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G = C23.556C24order 128 = 27

273rd central stem extension by C23 of C24

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

Aliases: C24.51C23, C23.556C24, C22.3312+ 1+4, C22.2472- 1+4, (C22×C4)⋊38D4, C232D428C2, C23.371(C2×D4), C22.1(C41D4), C23.4Q835C2, C2.31(C233D4), (C22×C4).855C23, (C23×C4).435C22, C22.368(C22×D4), (C22×D4).206C22, C2.C42.559C22, C2.32(C22.31C24), (C2×C4⋊D4)⋊25C2, (C2×C4).402(C2×D4), C2.13(C2×C41D4), (C2×C4⋊C4).380C22, (C2×C2.C42)⋊39C2, (C2×C22⋊C4).236C22, SmallGroup(128,1388)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.556C24
C1C2C22C23C24C22×D4C2×C4⋊D4 — C23.556C24
C1C23 — C23.556C24
C1C23 — C23.556C24
C1C23 — C23.556C24

Generators and relations for C23.556C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=b, e2=c, ab=ba, ac=ca, ede-1=ad=da, ae=ea, gfg=af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, dg=gd, eg=ge >

Subgroups: 916 in 402 conjugacy classes, 116 normal (7 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×16], C22, C22 [×10], C22 [×40], C2×C4 [×12], C2×C4 [×36], D4 [×40], C23, C23 [×6], C23 [×32], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4 [×22], C22×C4 [×6], C2×D4 [×48], C24, C24 [×4], C2.C42 [×4], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C4⋊D4 [×24], C23×C4 [×3], C22×D4 [×10], C2×C2.C42, C232D4 [×4], C23.4Q8 [×4], C2×C4⋊D4 [×6], C23.556C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C22×D4 [×3], 2+ 1+4 [×3], 2- 1+4, C2×C41D4, C233D4 [×3], C22.31C24 [×3], C23.556C24

Smallest permutation representation of C23.556C24
On 64 points
Generators in S64
(1 39)(2 40)(3 37)(4 38)(5 17)(6 18)(7 19)(8 20)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 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 62)(34 63)(35 64)(36 61)
(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 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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 59 51 15)(2 32 52 48)(3 57 49 13)(4 30 50 46)(5 41 36 25)(6 10 33 54)(7 43 34 27)(8 12 35 56)(9 61 53 17)(11 63 55 19)(14 38 58 22)(16 40 60 24)(18 42 62 26)(20 44 64 28)(21 45 37 29)(23 47 39 31)
(2 4)(5 61)(6 64)(7 63)(8 62)(9 41)(10 44)(11 43)(12 42)(13 57)(14 60)(15 59)(16 58)(17 36)(18 35)(19 34)(20 33)(22 24)(25 53)(26 56)(27 55)(28 54)(29 45)(30 48)(31 47)(32 46)(38 40)(50 52)
(1 11)(2 12)(3 9)(4 10)(5 45)(6 46)(7 47)(8 48)(13 17)(14 18)(15 19)(16 20)(21 25)(22 26)(23 27)(24 28)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4L4M4N4O4P
order12···2222222224···44444
size11···1222288884···48888

32 irreducible representations

dim11111244
type+++++++-
imageC1C2C2C2C2D42+ 1+42- 1+4
kernelC23.556C24C2×C2.C42C232D4C23.4Q8C2×C4⋊D4C22×C4C22C22
# reps114461231

Matrix representation of C23.556C24 in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00120000
00440000
00000300
00003000
00000002
00000020
,
43000000
11000000
00430000
00110000
00000010
00000001
00004000
00000400
,
10000000
44000000
00400000
00110000
00001000
00000400
00000040
00000001
,
40000000
04000000
00100000
00010000
00000100
00001000
00000001
00000010

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

C23.556C24 in GAP, Magma, Sage, TeX

C_2^3._{556}C_2^4
% in TeX

G:=Group("C2^3.556C2^4");
// GroupNames label

G:=SmallGroup(128,1388);
// by ID

G=gap.SmallGroup(128,1388);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,185]);
// Polycyclic

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

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