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G = C23.348C24  order 128 = 27

65th central stem extension by C23 of C24

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

Aliases: C24.8C23, C23.348C24, C22.1562+ 1+4, C22.1152- 1+4, (C2×Q8).224D4, C23.4Q8⋊6C2, C2.22(Q8⋊5D4), C2.14(Q8⋊6D4), C23.11D4⋊14C2, (C22×C4).804C23, (C2×C42).491C22, C22.228(C22×D4), C24.C22⋊39C2, C4.81(C22.D4), (C22×D4).133C22, (C22×Q8).425C22, C23.67C23⋊42C2, C23.65C23⋊54C2, C24.3C22.33C2, C2.C42.105C22, C2.8(C22.53C24), C2.26(C23.36C23), C2.14(C22.50C24), C2.17(C22.36C24), (C4×C4⋊C4)⋊57C2, (C2×C4×Q8)⋊17C2, (C2×C4).328(C2×D4), (C2×C4).105(C4○D4), (C2×C4⋊C4).230C22, (C2×C4.4D4).21C2, C22.225(C2×C4○D4), C2.26(C2×C22.D4), (C2×C22⋊C4).127C22, SmallGroup(128,1180)

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

Derived series
C1 — C23 — C23.348C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.348C24
Lower central
C1 — C23 — C23.348C24
Upper central
C1 — C23 — C23.348C24
Jennings
C1 — C23 — C23.348C24

Generators and relations for C23.348C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=g2=a, e2=b, f2=ba=ab, ac=ca, ede-1=gdg-1=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: 484 in 256 conjugacy classes, 104 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C4.4D4, C22×D4, C22×Q8, C4×C4⋊C4, C24.C22, C23.65C23, C24.3C22, C23.67C23, C23.11D4, C23.4Q8, C2×C4×Q8, C2×C4.4D4, C23.348C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22.D4, C23.36C23, C22.36C24, Q8⋊5D4, Q8⋊6D4, C22.50C24, C22.53C24, C23.348C24

Smallest permutation representation of C23.348C24
►On 64 points
Generators in S64
(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 10)(2 11)(3 12)(4 9)(5 19)(6 20)(7 17)(8 18)(13 25)(14 26)(15 27)(16 28)(21 62)(22 63)(23 64)(24 61)(29 35)(30 36)(31 33)(32 34)(37 51)(38 52)(39 49)(40 50)(41 57)(42 58)(43 59)(44 60)(45 53)(46 54)(47 55)(48 56)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4Z4AA4AB
order12···2224···44···444
size11···1882···24···488

38 irreducible representations

dim11111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.348C24C4×C4⋊C4C24.C22C23.65C23C24.3C22C23.67C23C23.11D4C23.4Q8C2×C4×Q8C2×C4.4D4C2×Q8C2×C4C22C22
# reps114121221141611

Matrix representation of C23.348C24 ►in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
220000
030000
003000
002200
000010
000001
,
110000
340000
002400
003300
000013
000004
,
330000
420000
002000
000200
000010
000014
,
110000
340000
001200
004400
000010
000001

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

C23.348C24 in GAP, Magma, Sage, TeX 

C_2^3._{348}C_2^4
 
% in TeX
 
G:=Group("C2^3.348C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1180);
 
// by ID
 
G=gap.SmallGroup(128,1180);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,268,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=g^2=a,e^2=b,f^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=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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