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G = C23.344C24order 128 = 27

61st central stem extension by C23 of C24

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

Aliases: C23.344C24, C24.570C23, C22.1532+ 1+4, (C2×D4).287D4, C23.165(C2×D4), C2.36(D45D4), (C23×C4).76C22, (C22×C4).61C23, C23.8Q839C2, C23.232(C4○D4), C23.10D424C2, C23.23D439C2, C23.11D412C2, C23.34D423C2, (C2×C42).487C22, C22.224(C22×D4), C24.C2238C2, C23.83C237C2, (C22×D4).131C22, C23.63C2335C2, C2.12(C22.32C24), C2.C42.101C22, C22.22(C22.D4), C2.19(C22.47C24), C2.24(C23.36C23), (C2×C4×D4)⋊31C2, (C2×C4).326(C2×D4), (C2×C4⋊D4).29C2, (C2×C4).730(C4○D4), (C2×C4⋊C4).226C22, C22.221(C2×C4○D4), (C2×C2.C42)⋊31C2, C2.22(C2×C22.D4), (C2×C22⋊C4).453C22, SmallGroup(128,1176)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.344C24
C1C2C22C23C24C23×C4C2×C2.C42 — C23.344C24
C1C23 — C23.344C24
C1C23 — C23.344C24
C1C23 — C23.344C24

Generators and relations for C23.344C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=ba=ab, f2=b, ac=ca, ede-1=gdg=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: 628 in 305 conjugacy classes, 104 normal (82 characteristic)
C1, C2 [×7], C2 [×7], C4 [×15], C22 [×7], C22 [×4], C22 [×29], C2×C4 [×6], C2×C4 [×49], D4 [×16], C23, C23 [×8], C23 [×17], C42 [×2], C22⋊C4 [×17], C4⋊C4 [×7], C22×C4 [×12], C22×C4 [×19], C2×D4 [×4], C2×D4 [×15], C24 [×3], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×11], C2×C4⋊C4 [×4], C4×D4 [×4], C4⋊D4 [×4], C23×C4 [×4], C22×D4 [×3], C2×C2.C42, C23.34D4, C23.8Q8, C23.23D4 [×4], C23.63C23, C24.C22, C23.10D4, C23.11D4 [×2], C23.83C23, C2×C4×D4, C2×C4⋊D4, C23.344C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C2×C22.D4, C23.36C23, C22.32C24, D45D4 [×2], C22.47C24 [×2], C23.344C24

Smallest permutation representation of C23.344C24
On 64 points
Generators in S64
(1 41)(2 42)(3 43)(4 44)(5 62)(6 63)(7 64)(8 61)(9 23)(10 24)(11 21)(12 22)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(25 31)(26 32)(27 29)(28 30)(33 39)(34 40)(35 37)(36 38)(53 59)(54 60)(55 57)(56 58)
(1 43)(2 44)(3 41)(4 42)(5 64)(6 61)(7 62)(8 63)(9 21)(10 22)(11 23)(12 24)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(53 57)(54 58)(55 59)(56 60)
(1 57)(2 58)(3 59)(4 60)(5 20)(6 17)(7 18)(8 19)(9 38)(10 39)(11 40)(12 37)(13 27)(14 28)(15 25)(16 26)(21 34)(22 35)(23 36)(24 33)(29 45)(30 46)(31 47)(32 48)(41 55)(42 56)(43 53)(44 54)(49 63)(50 64)(51 61)(52 62)
(1 17)(2 50)(3 19)(4 52)(5 54)(6 57)(7 56)(8 59)(9 15)(10 48)(11 13)(12 46)(14 22)(16 24)(18 42)(20 44)(21 45)(23 47)(25 38)(26 33)(27 40)(28 35)(29 34)(30 37)(31 36)(32 39)(41 49)(43 51)(53 61)(55 63)(58 64)(60 62)
(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 51 43 17)(2 62 44 7)(3 49 41 19)(4 64 42 5)(6 57 61 53)(8 59 63 55)(9 47 21 13)(10 32 22 28)(11 45 23 15)(12 30 24 26)(14 39 48 35)(16 37 46 33)(18 58 52 54)(20 60 50 56)(25 40 29 36)(27 38 31 34)
(1 47)(2 48)(3 45)(4 46)(5 37)(6 38)(7 39)(8 40)(9 17)(10 18)(11 19)(12 20)(13 43)(14 44)(15 41)(16 42)(21 51)(22 52)(23 49)(24 50)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 64)(34 61)(35 62)(36 63)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N4A4B4C4D4E···4T4U4V4W
order12···2222222244444···4444
size11···1222244822224···4888

38 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC23.344C24C2×C2.C42C23.34D4C23.8Q8C23.23D4C23.63C23C24.C22C23.10D4C23.11D4C23.83C23C2×C4×D4C2×C4⋊D4C2×D4C2×C4C23C22
# reps1111411121114882

Matrix representation of C23.344C24 in GL6(𝔽5)

400000
040000
004000
000400
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
010000
000300
002000
000040
000004
,
030000
300000
000400
004000
000022
000013
,
400000
040000
000300
003000
000040
000021
,
010000
100000
000100
001000
000040
000004

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

C23.344C24 in GAP, Magma, Sage, TeX

C_2^3._{344}C_2^4
% in TeX

G:=Group("C2^3.344C2^4");
// GroupNames label

G:=SmallGroup(128,1176);
// by ID

G=gap.SmallGroup(128,1176);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,723,184,675]);
// Polycyclic

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