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G = C23.524C24order 128 = 27

241st central stem extension by C23 of C24

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

Aliases: C23.524C24, C24.365C23, C22.3012+ 1+4, C22.2202- 1+4, C23.195(C2×D4), (C22×C4).402D4, C23.4Q827C2, C23.7Q878C2, C23.10D459C2, C23.11D458C2, (C22×C4).134C23, (C23×C4).426C22, (C2×C42).603C22, C22.349(C22×D4), C24.3C2266C2, C4.96(C22.D4), (C22×D4).195C22, C2.37(C22.29C24), C23.65C23102C2, C2.C42.250C22, C2.45(C22.36C24), C2.25(C22.31C24), C2.25(C22.34C24), (C2×C4).383(C2×D4), (C2×C4⋊D4).40C2, (C2×C42⋊C2)⋊38C2, (C2×C4).658(C4○D4), (C2×C4⋊C4).355C22, C22.396(C2×C4○D4), C2.42(C2×C22.D4), (C2×C22⋊C4).215C22, SmallGroup(128,1356)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.524C24
C1C2C22C23C22×C4C22×D4C24.3C22 — C23.524C24
C1C23 — C23.524C24
C1C23 — C23.524C24
C1C23 — C23.524C24

Generators and relations for C23.524C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=b, eae-1=ab=ba, faf=ac=ca, ad=da, ag=ga, bc=cb, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge >

Subgroups: 580 in 274 conjugacy classes, 100 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×4], C22 [×24], C2×C4 [×8], C2×C4 [×40], D4 [×12], C23, C23 [×2], C23 [×20], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×14], C22×C4 [×2], C22×C4 [×14], C22×C4 [×4], C2×D4 [×14], C24, C24 [×2], C2.C42 [×6], C2×C42 [×2], C2×C22⋊C4 [×14], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C42⋊C2 [×4], C4⋊D4 [×4], C23×C4, C22×D4, C22×D4 [×2], C23.7Q8, C23.65C23 [×2], C24.3C22 [×2], C23.10D4 [×4], C23.11D4 [×2], C23.4Q8 [×2], C2×C42⋊C2, C2×C4⋊D4, C23.524C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×3], 2- 1+4, C2×C22.D4, C22.29C24, C22.31C24, C22.34C24 [×2], C22.36C24 [×2], C23.524C24

Smallest permutation representation of C23.524C24
On 64 points
Generators in S64
(1 47)(2 35)(3 45)(4 33)(5 55)(6 51)(7 53)(8 49)(9 52)(10 54)(11 50)(12 56)(13 57)(14 42)(15 59)(16 44)(17 38)(18 31)(19 40)(20 29)(21 36)(22 46)(23 34)(24 48)(25 41)(26 58)(27 43)(28 60)(30 62)(32 64)(37 61)(39 63)
(1 23)(2 24)(3 21)(4 22)(5 11)(6 12)(7 9)(8 10)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 37)(30 38)(31 39)(32 40)(33 46)(34 47)(35 48)(36 45)(41 57)(42 58)(43 59)(44 60)(49 54)(50 55)(51 56)(52 53)
(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 27)(2 28)(3 25)(4 26)(5 39)(6 40)(7 37)(8 38)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 49)(18 50)(19 51)(20 52)(33 58)(34 59)(35 60)(36 57)(41 45)(42 46)(43 47)(44 48)(53 61)(54 62)(55 63)(56 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)
(2 28)(4 26)(5 11)(6 32)(7 9)(8 30)(10 38)(12 40)(14 22)(16 24)(17 56)(18 61)(19 54)(20 63)(29 37)(31 39)(33 60)(34 36)(35 58)(41 43)(42 48)(44 46)(45 47)(49 64)(50 53)(51 62)(52 55)(57 59)
(1 5 23 11)(2 6 24 12)(3 7 21 9)(4 8 22 10)(13 29 25 37)(14 30 26 38)(15 31 27 39)(16 32 28 40)(17 42 62 58)(18 43 63 59)(19 44 64 60)(20 41 61 57)(33 49 46 54)(34 50 47 55)(35 51 48 56)(36 52 45 53)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O···4T
order12···2222244444···44···4
size11···1448822224···48···8

32 irreducible representations

dim1111111112244
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.524C24C23.7Q8C23.65C23C24.3C22C23.10D4C23.11D4C23.4Q8C2×C42⋊C2C2×C4⋊D4C22×C4C2×C4C22C22
# reps1122422114831

Matrix representation of C23.524C24 in GL8(𝔽5)

01000000
10000000
00400000
00040000
00003100
00002200
00000024
00000033
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
30000000
03000000
00010000
00100000
00000010
00000001
00004000
00000400
,
10000000
04000000
00100000
00040000
00001000
00004400
00000040
00000011
,
10000000
01000000
00100000
00010000
00004300
00001100
00000043
00000011

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

C23.524C24 in GAP, Magma, Sage, TeX

C_2^3._{524}C_2^4
% in TeX

G:=Group("C2^3.524C2^4");
// GroupNames label

G:=SmallGroup(128,1356);
// by ID

G=gap.SmallGroup(128,1356);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,185,80]);
// Polycyclic

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

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