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G = C23.358C24  order 128 = 27

75th central stem extension by C23 of C24

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

Aliases: C23.358C24, C24.279C23, C22.1212- (1+4), C22.1652+ (1+4), C4⋊C4.328D4, C23⋊Q8⋊10C2, C2.44(D4⋊5D4), C2.25(Q8⋊5D4), (C23×C4).84C22, (C22×C4).63C23, C23.8Q8⋊47C2, C23.139(C4○D4), (C2×C42).501C22, C23.10D4.4C2, C22.238(C22×D4), C24.C22⋊43C2, (C22×D4).134C22, (C22×Q8).428C22, C23.63C23⋊40C2, C23.81C23⋊15C2, C2.30(C22.19C24), C24.3C22.36C2, C2.C42.115C22, C2.21(C22.46C24), C2.10(C22.53C24), C2.31(C23.36C23), C2.20(C22.36C24), (C2×C4×Q8)⋊18C2, (C4×C22⋊C4)⋊61C2, (C2×C4).336(C2×D4), (C2×C4).855(C4○D4), (C2×C4⋊C4).239C22, C22.235(C2×C4○D4), (C2×C22⋊C4).135C22, (C2×C22.D4).12C2, SmallGroup(128,1190)

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

Derived series
C1 — C23 — C23.358C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C4×C22⋊C4 — C23.358C24
Lower central
C1 — C23 — C23.358C24
Upper central
C1 — C23 — C23.358C24
Jennings
C1 — C23 — C23.358C24

Subgroups: 500 in 263 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×3], C4 [×19], C22 [×7], C22 [×17], C2×C4 [×12], C2×C4 [×37], D4 [×4], Q8 [×8], C23, C23 [×2], C23 [×13], C42 [×7], C22⋊C4 [×21], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×13], C22×C4 [×5], C2×D4 [×5], C2×Q8 [×6], C24 [×2], C2.C42 [×10], C2×C42 [×4], C2×C22⋊C4 [×11], C2×C4⋊C4 [×7], C4×Q8 [×4], C22.D4 [×4], C23×C4, C22×D4, C22×Q8, C4×C22⋊C4, C23.8Q8, C23.63C23 [×2], C24.C22 [×4], C24.3C22, C23⋊Q8 [×2], C23.10D4, C23.81C23, C2×C4×Q8, C2×C22.D4, C23.358C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ (1+4), 2- (1+4), C22.19C24, C23.36C23, C22.36C24, D4⋊5D4, Q8⋊5D4, C22.46C24, C22.53C24, C23.358C24

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=a, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=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 11)(2 12)(3 9)(4 10)(5 27)(6 28)(7 25)(8 26)(13 19)(14 20)(15 17)(16 18)(21 45)(22 46)(23 47)(24 48)(29 62)(30 63)(31 64)(32 61)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000040
000004
,
100000
010000
001000
000100
000040
000004
,
240000
330000
003000
001200
000031
000022
,
400000
040000
001400
002400
000031
000022
,
120000
040000
002300
004300
000012
000004
,
200000
020000
001400
002400
000030
000003

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

38 conjugacy classes

class 1 2A···2G2H2I2J4A···4H4I···4X4Y4Z4AA
order12···22224···44···4444
size11···14482···24···4888

38 irreducible representations

dim1111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.358C24C4×C22⋊C4C23.8Q8C23.63C23C24.C22C24.3C22C23⋊Q8C23.10D4C23.81C23C2×C4×Q8C2×C22.D4C4⋊C4C2×C4C23C22C22
# reps11124121111412411

In GAP, Magma, Sage, TeX 

C_2^3._{358}C_2^4
 
% in TeX
 
G:=Group("C2^3.358C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1190);
 
// by ID
 
G=gap.SmallGroup(128,1190);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,675,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=a,g^2=b,a*b=b*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=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=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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