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G = C23.352C24  order 128 = 27

69th central stem extension by C23 of C24

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

Aliases: C23.352C24, C24.274C23, C22.1182- (1+4), C22.1602+ (1+4), C4⋊C4⋊22D4, C2.20(D42), (C2×D4)⋊14Q8, C2.15(D4×Q8), C4⋊2(C22⋊Q8), C23⋊Q8⋊9C2, C23.13(C2×Q8), C2.23(Q8⋊5D4), C2.14(D4⋊3Q8), C23.8Q8⋊43C2, C23.Q8⋊11C2, C22.78(C22×Q8), (C23×C4).356C22, (C2×C42).495C22, (C22×C4).807C23, C22.232(C22×D4), (C22×D4).512C22, (C22×Q8).107C22, C23.65C23⋊56C2, C24.3C22.34C2, C2.C42.109C22, C2.18(C22.26C24), C2.19(C22.36C24), (C2×C4⋊Q8)⋊9C2, (C4×C4⋊C4)⋊58C2, (C2×C4×D4).50C2, (C2×C4).54(C2×D4), (C2×C22⋊Q8)⋊13C2, (C2×C4).306(C2×Q8), C2.25(C2×C22⋊Q8), (C2×C4).107(C4○D4), (C2×C4⋊C4).234C22, C22.229(C2×C4○D4), (C2×C22⋊C4).130C22, SmallGroup(128,1184)

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

Derived series
C1 — C23 — C23.352C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C23.352C24
Lower central
C1 — C23 — C23.352C24
Upper central
C1 — C23 — C23.352C24
Jennings
C1 — C23 — C23.352C24

Subgroups: 580 in 310 conjugacy classes, 120 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×18], C22 [×7], C22 [×20], C2×C4 [×18], C2×C4 [×38], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×6], C22⋊C4 [×18], C4⋊C4 [×8], C4⋊C4 [×18], C22×C4 [×7], C22×C4 [×6], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×10], C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×5], C2×C4⋊C4 [×6], C4×D4 [×4], C22⋊Q8 [×8], C4⋊Q8 [×4], C23×C4 [×2], C22×D4, C22×Q8 [×2], C4×C4⋊C4, C23.8Q8 [×2], C23.65C23 [×2], C24.3C22 [×2], C23⋊Q8 [×2], C23.Q8 [×2], C2×C4×D4, C2×C22⋊Q8 [×2], C2×C4⋊Q8, C23.352C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×4], C23 [×15], C2×D4 [×12], C2×Q8 [×6], C4○D4 [×4], C24, C22⋊Q8 [×4], C22×D4 [×2], C22×Q8, C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C2×C22⋊Q8, C22.26C24, C22.36C24, D42, Q8⋊5D4, D4×Q8, D4⋊3Q8, C23.352C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ba=ab, e2=f2=b, g2=a, 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 33)(2 34)(3 35)(4 36)(5 62)(6 63)(7 64)(8 61)(9 14)(10 15)(11 16)(12 13)(17 23)(18 24)(19 21)(20 22)(25 31)(26 32)(27 29)(28 30)(37 51)(38 52)(39 49)(40 50)(41 47)(42 48)(43 45)(44 46)(53 59)(54 60)(55 57)(56 58)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
440000
210000
003000
000200
000040
000041
,
440000
210000
002000
000200
000013
000014
,
300000
420000
000100
004000
000013
000014
,
400000
040000
001000
000100
000013
000014

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim111111111122244
type+++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2D4Q8C4○D42+ (1+4)2- (1+4)
kernelC23.352C24C4×C4⋊C4C23.8Q8C23.65C23C24.3C22C23⋊Q8C23.Q8C2×C4×D4C2×C22⋊Q8C2×C4⋊Q8C4⋊C4C2×D4C2×C4C22C22
# reps112222212184811

In GAP, Magma, Sage, TeX 

C_2^3._{352}C_2^4
 
% in TeX
 
G:=Group("C2^3.352C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1184);
 
// by ID
 
G=gap.SmallGroup(128,1184);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=b*a=a*b,e^2=f^2=b,g^2=a,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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