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G = C24.278C23order 128 = 27

118th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.278C23, C23.357C24, C22.1642+ 1+4, C22⋊C439D4, C232D4.4C2, C23.170(C2×D4), C2.43(D45D4), C23.28(C4○D4), (C23×C4).83C22, (C22×C4).62C23, C23.Q814C2, C23.7Q847C2, C23.34D425C2, C23.10D427C2, C23.23D442C2, (C2×C42).500C22, C22.237(C22×D4), C24.C2242C2, C23.83C239C2, (C22×D4).516C22, C23.63C2339C2, C2.29(C22.19C24), C2.16(C22.45C24), C2.C42.114C22, C2.30(C23.36C23), C2.21(C22.47C24), C2.12(C22.34C24), (C2×C4×D4)⋊35C2, (C4×C22⋊C4)⋊60C2, (C2×C4).895(C2×D4), (C2×C4).731(C4○D4), (C2×C4⋊C4).849C22, C22.234(C2×C4○D4), (C2×C22.D4)⋊14C2, (C2×C22⋊C4).134C22, SmallGroup(128,1189)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.278C23
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C24.278C23
C1C23 — C24.278C23
C1C23 — C24.278C23
C1C23 — C24.278C23

Generators and relations for C24.278C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 612 in 298 conjugacy classes, 100 normal (82 characteristic)
C1, C2 [×7], C2 [×6], C4 [×16], C22 [×7], C22 [×30], C2×C4 [×8], C2×C4 [×44], D4 [×16], C23, C23 [×6], C23 [×18], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×17], C4⋊C4 [×9], C22×C4 [×12], C22×C4 [×15], C2×D4 [×16], C24 [×3], C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4 [×5], C4×D4 [×4], C22.D4 [×4], C23×C4 [×3], C22×D4 [×3], C4×C22⋊C4, C23.7Q8, C23.34D4, C23.23D4 [×3], C23.63C23, C24.C22 [×2], C232D4, C23.10D4, C23.Q8, C23.83C23, C2×C4×D4, C2×C22.D4, C24.278C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C23.36C23, C22.34C24, D45D4 [×2], C22.45C24, C22.47C24, C24.278C23

Smallest permutation representation of C24.278C23
On 64 points
Generators in S64
(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 58)(6 59)(7 60)(8 57)(9 49)(10 50)(11 51)(12 52)(13 37)(14 38)(15 39)(16 40)(17 48)(18 45)(19 46)(20 47)(21 42)(22 43)(23 44)(24 41)(29 33)(30 34)(31 35)(32 36)(53 61)(54 62)(55 63)(56 64)
(1 11)(2 12)(3 9)(4 10)(5 54)(6 55)(7 56)(8 53)(13 35)(14 36)(15 33)(16 34)(17 44)(18 41)(19 42)(20 43)(21 46)(22 47)(23 48)(24 45)(25 51)(26 52)(27 49)(28 50)(29 39)(30 40)(31 37)(32 38)(57 61)(58 62)(59 63)(60 64)
(1 63)(2 62)(3 61)(4 64)(5 52)(6 51)(7 50)(8 49)(9 57)(10 60)(11 59)(12 58)(13 47)(14 46)(15 45)(16 48)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 55)(26 54)(27 53)(28 56)(29 41)(30 44)(31 43)(32 42)
(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)
(2 12)(4 10)(5 62)(6 59)(7 64)(8 57)(14 36)(16 34)(17 23)(18 45)(19 21)(20 47)(22 43)(24 41)(26 52)(28 50)(30 40)(32 38)(42 46)(44 48)(53 61)(54 58)(55 63)(56 60)
(1 13 25 37)(2 14 26 38)(3 15 27 39)(4 16 28 40)(5 44 58 23)(6 41 59 24)(7 42 60 21)(8 43 57 22)(9 33 49 29)(10 34 50 30)(11 35 51 31)(12 36 52 32)(17 62 48 54)(18 63 45 55)(19 64 46 56)(20 61 47 53)

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4T4U4V4W4X
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim11111111111112224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC24.278C23C4×C22⋊C4C23.7Q8C23.34D4C23.23D4C23.63C23C24.C22C232D4C23.10D4C23.Q8C23.83C23C2×C4×D4C2×C22.D4C22⋊C4C2×C4C23C22
# reps11113121111114882

Matrix representation of C24.278C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
100000
140000
001000
000100
000013
000004
,
210000
030000
000100
001000
000010
000001
,
100000
010000
001000
000400
000010
000014
,
420000
010000
001000
000100
000020
000002

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

C24.278C23 in GAP, Magma, Sage, TeX

C_2^4._{278}C_2^3
% in TeX

G:=Group("C2^4.278C2^3");
// GroupNames label

G:=SmallGroup(128,1189);
// by ID

G=gap.SmallGroup(128,1189);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675,136]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=a,g^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations

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