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G = C24.542C23order 128 = 27

23rd non-split extension by C24 of C23 acting via C23/C22=C2

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

Aliases: C24.542C23, C23.193C24, C22.322+ 1+4, C22.192- 1+4, C23.364(C2×D4), (C22×C4).360D4, C23.23D44C2, C22.84(C23×C4), C23.80(C22×C4), (C23×C4).40C22, C23.7Q811C2, C22.87(C22×D4), (C22×C4).458C23, (C2×C42).406C22, C24.3C2212C2, C2.2(C22.29C24), (C22×D4).474C22, (C22×Q8).396C22, C23.67C2312C2, C2.C42.32C22, C2.2(C23.38C23), C2.1(C22.31C24), C2.5(C23.33C23), (C2×C4○D4)⋊14C4, (C2×D4)⋊40(C2×C4), (C22×C4⋊C4)⋊7C2, (C2×Q8)⋊33(C2×C4), (C2×C4)⋊7(C22⋊C4), (C22×C4)⋊20(C2×C4), C4.65(C2×C22⋊C4), (C2×C42⋊C2)⋊9C2, (C2×C4).1392(C2×D4), (C22×C4○D4).7C2, (C2×C4⋊C4).805C22, (C2×C4).216(C22×C4), C22.16(C2×C22⋊C4), C2.15(C22×C22⋊C4), (C2×C22⋊C4).22C22, SmallGroup(128,1043)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C24.542C23
C1C2C22C23C22×C4C23×C4C22×C4○D4 — C24.542C23
C1C22 — C24.542C23
C1C23 — C24.542C23
C1C23 — C24.542C23

Generators and relations for C24.542C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=d, g2=b, eae-1=faf=ab=ba, ac=ca, ad=da, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 780 in 440 conjugacy classes, 180 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×8], C4 [×12], C22 [×3], C22 [×8], C22 [×32], C2×C4 [×32], C2×C4 [×52], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×16], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×30], C22×C4 [×16], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×6], C2×C4⋊C4 [×4], C42⋊C2 [×4], C23×C4, C23×C4 [×4], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C23.7Q8 [×4], C23.23D4 [×4], C24.3C22 [×2], C23.67C23 [×2], C22×C4⋊C4, C2×C42⋊C2, C22×C4○D4, C24.542C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C22×C22⋊C4, C23.33C23 [×2], C22.29C24, C23.38C23, C22.31C24 [×2], C24.542C23

Smallest permutation representation of C24.542C23
On 64 points
Generators in S64
(1 39)(2 7)(3 37)(4 5)(6 11)(8 9)(10 38)(12 40)(13 45)(14 18)(15 47)(16 20)(17 41)(19 43)(21 35)(22 62)(23 33)(24 64)(25 57)(26 30)(27 59)(28 32)(29 53)(31 55)(34 52)(36 50)(42 46)(44 48)(49 61)(51 63)(54 58)(56 60)
(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 47)(2 60)(3 45)(4 58)(5 26)(6 15)(7 28)(8 13)(9 17)(10 30)(11 19)(12 32)(14 36)(16 34)(18 22)(20 24)(21 29)(23 31)(25 35)(27 33)(37 41)(38 54)(39 43)(40 56)(42 62)(44 64)(46 50)(48 52)(49 57)(51 59)(53 61)(55 63)
(1 15 11 43)(2 44 12 16)(3 13 9 41)(4 42 10 14)(5 46 38 18)(6 19 39 47)(7 48 40 20)(8 17 37 45)(21 53 49 25)(22 26 50 54)(23 55 51 27)(24 28 52 56)(29 61 57 35)(30 36 58 62)(31 63 59 33)(32 34 60 64)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB
order12···2222222224···44···4
size11···1222244442···24···4

44 irreducible representations

dim111111111244
type++++++++++-
imageC1C2C2C2C2C2C2C2C4D42+ 1+42- 1+4
kernelC24.542C23C23.7Q8C23.23D4C24.3C22C23.67C23C22×C4⋊C4C2×C42⋊C2C22×C4○D4C2×C4○D4C22×C4C22C22
# reps1442211116822

Matrix representation of C24.542C23 in GL8(𝔽5)

40000000
04000000
00400000
00040000
00000200
00003000
00000003
00000020
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
40000000
00010000
00100000
00000010
00000001
00004000
00000400
,
40000000
01000000
00100000
00040000
00000100
00001000
00000004
00000040
,
10000000
01000000
00400000
00040000
00002000
00000200
00000030
00000003

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3] >;

C24.542C23 in GAP, Magma, Sage, TeX

C_2^4._{542}C_2^3
% in TeX

G:=Group("C2^4.542C2^3");
// GroupNames label

G:=SmallGroup(128,1043);
// by ID

G=gap.SmallGroup(128,1043);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,675,80]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=d,g^2=b,e*a*e^-1=f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,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=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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