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G = C24.593C23  order 128 = 27

74th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.593C23, C23.546C24, C22.3212+ (1+4), C22.2382- (1+4), (C22×C4).61Q8, C23.101(C2×Q8), C2.9(C23⋊2Q8), C23.246(C4○D4), (C23×C4).142C22, (C22×C4).156C23, C23.8Q8.42C2, C23.7Q8.60C2, C22.5(C42.C2), C22.135(C22×Q8), C23.83C23⋊65C2, C23.81C23⋊66C2, C2.45(C22.32C24), C2.C42.556C22, C2.45(C22.33C24), C2.22(C23.41C23), (C2×C4).132(C2×Q8), C2.19(C2×C42.C2), (C2×C4⋊C4).372C22, C22.418(C2×C4○D4), (C2×C22⋊C4).232C22, (C2×C2.C42).30C2, SmallGroup(128,1378)

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

Derived series
C1 — C23 — C24.593C23
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.8Q8 — C24.593C23
Lower central
C1 — C23 — C24.593C23
Upper central
C1 — C23 — C24.593C23
Jennings
C1 — C23 — C24.593C23

Subgroups: 420 in 218 conjugacy classes, 100 normal (12 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×4], C2×C4 [×52], C23, C23 [×6], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×18], C22×C4 [×10], C24, C2.C42 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C2×C2.C42, C23.7Q8 [×2], C23.8Q8 [×4], C23.81C23 [×4], C23.83C23 [×4], C24.593C23

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C2×C42.C2, C22.32C24 [×2], C22.33C24 [×2], C23⋊2Q8, C23.41C23, C24.593C23

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

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

10000000
01000000
00100000
00010000
00001000
00000400
00000010
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
14000000
24000000
00300000
00030000
00000030
00000003
00002000
00000200
,
14000000
24000000
00320000
00120000
00000010
00000001
00001000
00000100
,
32000000
02000000
00100000
00240000
00000100
00001000
00000001
00000010

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[1,2,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4L4M···4T
order12···222224···44···4
size11···122224···48···8

32 irreducible representations

dim1111112244
type++++++-+-
imageC1C2C2C2C2C2Q8C4○D42+ (1+4)2- (1+4)
kernelC24.593C23C2×C2.C42C23.7Q8C23.8Q8C23.81C23C23.83C23C22×C4C23C22C22
# reps1124444831

In GAP, Magma, Sage, TeX 

C_2^4._{593}C_2^3
 
% in TeX
 
G:=Group("C2^4.593C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1378);
 
// by ID
 
G=gap.SmallGroup(128,1378);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,723,184,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=g^2=c,g*a*g^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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