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G = C23.479C24  order 128 = 27

196th central stem extension by C23 of C24

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

Aliases: C23.479C24, C24.343C23, C22.2622+ 1+4, (C2×D4).32Q8, C23.27(C2×Q8), (C22×C4).394D4, C23.620(C2×D4), C2.39(D4⋊3Q8), C23.7Q8⋊73C2, C23.Q8⋊33C2, C23.8Q8⋊72C2, C23.332(C4○D4), C2.16(C23⋊3D4), (C2×C42).573C22, (C22×C4).546C23, (C23×C4).124C22, C22.320(C22×D4), C22.53(C22⋊Q8), C22.114(C22×Q8), C23.23D4.40C2, (C22×D4).535C22, C23.63C23⋊96C2, C23.81C23⋊45C2, C2.64(C22.19C24), C2.C42.213C22, C2.65(C22.47C24), (C2×C4×D4).66C2, (C2×C4).57(C2×Q8), (C22×C4⋊C4)⋊27C2, (C2×C4).362(C2×D4), C2.37(C2×C22⋊Q8), (C2×C4).397(C4○D4), (C2×C4⋊C4).877C22, C22.355(C2×C4○D4), (C2×C22⋊C4).194C22, SmallGroup(128,1311)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.479C24
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4 — C23.479C24
Lower central
C1 — C23 — C23.479C24
Upper central
C1 — C23 — C23.479C24
Jennings
C1 — C23 — C23.479C24

Generators and relations for C23.479C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=db=bd, f2=d, eae-1=gag=ab=ba, faf-1=ac=ca, ad=da, bc=cb, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 548 in 292 conjugacy classes, 112 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C23.7Q8, C23.8Q8, C23.8Q8, C23.23D4, C23.63C23, C23.Q8, C23.81C23, C22×C4⋊C4, C2×C4×D4, C23.479C24
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, 2+ 1+4, C2×C22⋊Q8, C22.19C24, C23⋊3D4, C22.47C24, D4⋊3Q8, C23.479C24

Smallest permutation representation of C23.479C24
►On 64 points
Generators in S64
(2 46)(4 48)(5 52)(6 41)(7 50)(8 43)(9 39)(10 35)(11 37)(12 33)(13 20)(15 18)(21 36)(22 38)(23 34)(24 40)(26 31)(28 29)(42 61)(44 63)(49 64)(51 62)(54 57)(56 59)

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4T4U4V4W4X
order12···222222244444···44444
size11···122224422224···48888

38 irreducible representations

dim11111111122224
type++++++++++-+
imageC1C2C2C2C2C2C2C2C2D4Q8C4○D4C4○D42+ 1+4
kernelC23.479C24C23.7Q8C23.8Q8C23.23D4C23.63C23C23.Q8C23.81C23C22×C4⋊C4C2×C4×D4C22×C4C2×D4C2×C4C23C22
# reps11422221144842

Matrix representation of C23.479C24 ►in GL6(𝔽5)

100000
010000
001000
002400
000010
000004
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
300000
320000
004000
000400
000003
000030
,
130000
140000
001400
000400
000001
000010
,
100000
010000
004000
000400
000001
000010

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

C23.479C24 in GAP, Magma, Sage, TeX 

C_2^3._{479}C_2^4
 
% in TeX
 
G:=Group("C2^3.479C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1311);
 
// by ID
 
G=gap.SmallGroup(128,1311);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,352,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,e^2=d*b=b*d,f^2=d,e*a*e^-1=g*a*g=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*e=e*b,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^-1=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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