 
G:=Group("C2^3.350C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1182);
 
// by ID
 
G=gap.SmallGroup(128,1182);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,344,758,723,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=f^2=b,e^2=b*a=a*b,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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