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G = C23.244C24  order 128 = 27

97th central extension by C23 of C24

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

Aliases: C23.244C24, C24.216C23, C22.582- 1+4, C4.74(C4×D4), C4⋊C4.395D4, C22⋊Q8⋊20C4, C2.7(D4⋊6D4), C2.6(Q8⋊5D4), (C2×C42).23C22, C23.94(C22×C4), (C23×C4).309C22, (C22×C4).766C23, C22.135(C23×C4), C22.115(C22×D4), C24.C22.7C2, (C22×Q8).405C22, C23.67C23⋊21C2, C23.65C23⋊27C2, C23.63C23⋊19C2, C2.C42.63C22, C2.8(C22.46C24), C2.5(C22.50C24), C2.17(C23.32C23), (C2×C4×Q8)⋊9C2, (C4×C4⋊C4)⋊44C2, C2.38(C2×C4×D4), C2.34(C4×C4○D4), C4⋊C4.157(C2×C4), (C2×C4).890(C2×D4), C22⋊C4.11(C2×C4), (C4×C22⋊C4).30C2, (C2×C4).43(C22×C4), (C2×Q8).150(C2×C4), (C2×C22⋊Q8).19C2, (C2×C4).800(C4○D4), (C2×C4⋊C4).976C22, (C22×C4).315(C2×C4), C22.129(C2×C4○D4), (C2×C42⋊C2).34C2, (C2×C22⋊C4).443C22, SmallGroup(128,1094)

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

Derived series
C1 — C22 — C23.244C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C23.244C24
Lower central
C1 — C22 — C23.244C24
Upper central
C1 — C23 — C23.244C24
Jennings
C1 — C23 — C23.244C24

Generators and relations for C23.244C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=d, g2=b, faf-1=ab=ba, eae-1=ac=ca, ad=da, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 444 in 288 conjugacy classes, 152 normal (42 characteristic)
C1, C2, C2, C4, C4, 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×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C23×C4, C22×Q8, C4×C22⋊C4, C4×C4⋊C4, C4×C4⋊C4, C23.63C23, C24.C22, C23.65C23, C23.67C23, C2×C42⋊C2, C2×C4×Q8, C2×C22⋊Q8, C23.244C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2- 1+4, C2×C4×D4, C4×C4○D4, C23.32C23, D4⋊6D4, Q8⋊5D4, C22.46C24, C22.50C24, C23.244C24

Smallest permutation representation of C23.244C24
►On 64 points
Generators in S64
(2 52)(4 50)(5 36)(6 39)(7 34)(8 37)(10 24)(12 22)(14 28)(16 26)(17 45)(18 60)(19 47)(20 58)(29 57)(30 48)(31 59)(32 46)(33 63)(35 61)(38 62)(40 64)(42 56)(44 54)

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

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

50 conjugacy classes

class 1 2A···2G2H2I4A···4X4Y···4AN
order12···2224···44···4
size11···1442···24···4

50 irreducible representations

dim11111111111224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42- 1+4
kernelC23.244C24C4×C22⋊C4C4×C4⋊C4C23.63C23C24.C22C23.65C23C23.67C23C2×C42⋊C2C2×C4×Q8C2×C22⋊Q8C22⋊Q8C4⋊C4C2×C4C22
# reps1132411111164122

Matrix representation of C23.244C24 ►in GL6(𝔽5)

100000
040000
001000
001400
000010
000044
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000040
000004
,
010000
400000
001300
000400
000020
000033
,
200000
020000
001000
000100
000043
000011
,
400000
010000
001000
001400
000020
000002

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,2,3,0,0,0,0,0,3],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,3,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;
 

C23.244C24 in GAP, Magma, Sage, TeX 

C_2^3._{244}C_2^4
 
% in TeX
 
G:=Group("C2^3.244C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1094);
 
// by ID
 
G=gap.SmallGroup(128,1094);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=d,g^2=b,f*a*f^-1=a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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