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G = C24.457C23  order 128 = 27

297th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.457C23, C23.707C24, C22.3672- (1+4), C22.4802+ (1+4), C22⋊C4.4Q8, C23.47(C2×Q8), C2.70(D4⋊3Q8), (C23×C4).179C22, (C22×C4).613C23, (C2×C42).727C22, C23.4Q8.30C2, C23.8Q8.67C2, C22.169(C22×Q8), C24.C22.81C2, C23.83C23⋊129C2, C23.65C23⋊162C2, C23.81C23⋊131C2, C2.41(C22.54C24), C2.C42.411C22, C2.48(C23.41C23), C2.48(C22.53C24), C2.118(C22.33C24), (C2×C4).93(C2×Q8), (C2×C4).248(C4○D4), (C2×C4⋊C4).517C22, C22.568(C2×C4○D4), (C2×C22⋊C4).330C22, SmallGroup(128,1539)

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

Derived series
C1 — C23 — C24.457C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C24.C22 — C24.457C23
Lower central
C1 — C23 — C24.457C23
Upper central
C1 — C23 — C24.457C23
Jennings
C1 — C23 — C24.457C23

Subgroups: 388 in 204 conjugacy classes, 96 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×18], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×8], C2×C4 [×42], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×7], C4⋊C4 [×18], C22×C4 [×2], C22×C4 [×12], C22×C4 [×3], C24, C2.C42 [×2], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×12], C23×C4, C23.8Q8, C23.8Q8 [×2], C24.C22 [×2], C23.65C23 [×4], C23.81C23 [×2], C23.4Q8 [×2], C23.83C23 [×2], C24.457C23

Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×4], C24, C22×Q8, C2×C4○D4 [×2], 2+ (1+4) [×3], 2- (1+4), C22.33C24 [×2], C23.41C23, D4⋊3Q8 [×2], C22.53C24, C22.54C24, C24.457C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(2 56)(4 54)(5 18)(6 51)(7 20)(8 49)(9 43)(11 41)(13 45)(15 47)(17 64)(19 62)(21 37)(22 35)(23 39)(24 33)(26 29)(28 31)(34 58)(36 60)(38 59)(40 57)(50 61)(52 63)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
001000
003400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
300000
030000
003300
000200
000030
000032
,
010000
400000
003000
000300
000013
000014
,
400000
040000
004400
002100
000042
000041

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim11111112244
type+++++++-+-
imageC1C2C2C2C2C2C2Q8C4○D42+ (1+4)2- (1+4)
kernelC24.457C23C23.8Q8C24.C22C23.65C23C23.81C23C23.4Q8C23.83C23C22⋊C4C2×C4C22C22
# reps13242224831

In GAP, Magma, Sage, TeX 

C_2^4._{457}C_2^3
 
% in TeX
 
G:=Group("C2^4.457C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1539);
 
// by ID
 
G=gap.SmallGroup(128,1539);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,758,723,604,1571,346,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=f^2=b*c*d,g^2=c*b=b*c,e*a*e^-1=g*a*g^-1=a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,b*d=d*b,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,g*e*g^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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