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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, C22.1152- (1+4), C22.1562+ (1+4), (C2×Q8).224D4, C23.4Q8⋊6C2, C2.22(Q8⋊5D4), C2.14(Q8⋊6D4), C23.11D4⋊14C2, (C2×C42).491C22, (C22×C4).804C23, C22.228(C22×D4), C24.C22⋊39C2, C4.81(C22.D4), (C22×D4).133C22, (C22×Q8).425C22, C23.65C23⋊54C2, C23.67C23⋊42C2, C24.3C22.33C2, C2.C42.105C22, C2.8(C22.53C24), C2.17(C22.36C24), C2.14(C22.50C24), C2.26(C23.36C23), (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

Subgroups: 484 in 256 conjugacy classes, 104 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×16], C22 [×7], C22 [×14], C2×C4 [×14], C2×C4 [×32], D4 [×4], Q8 [×8], C23, C23 [×14], C42 [×10], C22⋊C4 [×20], C4⋊C4 [×14], C22×C4 [×7], C22×C4 [×6], C2×D4 [×6], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42 [×3], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×4], C2×C4⋊C4 [×4], C4×Q8 [×4], C4.4D4 [×4], C22×D4, C22×Q8, C4×C4⋊C4, C24.C22 [×4], C23.65C23, C24.3C22 [×2], C23.67C23, C23.11D4 [×2], C23.4Q8 [×2], C2×C4×Q8, C2×C4.4D4, C23.348C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C2×C22.D4, C23.36C23, C22.36C24, Q8⋊5D4, Q8⋊6D4, C22.50C24, C22.53C24, C23.348C24

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)2- (1+4)
kernelC23.348C24C4×C4⋊C4C24.C22C23.65C23C24.3C22C23.67C23C23.11D4C23.4Q8C2×C4×Q8C2×C4.4D4C2×Q8C2×C4C22C22
# reps114121221141611

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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