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G = C24.405C23order 128 = 27

245th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.405C23, C23.598C24, C22.3722+ 1+4, C22.2772- 1+4, C22⋊C4.16D4, C428C455C2, C23.213(C2×D4), C2.62(D46D4), C2.103(D45D4), (C22×C4).875C23, (C23×C4).460C22, (C2×C42).650C22, C22.407(C22×D4), C23.8Q8.47C2, C23.34D4.27C2, C23.11D4.35C2, (C22×Q8).185C22, C23.81C2386C2, C23.67C2381C2, C23.83C2381C2, C23.78C2345C2, C24.C22.50C2, C23.65C23121C2, C23.63C23135C2, C2.C42.304C22, C2.40(C22.35C24), C2.73(C22.36C24), C2.14(C22.57C24), C2.60(C23.38C23), C2.84(C22.46C24), (C2×C4).100(C2×D4), (C2×C22⋊Q8).44C2, (C2×C4).425(C4○D4), (C2×C4⋊C4).411C22, C22.460(C2×C4○D4), (C2×C422C2).13C2, (C2×C22⋊C4).264C22, SmallGroup(128,1430)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.405C23
C1C2C22C23C24C23×C4C23.34D4 — C24.405C23
C1C23 — C24.405C23
C1C23 — C24.405C23
C1C23 — C24.405C23

Generators and relations for C24.405C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=f2=c, g2=b, faf-1=ab=ba, ac=ca, ad=da, eae-1=abc, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, gfg-1=df=fd, dg=gd >

Subgroups: 420 in 228 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×8], C2×C4 [×42], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×9], C4⋊C4 [×19], C22×C4 [×14], C22×C4 [×5], C2×Q8 [×5], C24, C2.C42 [×14], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×11], C22⋊Q8 [×4], C422C2 [×4], C23×C4, C22×Q8, C23.34D4, C428C4, C23.8Q8, C23.63C23, C24.C22, C23.65C23, C23.67C23, C23.78C23, C23.11D4 [×2], C23.81C23 [×2], C23.83C23, C2×C22⋊Q8, C2×C422C2, C24.405C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ 1+4, 2- 1+4 [×3], C23.38C23, C22.35C24, C22.36C24, D45D4, D46D4, C22.46C24, C22.57C24, C24.405C23

Smallest permutation representation of C24.405C23
On 64 points
Generators in S64
(2 38)(4 40)(5 17)(6 8)(7 19)(10 44)(12 42)(13 45)(14 16)(15 47)(18 20)(22 52)(24 50)(26 56)(28 54)(29 57)(30 32)(31 59)(33 35)(34 63)(36 61)(46 48)(58 60)(62 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 13 3 15)(2 46 4 48)(5 41 7 43)(6 10 8 12)(9 19 11 17)(14 38 16 40)(18 42 20 44)(21 31 23 29)(22 60 24 58)(25 34 27 36)(26 64 28 62)(30 50 32 52)(33 54 35 56)(37 47 39 45)(49 59 51 57)(53 63 55 61)
(1 43 39 11)(2 42 40 10)(3 41 37 9)(4 44 38 12)(5 29 17 57)(6 32 18 60)(7 31 19 59)(8 30 20 58)(13 36 45 61)(14 35 46 64)(15 34 47 63)(16 33 48 62)(21 53 49 25)(22 56 50 28)(23 55 51 27)(24 54 52 26)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A···4P4Q···4V
order12···2224···44···4
size11···1444···48···8

32 irreducible representations

dim111111111111112244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC24.405C23C23.34D4C428C4C23.8Q8C23.63C23C24.C22C23.65C23C23.67C23C23.78C23C23.11D4C23.81C23C23.83C23C2×C22⋊Q8C2×C422C2C22⋊C4C2×C4C22C22
# reps111111111221114813

Matrix representation of C24.405C23 in GL6(𝔽5)

100000
010000
001000
000400
000010
000034
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
000200
003000
000022
000003
,
010000
100000
000100
001000
000030
000003
,
100000
040000
003000
000300
000040
000021

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

C24.405C23 in GAP, Magma, Sage, TeX

C_2^4._{405}C_2^3
% in TeX

G:=Group("C2^4.405C2^3");
// GroupNames label

G:=SmallGroup(128,1430);
// by ID

G=gap.SmallGroup(128,1430);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,232,758,723,100,1571,346]);
// Polycyclic

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

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