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G = C23.405C24  order 128 = 27

122nd central stem extension by C23 of C24

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

Aliases: C23.405C24, C24.580C23, C22.2012+ 1+4, C22⋊C4.8Q8, C42⋊8C4⋊33C2, C23.121(C2×Q8), C2.22(D4⋊3Q8), (C22×C4).80C23, C23.312(C4○D4), C22.88(C22×Q8), (C23×C4).102C22, (C2×C42).525C22, C23.7Q8.49C2, C23.8Q8.22C2, C22.14(C42.C2), C23.63C23⋊67C2, C23.81C23⋊27C2, C23.65C23⋊74C2, C23.83C23⋊27C2, C2.19(C22.32C24), C2.30(C22.45C24), C2.C42.156C22, C2.52(C23.36C23), (C2×C4).41(C2×Q8), (C4×C22⋊C4).54C2, C2.10(C2×C42.C2), (C2×C4).377(C4○D4), (C2×C4⋊C4).271C22, C22.282(C2×C4○D4), (C2×C22⋊C4).502C22, (C2×C2.C42).27C2, SmallGroup(128,1237)

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

Derived series
C1 — C23 — C23.405C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42 — C23.405C24
Lower central
C1 — C23 — C23.405C24
Upper central
C1 — C23 — C23.405C24
Jennings
C1 — C23 — C23.405C24

Generators and relations for C23.405C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=ba=ab, f2=b, 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: 420 in 228 conjugacy classes, 104 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C2×C2.C42, C4×C22⋊C4, C23.7Q8, C42⋊8C4, C23.8Q8, C23.63C23, C23.65C23, C23.81C23, C23.83C23, C23.405C24
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, C24, C42.C2, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C42.C2, C23.36C23, C22.32C24, C22.45C24, D4⋊3Q8, C23.405C24

Smallest permutation representation of C23.405C24
►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 31 23 45)(2 60 24 18)(3 29 21 47)(4 58 22 20)(5 16 36 54)(6 41 33 27)(7 14 34 56)(8 43 35 25)(9 59 51 17)(10 32 52 46)(11 57 49 19)(12 30 50 48)(13 63 55 39)(15 61 53 37)(26 38 44 62)(28 40 42 64)

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4V4W4X4Y4Z
order12···2222244444···44444
size11···1222222224···48888

38 irreducible representations

dim11111111112224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2Q8C4○D4C4○D42+ 1+4
kernelC23.405C24C2×C2.C42C4×C22⋊C4C23.7Q8C42⋊8C4C23.8Q8C23.63C23C23.65C23C23.81C23C23.83C23C22⋊C4C2×C4C23C22
# reps11111421134882

Matrix representation of C23.405C24 ►in GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
200000
020000
002200
001300
000040
000031
,
420000
010000
001000
003400
000030
000003
,
400000
410000
003000
004200
000032
000002
,
400000
040000
001000
003400
000010
000001

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

C23.405C24 in GAP, Magma, Sage, TeX 

C_2^3._{405}C_2^4
 
% in TeX
 
G:=Group("C2^3.405C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1237);
 
// by ID
 
G=gap.SmallGroup(128,1237);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,344,758,723,184,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=b*a=a*b,f^2=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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