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G = C23.354C24order 128 = 27

71st central stem extension by C23 of C24

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

Aliases: C23.354C24, C24.275C23, C22.1612+ 1+4, C4⋊C4.327D4, (C2×D4).29Q8, C23.14(C2×Q8), C4.55(C22⋊Q8), C2.41(D45D4), C23.4Q88C2, C2.15(Q86D4), C2.15(D43Q8), C23.8Q844C2, C23.Q812C2, C23.7Q846C2, C22.80(C22×Q8), (C22×C4).809C23, (C2×C42).497C22, (C23×C4).357C22, C22.234(C22×D4), (C22×D4).513C22, C23.65C2357C2, C24.3C22.35C2, C2.C42.111C22, C2.11(C22.34C24), C2.29(C23.36C23), (C4×C4⋊C4)⋊59C2, (C2×C4×D4).51C2, (C2×C4).333(C2×D4), (C2×C42.C2)⋊4C2, (C2×C4).307(C2×Q8), C2.27(C2×C22⋊Q8), (C2×C4).109(C4○D4), (C2×C4⋊C4).236C22, C22.231(C2×C4○D4), (C2×C22⋊C4).131C22, SmallGroup(128,1186)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.354C24
C1C2C22C23C22×C4C2×C42C4×C4⋊C4 — C23.354C24
C1C23 — C23.354C24
C1C23 — C23.354C24
C1C23 — C23.354C24

Generators and relations for C23.354C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=c, f2=cb=bc, eae-1=gag=ab=ba, ac=ca, ad=da, faf-1=acd, bd=db, geg=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, gfg=cdf >

Subgroups: 516 in 274 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×16], C22 [×7], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×6], C22⋊C4 [×14], C4⋊C4 [×4], C4⋊C4 [×20], C22×C4 [×7], C22×C4 [×6], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×2], C2.C42 [×4], C2×C42 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×5], C2×C4⋊C4 [×8], C4×D4 [×4], C42.C2 [×4], C23×C4 [×2], C22×D4, C4×C4⋊C4, C23.7Q8 [×2], C23.8Q8 [×2], C23.65C23 [×2], C24.3C22 [×2], C23.Q8 [×2], C23.4Q8 [×2], C2×C4×D4, C2×C42.C2, C23.354C24
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, C23.36C23, C22.34C24, D45D4, Q86D4, D43Q8 [×2], C23.354C24

Smallest permutation representation of C23.354C24
On 64 points
Generators in S64
(2 40)(4 38)(5 61)(6 35)(7 63)(8 33)(9 43)(11 41)(13 59)(14 30)(15 57)(16 32)(17 36)(18 64)(19 34)(20 62)(22 52)(24 50)(25 55)(27 53)(29 47)(31 45)(46 58)(48 60)
(1 39)(2 40)(3 37)(4 38)(5 19)(6 20)(7 17)(8 18)(9 43)(10 44)(11 41)(12 42)(13 47)(14 48)(15 45)(16 46)(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)
(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 49)(2 50)(3 51)(4 52)(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 45 37 13)(2 30 38 58)(3 47 39 15)(4 32 40 60)(5 41 17 9)(6 26 18 54)(7 43 19 11)(8 28 20 56)(10 33 42 62)(12 35 44 64)(14 50 46 22)(16 52 48 24)(21 57 49 29)(23 59 51 31)(25 61 53 36)(27 63 55 34)
(1 43)(2 10)(3 41)(4 12)(5 57)(6 32)(7 59)(8 30)(9 39)(11 37)(13 36)(14 64)(15 34)(16 62)(17 29)(18 60)(19 31)(20 58)(21 55)(22 26)(23 53)(24 28)(25 51)(27 49)(33 48)(35 46)(38 42)(40 44)(45 61)(47 63)(50 54)(52 56)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim11111111112224
type+++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2D4Q8C4○D42+ 1+4
kernelC23.354C24C4×C4⋊C4C23.7Q8C23.8Q8C23.65C23C24.3C22C23.Q8C23.4Q8C2×C4×D4C2×C42.C2C4⋊C4C2×D4C2×C4C22
# reps112222221144122

Matrix representation of C23.354C24 in GL6(𝔽5)

400000
410000
001000
001400
000010
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
130000
140000
002000
000200
000040
000001
,
210000
230000
001300
001400
000001
000010
,
130000
040000
004000
004100
000040
000001

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

C23.354C24 in GAP, Magma, Sage, TeX

C_2^3._{354}C_2^4
% in TeX

G:=Group("C2^3.354C2^4");
// GroupNames label

G:=SmallGroup(128,1186);
// by ID

G=gap.SmallGroup(128,1186);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,232,758,723,268,675,80]);
// Polycyclic

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

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