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G = C23.307C24  order 128 = 27

24th central stem extension by C23 of C24

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

Aliases: C23.307C24, C24.244C23, C22.892- (1+4), C22.1242+ (1+4), (C2×D4)⋊42D4, (C2×Q8)⋊30D4, (C22×C4)⋊20D4, C4.79C22≀C2, C23⋊2(C4○D4), C23⋊2D4⋊5C2, C2.8(Q8⋊5D4), C2.6(Q8⋊6D4), C23.150(C2×D4), C2.14(D4⋊5D4), C2.10(D4⋊6D4), C23.10D4⋊5C2, C23.7Q8⋊30C2, C23.8Q8⋊24C2, C23.23D4⋊24C2, (C2×C42).460C22, (C22×C4).788C23, (C23×C4).327C22, C22.187(C22×D4), C24.3C22⋊25C2, (C22×D4).500C22, (C22×Q8).414C22, C23.67C23⋊27C2, C2.18(C22.19C24), C2.C42.75C22, C2.6(C22.31C24), (C2×C4×D4)⋊19C2, (C2×C4)⋊2(C4○D4), (C2×C4⋊D4)⋊2C2, (C2×C22⋊Q8)⋊2C2, (C2×C4).303(C2×D4), (C22×C4○D4)⋊3C2, C2.14(C2×C22≀C2), (C2×C4⋊C4).202C22, C22.186(C2×C4○D4), (C2×C22⋊C4).106C22, SmallGroup(128,1139)

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

Derived series
C1 — C23 — C23.307C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C23.307C24
Lower central
C1 — C23 — C23.307C24
Upper central
C1 — C23 — C23.307C24
Jennings
C1 — C23 — C23.307C24

Subgroups: 884 in 454 conjugacy classes, 120 normal (42 characteristic)
C1, C2 [×7], C2 [×8], C4 [×4], C4 [×14], C22 [×7], C22 [×40], C2×C4 [×14], C2×C4 [×54], D4 [×40], Q8 [×8], C23, C23 [×8], C23 [×24], C42 [×2], C22⋊C4 [×20], C4⋊C4 [×10], C22×C4 [×5], C22×C4 [×10], C22×C4 [×22], C2×D4 [×4], C2×D4 [×32], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24 [×2], C24 [×2], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C4×D4 [×4], C4⋊D4 [×4], C22⋊Q8 [×4], C23×C4 [×2], C23×C4 [×2], C22×D4 [×2], C22×D4 [×4], C22×Q8, C2×C4○D4 [×12], C23.7Q8, C23.8Q8 [×2], C23.23D4 [×2], C24.3C22, C23.67C23, C23⋊2D4 [×2], C23.10D4 [×2], C2×C4×D4, C2×C4⋊D4, C2×C22⋊Q8, C22×C4○D4, C23.307C24

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22≀C2 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ (1+4), 2- (1+4), C2×C22≀C2, C22.19C24, C22.31C24, D4⋊5D4, D4⋊6D4, Q8⋊5D4, Q8⋊6D4, C23.307C24

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000400
000043
000001
,
030000
200000
004000
000100
000024
000033
,
010000
100000
000400
004000
000040
000004
,
400000
040000
001000
000100
000043
000011

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

38 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4R4S4T4U4V
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim1111111111112222244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D42+ (1+4)2- (1+4)
kernelC23.307C24C23.7Q8C23.8Q8C23.23D4C24.3C22C23.67C23C23⋊2D4C23.10D4C2×C4×D4C2×C4⋊D4C2×C22⋊Q8C22×C4○D4C22×C4C2×D4C2×Q8C2×C4C23C22C22
# reps1122112211114444411

In GAP, Magma, Sage, TeX 

C_2^3._{307}C_2^4
 
% in TeX
 
G:=Group("C2^3.307C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1139);
 
// by ID
 
G=gap.SmallGroup(128,1139);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=a,a*b=b*a,a*c=c*a,e*d*e=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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