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G = C23.479C24order 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(D43Q8), C23.7Q873C2, C23.Q833C2, C23.8Q872C2, C23.332(C4○D4), C2.16(C233D4), (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, C23.63C2396C2, C23.81C2345C2, 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

C1C23 — C23.479C24
C1C2C22C23C24C23×C4C22×C4⋊C4 — C23.479C24
C1C23 — C23.479C24
C1C23 — C23.479C24
C1C23 — 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 [×3], C2 [×4], C2 [×6], C4 [×18], C22 [×3], C22 [×8], C22 [×22], C2×C4 [×10], C2×C4 [×54], D4 [×8], C23, C23 [×8], C23 [×10], C42 [×2], C22⋊C4 [×14], C4⋊C4 [×18], C22×C4 [×5], C22×C4 [×12], C22×C4 [×16], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×10], C2×C4⋊C4 [×4], C4×D4 [×4], C23×C4 [×4], C22×D4, C23.7Q8, C23.8Q8 [×2], C23.8Q8 [×2], C23.23D4 [×2], C23.63C23 [×2], C23.Q8 [×2], C23.81C23 [×2], C22×C4⋊C4, C2×C4×D4, C23.479C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C22⋊Q8, C22.19C24, C233D4, C22.47C24 [×2], D43Q8 [×2], C23.479C24

Smallest permutation representation of C23.479C24
On 64 points
Generators in S64
(2 11)(4 9)(5 41)(6 24)(7 43)(8 22)(13 52)(15 50)(17 35)(18 63)(19 33)(20 61)(21 39)(23 37)(25 54)(27 56)(29 60)(31 58)(34 45)(36 47)(38 42)(40 44)(46 62)(48 64)
(1 10)(2 11)(3 12)(4 9)(5 37)(6 38)(7 39)(8 40)(13 52)(14 49)(15 50)(16 51)(17 46)(18 47)(19 48)(20 45)(21 43)(22 44)(23 41)(24 42)(25 54)(26 55)(27 56)(28 53)(29 60)(30 57)(31 58)(32 59)(33 64)(34 61)(35 62)(36 63)
(1 26)(2 27)(3 28)(4 25)(5 23)(6 24)(7 21)(8 22)(9 54)(10 55)(11 56)(12 53)(13 60)(14 57)(15 58)(16 59)(17 62)(18 63)(19 64)(20 61)(29 52)(30 49)(31 50)(32 51)(33 48)(34 45)(35 46)(36 47)(37 41)(38 42)(39 43)(40 44)
(1 12)(2 9)(3 10)(4 11)(5 39)(6 40)(7 37)(8 38)(13 50)(14 51)(15 52)(16 49)(17 48)(18 45)(19 46)(20 47)(21 41)(22 42)(23 43)(24 44)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(33 62)(34 63)(35 64)(36 61)
(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 20 12 47)(2 48 9 17)(3 18 10 45)(4 46 11 19)(5 58 39 29)(6 30 40 59)(7 60 37 31)(8 32 38 57)(13 41 50 21)(14 22 51 42)(15 43 52 23)(16 24 49 44)(25 35 56 64)(26 61 53 36)(27 33 54 62)(28 63 55 34)
(1 50)(2 51)(3 52)(4 49)(5 63)(6 64)(7 61)(8 62)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 47)(42 48)(43 45)(44 46)(53 60)(54 57)(55 58)(56 59)

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

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

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

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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