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G = C24.264C23  order 128 = 27

104th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.264C23, C23.332C24, C22.1042- (1+4), C4.29C22≀C2, (C2×D4).284D4, (C2×Q8).221D4, C23.161(C2×D4), (C22×C4).376D4, C2.20(D4⋊6D4), C2.16(Q8⋊5D4), C23.8Q8⋊36C2, C23.10D4⋊21C2, (C22×C4).799C23, (C23×C4).345C22, (C2×C42).478C22, C22.212(C22×D4), C24.3C22⋊36C2, (C22×D4).507C22, (C22×Q8).422C22, C23.67C23⋊36C2, C2.C42.93C22, C2.16(C22.26C24), C2.12(C23.38C23), (C2×C4⋊Q8)⋊5C2, (C2×C4)⋊4(C4○D4), (C2×C22⋊Q8)⋊8C2, (C4×C22⋊C4)⋊56C2, (C2×C4.4D4)⋊7C2, (C2×C4).317(C2×D4), C2.20(C2×C22≀C2), (C2×C4⋊C4).218C22, (C22×C4○D4).11C2, C22.211(C2×C4○D4), (C2×C22⋊C4).122C22, SmallGroup(128,1164)

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

Derived series
C1 — C23 — C24.264C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.264C23
Lower central
C1 — C23 — C24.264C23
Upper central
C1 — C23 — C24.264C23
Jennings
C1 — C23 — C24.264C23

Subgroups: 772 in 418 conjugacy classes, 120 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×30], C2×C4 [×16], C2×C4 [×52], D4 [×24], Q8 [×12], C23, C23 [×6], C23 [×18], C42 [×4], C22⋊C4 [×24], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×14], C22×C4 [×16], C2×D4 [×4], C2×D4 [×16], C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×32], C24, C24 [×2], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×6], C22⋊Q8 [×4], C4.4D4 [×4], C4⋊Q8 [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×12], C4×C22⋊C4, C23.8Q8 [×4], C24.3C22, C23.67C23, C23.10D4 [×4], C2×C22⋊Q8, C2×C4.4D4, C2×C4⋊Q8, C22×C4○D4, C24.264C23

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], C2×C22≀C2, C22.26C24, C23.38C23, D4⋊6D4 [×2], Q8⋊5D4 [×2], C24.264C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=c, eae-1=gag-1=ab=ba, ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Smallest permutation representation
►On 64 points
Generators in S64
(1 55)(2 54)(3 53)(4 56)(5 45)(6 48)(7 47)(8 46)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 58)(18 57)(19 60)(20 59)(21 38)(22 37)(23 40)(24 39)(33 51)(34 50)(35 49)(36 52)(41 64)(42 63)(43 62)(44 61)

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

040000
400000
004000
000100
000010
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
200000
030000
001000
000100
000040
000001
,
010000
400000
000100
001000
000001
000040
,
010000
400000
004000
000400
000010
000001

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4T4U4V4W4X
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim111111111122224
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D42- (1+4)
kernelC24.264C23C4×C22⋊C4C23.8Q8C24.3C22C23.67C23C23.10D4C2×C22⋊Q8C2×C4.4D4C2×C4⋊Q8C22×C4○D4C22×C4C2×D4C2×Q8C2×C4C22
# reps114114111144482

In GAP, Magma, Sage, TeX 

C_2^4._{264}C_2^3
 
% in TeX
 
G:=Group("C2^4.264C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1164);
 
// by ID
 
G=gap.SmallGroup(128,1164);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,b*d=d*b,g*e*g^-1=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,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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