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G = C23.321C24  order 128 = 27

38th central stem extension by C23 of C24

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

Aliases: C23.321C24, C24.255C23, C22.962- 1+4, (C2×Q8).219D4, C42⋊8C4⋊22C2, C2.12(Q8⋊5D4), C23.23(C4○D4), (C22×C4).55C23, (C23×C4).336C22, (C2×C42).469C22, C23.11D4.3C2, C22.201(C22×D4), C23.7Q8.34C2, C23.83C23⋊4C2, C4.48(C22.D4), (C22×Q8).416C22, C23.67C23⋊31C2, C23.63C23⋊29C2, C23.65C23⋊39C2, C24.C22.11C2, C2.C42.84C22, C2.8(C22.35C24), C2.16(C22.46C24), C2.10(C22.50C24), C2.20(C23.36C23), (C2×C4×Q8)⋊13C2, (C2×C4).311(C2×D4), (C4×C22⋊C4).35C2, (C2×C22⋊Q8).22C2, (C2×C4).854(C4○D4), (C2×C4⋊C4).846C22, C22.200(C2×C4○D4), C2.18(C2×C22.D4), (C2×C22⋊C4).113C22, SmallGroup(128,1153)

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

Derived series
C1 — C23 — C23.321C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.321C24
Lower central
C1 — C23 — C23.321C24
Upper central
C1 — C23 — C23.321C24
Jennings
C1 — C23 — C23.321C24

Generators and relations for C23.321C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=c, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, bc=cb, bd=db, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 420 in 242 conjugacy classes, 104 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C22⋊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, C4×Q8, C22⋊Q8, C23×C4, C22×Q8, C4×C22⋊C4, C23.7Q8, C42⋊8C4, C23.63C23, C24.C22, C23.65C23, C23.67C23, C23.11D4, C23.83C23, C2×C4×Q8, C2×C22⋊Q8, C23.321C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2- 1+4, C2×C22.D4, C23.36C23, C22.35C24, Q8⋊5D4, C22.46C24, C22.50C24, C23.321C24

Smallest permutation representation of C23.321C24
►On 64 points
Generators in S64
(2 42)(4 44)(5 62)(6 51)(7 64)(8 49)(10 58)(12 60)(14 26)(16 28)(17 63)(18 52)(19 61)(20 50)(21 40)(22 56)(23 38)(24 54)(30 46)(32 48)(33 39)(34 55)(35 37)(36 53)

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim1111111111112224
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42- 1+4
kernelC23.321C24C4×C22⋊C4C23.7Q8C42⋊8C4C23.63C23C24.C22C23.65C23C23.67C23C23.11D4C23.83C23C2×C4×Q8C2×C22⋊Q8C2×Q8C2×C4C23C22
# reps11112211221141242

Matrix representation of C23.321C24 ►in GL6(𝔽5)

100000
340000
001000
001400
000010
000044
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
330000
020000
002100
002300
000040
000004
,
300000
420000
001300
001400
000043
000001
,
200000
130000
004000
000400
000010
000001

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

C23.321C24 in GAP, Magma, Sage, TeX 

C_2^3._{321}C_2^4
 
% in TeX
 
G:=Group("C2^3.321C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1153);
 
// by ID
 
G=gap.SmallGroup(128,1153);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=g*e*g^-1=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*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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