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G = C23.402C24order 128 = 27

119th central stem extension by C23 of C24

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

Aliases: C23.402C24, C24.578C23, C22.1992+ 1+4, C22⋊C416Q8, C429C424C2, (C22×C4).389D4, C23.616(C2×D4), C22.16(C4⋊Q8), C23.120(C2×Q8), C2.21(D43Q8), (C22×C4).79C23, (C23×C4).99C22, C22.87(C22×Q8), (C2×C42).522C22, C22.278(C22×D4), C23.8Q8.21C2, C4.52(C22.D4), (C22×Q8).119C22, C23.81C2326C2, C23.67C2352C2, C2.18(C22.29C24), C2.C42.153C22, C2.10(C2×C4⋊Q8), (C2×C4).40(C2×Q8), (C2×C4).349(C2×D4), (C22×C4⋊C4).37C2, (C4×C22⋊C4).52C2, (C2×C22⋊Q8).32C2, (C2×C4).814(C4○D4), (C2×C4⋊C4).859C22, C22.279(C2×C4○D4), C2.37(C2×C22.D4), (C2×C22⋊C4).501C22, SmallGroup(128,1234)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.402C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.402C24
C1C23 — C23.402C24
C1C23 — C23.402C24
C1C23 — C23.402C24

Generators and relations for C23.402C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=ca=ac, e2=g2=a, f2=b, ab=ba, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 484 in 272 conjugacy classes, 124 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×18], C22, C22 [×10], C22 [×12], C2×C4 [×16], C2×C4 [×50], Q8 [×4], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×22], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C2×Q8 [×6], C24, C2.C42 [×10], C2×C42 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×14], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C23×C4, C23×C4 [×2], C22×Q8, C4×C22⋊C4, C429C4 [×2], C23.8Q8 [×4], C23.67C23 [×2], C23.81C23 [×4], C22×C4⋊C4, C2×C22⋊Q8, C23.402C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C4○D4 [×4], C24, C22.D4 [×4], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C22.D4, C2×C4⋊Q8, C22.29C24, D43Q8 [×4], C23.402C24

Smallest permutation representation of C23.402C24
On 64 points
Generators in S64
(1 45)(2 46)(3 47)(4 48)(5 63)(6 64)(7 61)(8 62)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 49)(42 50)(43 51)(44 52)(53 60)(54 57)(55 58)(56 59)
(1 58)(2 59)(3 60)(4 57)(5 52)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 33)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)(21 37)(22 38)(23 39)(24 40)(41 64)(42 61)(43 62)(44 63)(45 55)(46 56)(47 53)(48 54)
(1 47)(2 48)(3 45)(4 46)(5 61)(6 62)(7 63)(8 64)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 51)(42 52)(43 49)(44 50)(53 58)(54 59)(55 60)(56 57)
(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 31 45 26)(2 27 46 32)(3 29 47 28)(4 25 48 30)(5 9 63 23)(6 24 64 10)(7 11 61 21)(8 22 62 12)(13 55 20 58)(14 59 17 56)(15 53 18 60)(16 57 19 54)(33 51 38 43)(34 44 39 52)(35 49 40 41)(36 42 37 50)
(1 5 58 52)(2 49 59 6)(3 7 60 50)(4 51 57 8)(9 18 34 28)(10 25 35 19)(11 20 36 26)(12 27 33 17)(13 37 31 21)(14 22 32 38)(15 39 29 23)(16 24 30 40)(41 56 64 46)(42 47 61 53)(43 54 62 48)(44 45 63 55)
(1 20 45 13)(2 14 46 17)(3 18 47 15)(4 16 48 19)(5 36 63 37)(6 38 64 33)(7 34 61 39)(8 40 62 35)(9 42 23 50)(10 51 24 43)(11 44 21 52)(12 49 22 41)(25 57 30 54)(26 55 31 58)(27 59 32 56)(28 53 29 60)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4V4W4X4Y4Z
order12···2222244444···44444
size11···1222222224···48888

38 irreducible representations

dim111111112224
type++++++++-++
imageC1C2C2C2C2C2C2C2Q8D4C4○D42+ 1+4
kernelC23.402C24C4×C22⋊C4C429C4C23.8Q8C23.67C23C23.81C23C22×C4⋊C4C2×C22⋊Q8C22⋊C4C22×C4C2×C4C22
# reps112424118482

Matrix representation of C23.402C24 in GL6(𝔽5)

100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000010
000001
,
200000
020000
002000
000300
000001
000040
,
100000
040000
001000
000400
000020
000003
,
010000
100000
000100
004000
000030
000002
,
100000
010000
004000
000400
000020
000003

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

C23.402C24 in GAP, Magma, Sage, TeX

C_2^3._{402}C_2^4
% in TeX

G:=Group("C2^3.402C2^4");
// GroupNames label

G:=SmallGroup(128,1234);
// by ID

G=gap.SmallGroup(128,1234);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,120,758,723,184,675,80]);
// Polycyclic

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

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