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G = C24.243C23  order 128 = 27

83rd non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.243C23, C23.306C24, C22.882- (1+4), (C2×D4).279D4, C2.9(D4⋊6D4), (C22×C4).371D4, C23.149(C2×D4), C22.16C22≀C2, C23.10D4⋊4C2, C23.8Q8⋊23C2, C23.34D4⋊18C2, C23.23D4⋊23C2, (C22×C4).787C23, (C23×C4).326C22, C22.186(C22×D4), C23.78C23⋊3C2, (C22×D4).499C22, (C22×Q8).413C22, C2.17(C22.19C24), C2.C42.74C22, C2.8(C23.38C23), (C2×C4)⋊9(C4○D4), (C22×C4⋊C4)⋊16C2, (C2×C4).302(C2×D4), C2.13(C2×C22≀C2), (C2×C4⋊C4).201C22, (C22×C4○D4).10C2, C22.185(C2×C4○D4), (C2×C22.D4)⋊3C2, (C2×C22⋊C4).105C22, SmallGroup(128,1138)

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

Derived series
C1 — C23 — C24.243C23
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C24.243C23
Lower central
C1 — C23 — C24.243C23
Upper central
C1 — C23 — C24.243C23
Jennings
C1 — C23 — C24.243C23

Subgroups: 788 in 432 conjugacy classes, 120 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×18], C22, C22 [×10], C22 [×32], C2×C4 [×12], C2×C4 [×66], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×16], C22⋊C4 [×20], C4⋊C4 [×16], C22×C4 [×2], C22×C4 [×14], C22×C4 [×28], C2×D4 [×8], C2×D4 [×14], C2×Q8 [×6], C4○D4 [×32], C24, C24 [×2], C2.C42 [×8], C2×C22⋊C4 [×10], C2×C4⋊C4 [×8], C2×C4⋊C4 [×4], C22.D4 [×8], C23×C4 [×3], C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×12], C23.34D4, C23.8Q8 [×4], C23.23D4 [×2], C23.10D4 [×2], C23.78C23 [×2], C22×C4⋊C4, C2×C22.D4 [×2], C22×C4○D4, C24.243C23

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, C22.19C24, C23.38C23, D4⋊6D4 [×4], C24.243C23

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=g2=1, f2=b, eae=gag=ab=ba, ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, geg=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
(2 52)(4 50)(5 64)(6 8)(7 62)(10 28)(12 26)(13 15)(14 30)(16 32)(17 23)(19 21)(29 31)(33 37)(34 36)(35 39)(38 40)(42 48)(44 46)(53 57)(54 56)(55 59)(58 60)(61 63)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
140000
001000
000400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
210000
230000
004000
000100
000040
000004
,
300000
030000
000100
001000
000001
000010
,
130000
040000
004000
000400
000010
000001

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D4E···4R4S4T4U4V
order12···22222222244444···44444
size11···12222444422224···48888

38 irreducible representations

dim1111111112224
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42- (1+4)
kernelC24.243C23C23.34D4C23.8Q8C23.23D4C23.10D4C23.78C23C22×C4⋊C4C2×C22.D4C22×C4○D4C22×C4C2×D4C2×C4C22
# reps1142221214882

In GAP, Magma, Sage, TeX 

C_2^4._{243}C_2^3
 
% in TeX
 
G:=Group("C2^4.243C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1138);
 
// by ID
 
G=gap.SmallGroup(128,1138);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,100,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=g^2=1,f^2=b,e*a*e=g*a*g=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=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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