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G = C24.579C23  order 128 = 27

60th non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.579C23, C23.403C24, C22.1522- (1+4), C42⋊8C4⋊31C2, C23.617(C2×D4), (C22×C4).390D4, C23.311(C4○D4), (C23×C4).100C22, (C2×C42).523C22, (C22×C4).524C23, C22.279(C22×D4), C23.34D4.16C2, C4.53(C22.D4), C22.29(C4.4D4), (C22×Q8).120C22, C23.83C23⋊26C2, C23.67C23⋊53C2, C2.C42.154C22, C2.19(C23.38C23), C2.43(C22.46C24), (C2×C4).832(C2×D4), (C22×C4⋊C4).38C2, (C4×C22⋊C4).53C2, C2.19(C2×C4.4D4), (C2×C22⋊Q8).33C2, (C2×C4).815(C4○D4), (C2×C4⋊C4).270C22, C22.280(C2×C4○D4), C2.38(C2×C22.D4), (C2×C22⋊C4).465C22, SmallGroup(128,1235)

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

Derived series
C1 — C23 — C24.579C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.579C23
Lower central
C1 — C23 — C24.579C23
Upper central
C1 — C23 — C24.579C23
Jennings
C1 — C23 — C24.579C23

Subgroups: 452 in 256 conjugacy classes, 108 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×14], C22, C22 [×10], C22 [×12], C2×C4 [×8], C2×C4 [×54], Q8 [×4], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×16], C22×C4 [×16], C2×Q8 [×6], C24, C2.C42 [×18], C2×C42 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×6], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C23×C4, C23×C4 [×2], C22×Q8, C4×C22⋊C4, C23.34D4 [×4], C42⋊8C4 [×2], C23.67C23 [×2], C23.83C23 [×4], C22×C4⋊C4, C2×C22⋊Q8, C24.579C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C4.4D4 [×4], C22×D4, C2×C4○D4 [×4], 2- (1+4) [×2], C2×C22.D4, C2×C4.4D4, C23.38C23, C22.46C24 [×4], C24.579C23

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
004000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
200000
020000
003000
000200
000012
000004
,
010000
100000
000100
004000
000024
000033
,
400000
010000
001000
000400
000043
000011

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4V4W4X4Y4Z
order12···2222244444···44444
size11···1222222224···48888

38 irreducible representations

dim111111112224
type+++++++++-
imageC1C2C2C2C2C2C2C2D4C4○D4C4○D42- (1+4)
kernelC24.579C23C4×C22⋊C4C23.34D4C42⋊8C4C23.67C23C23.83C23C22×C4⋊C4C2×C22⋊Q8C22×C4C2×C4C23C22
# reps114224114882

In GAP, Magma, Sage, TeX 

C_2^4._{579}C_2^3
 
% in TeX
 
G:=Group("C2^4.579C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1235);
 
// by ID
 
G=gap.SmallGroup(128,1235);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,723,268,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c*b=b*c,g^2=b,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,a*g=g*a,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,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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