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G = C24.428C23  order 128 = 27

268th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.428C23, C23.642C24, C22.4152+ (1+4), C22.3142- (1+4), C22⋊C4⋊6Q8, C23.38(C2×Q8), C2.54(D4⋊3Q8), C23⋊Q8.22C2, (C23×C4).484C22, (C22×C4).203C23, (C2×C42).688C22, C23.4Q8.23C2, C23.7Q8.69C2, C23.8Q8.54C2, C22.152(C22×Q8), (C22×Q8).205C22, C23.78C23⋊54C2, C23.83C23⋊94C2, C24.C22.59C2, C23.65C23⋊141C2, C23.63C23⋊158C2, C2.23(C22.54C24), C2.94(C22.45C24), C2.C42.346C22, C2.36(C23.41C23), C2.92(C22.36C24), (C2×C4).76(C2×Q8), (C2×C4).443(C4○D4), (C2×C4⋊C4).453C22, C22.503(C2×C4○D4), (C2×C22⋊C4).302C22, SmallGroup(128,1474)

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

Derived series
C1 — C23 — C24.428C23
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C24.428C23
Lower central
C1 — C23 — C24.428C23
Upper central
C1 — C23 — C24.428C23
Jennings
C1 — C23 — C24.428C23

Subgroups: 404 in 208 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], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×5], C4⋊C4 [×15], C22×C4 [×2], C22×C4 [×12], C22×C4 [×5], C2×Q8 [×3], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×10], C23×C4, C22×Q8, C23.7Q8 [×2], C23.8Q8, C23.63C23 [×2], C24.C22 [×2], C23.65C23 [×2], C23⋊Q8, C23.78C23 [×2], C23.4Q8, C23.83C23 [×2], C24.428C23

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), C22.36C24 [×2], C23.41C23, C22.45C24, D4⋊3Q8 [×2], C22.54C24, C24.428C23

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

Smallest permutation representation
►On 64 points
Generators in S64
(2 27)(4 25)(5 39)(6 44)(7 37)(8 42)(9 54)(11 56)(14 57)(16 59)(17 33)(18 45)(19 35)(20 47)(21 41)(22 38)(23 43)(24 40)(30 49)(32 51)(34 63)(36 61)(46 64)(48 62)

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
001000
000400
000010
000024
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
020000
200000
003000
000300
000032
000012
,
010000
400000
000100
001000
000030
000003
,
200000
030000
004000
000400
000040
000031

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

32 conjugacy classes

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

32 irreducible representations

dim11111111112244
type++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2Q8C4○D42+ (1+4)2- (1+4)
kernelC24.428C23C23.7Q8C23.8Q8C23.63C23C24.C22C23.65C23C23⋊Q8C23.78C23C23.4Q8C23.83C23C22⋊C4C2×C4C22C22
# reps12122212124831

In GAP, Magma, Sage, TeX 

C_2^4._{428}C_2^3
 
% in TeX
 
G:=Group("C2^4.428C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1474);
 
// by ID
 
G=gap.SmallGroup(128,1474);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d*b=b*d,f^2=c*b=b*c,g^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,f*a*f^-1=a*d=d*a,a*g=g*a,f*e*f^-1=b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,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>;
 
// generators/relations
 

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