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G = C23.250C24  order 128 = 27

103rd central extension by C23 of C24

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

Aliases: C23.250C24, C24.559C23, C22.822+ 1+4, C22⋊Q8⋊21C4, C22⋊C4⋊23Q8, C22.11(C4×Q8), C2.6(D4⋊3Q8), C23.116(C2×Q8), (C2×C42).26C22, C23.8Q8.8C2, C23.295(C4○D4), C22.45(C22×Q8), (C23×C4).312C22, (C22×C4).769C23, C22.141(C23×C4), C23.134(C22×C4), (C22×Q8).91C22, C23.63C23⋊21C2, C23.67C23⋊23C2, C2.6(C22.45C24), C2.33(C22.11C24), C2.C42.526C22, (C4×C4⋊C4)⋊48C2, C4⋊C4⋊17(C2×C4), C2.20(C2×C4×Q8), (C2×Q8)⋊15(C2×C4), C2.38(C4×C4○D4), (C2×C4).253(C2×Q8), C22⋊C4.33(C2×C4), (C4×C22⋊C4).31C2, (C2×C4).48(C22×C4), (C2×C22⋊Q8).20C2, (C2×C4).724(C4○D4), (C2×C4⋊C4).190C22, (C22×C4).316(C2×C4), C22.135(C2×C4○D4), (C2×C22⋊C4).558C22, C22⋊C4○4(C2.C42), (C2×C2.C42).21C2, SmallGroup(128,1100)

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

Derived series
C1 — C22 — C23.250C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C4×C22⋊C4 — C23.250C24
Lower central
C1 — C22 — C23.250C24
Upper central
C1 — C23 — C23.250C24
Jennings
C1 — C23 — C23.250C24

Generators and relations for C23.250C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=c, e2=cb=bc, f2=ba=ab, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 460 in 274 conjugacy classes, 152 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C23×C4, C22×Q8, C2×C2.C42, C4×C22⋊C4, C4×C22⋊C4, C4×C4⋊C4, C23.8Q8, C23.63C23, C23.67C23, C2×C22⋊Q8, C23.250C24
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, C24, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C4×Q8, C4×C4○D4, C22.11C24, C22.45C24, D4⋊3Q8, C23.250C24

Smallest permutation representation of C23.250C24
►On 64 points
Generators in S64
(1 11)(2 12)(3 9)(4 10)(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 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)

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

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

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

50 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4T4U···4AL
order12···222224···44···4
size11···122222···24···4

50 irreducible representations

dim1111111112224
type++++++++-+
imageC1C2C2C2C2C2C2C2C4Q8C4○D4C4○D42+ 1+4
kernelC23.250C24C2×C2.C42C4×C22⋊C4C4×C4⋊C4C23.8Q8C23.63C23C23.67C23C2×C22⋊Q8C22⋊Q8C22⋊C4C2×C4C23C22
# reps11322421164842

Matrix representation of C23.250C24 ►in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
010000
400000
000100
001000
000010
000034
,
100000
010000
003000
000300
000022
000013
,
300000
020000
000100
004000
000020
000002
,
400000
040000
001000
000100
000010
000034

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

C23.250C24 in GAP, Magma, Sage, TeX 

C_2^3._{250}C_2^4
 
% in TeX
 
G:=Group("C2^3.250C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1100);
 
// by ID
 
G=gap.SmallGroup(128,1100);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,680,758,268,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=c,e^2=c*b=b*c,f^2=b*a=a*b,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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