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G = C24.385C23order 128 = 27

225th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.385C23, C23.577C24, C22.3512+ 1+4, C22.2622- 1+4, C2.41D42, C22⋊C42Q8, C2.26(D4×Q8), C4⋊C4.116D4, C23.31(C2×Q8), C23⋊Q8.19C2, (C23×C4).447C22, (C22×C4).866C23, (C2×C42).636C22, C22.386(C22×D4), C23.8Q8.44C2, C23.4Q8.19C2, C22.142(C22×Q8), (C22×Q8).175C22, C23.78C2338C2, C23.81C2376C2, C24.C22.47C2, C23.63C23127C2, C2.C42.288C22, C2.8(C22.57C24), C2.29(C23.41C23), C2.53(C23.38C23), (C2×C4⋊Q8)⋊21C2, (C2×C4).85(C2×D4), (C2×C4).66(C2×Q8), (C2×C22⋊Q8).41C2, (C2×C4⋊C4).395C22, (C2×C22⋊C4).248C22, SmallGroup(128,1409)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.385C23
C1C2C22C23C22×C4C23×C4C2×C22⋊Q8 — C24.385C23
C1C23 — C24.385C23
C1C23 — C24.385C23
C1C23 — C24.385C23

Generators and relations for C24.385C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=g2=a, ab=ba, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >

Subgroups: 500 in 272 conjugacy classes, 112 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×22], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×16], C2×C4 [×38], Q8 [×12], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×9], C4⋊C4 [×8], C4⋊C4 [×21], C22×C4 [×2], C22×C4 [×12], C22×C4 [×5], C2×Q8 [×15], C24, C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×12], C22⋊Q8 [×8], C4⋊Q8 [×8], C23×C4, C22×Q8, C22×Q8 [×2], C23.8Q8, C23.63C23 [×2], C24.C22 [×2], C23⋊Q8, C23.78C23 [×2], C23.81C23 [×2], C23.4Q8, C2×C22⋊Q8 [×2], C2×C4⋊Q8 [×2], C24.385C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], Q8 [×4], C23 [×15], C2×D4 [×12], C2×Q8 [×6], C24, C22×D4 [×2], C22×Q8, 2+ 1+4, 2- 1+4 [×3], C23.38C23 [×2], C23.41C23, D42, D4×Q8 [×2], C22.57C24, C24.385C23

Smallest permutation representation of C24.385C23
On 64 points
Generators in S64
(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 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 47 3 45)(2 46 4 48)(5 58 7 60)(6 57 8 59)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)(25 36 27 34)(26 35 28 33)(29 40 31 38)(30 39 32 37)(41 51 43 49)(42 50 44 52)(53 63 55 61)(54 62 56 64)
(2 10)(4 12)(5 24)(6 49)(7 22)(8 51)(14 42)(16 44)(17 36)(18 62)(19 34)(20 64)(21 40)(23 38)(26 56)(28 54)(30 60)(32 58)(33 46)(35 48)(37 50)(39 52)(45 61)(47 63)
(1 43 3 41)(2 14 4 16)(5 35 7 33)(6 63 8 61)(9 15 11 13)(10 42 12 44)(17 21 19 23)(18 52 20 50)(22 46 24 48)(25 31 27 29)(26 60 28 58)(30 54 32 56)(34 38 36 40)(37 62 39 64)(45 49 47 51)(53 59 55 57)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim11111111112244
type++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2Q8D42+ 1+42- 1+4
kernelC24.385C23C23.8Q8C23.63C23C24.C22C23⋊Q8C23.78C23C23.81C23C23.4Q8C2×C22⋊Q8C2×C4⋊Q8C22⋊C4C4⋊C4C22C22
# reps11221221224813

Matrix representation of C24.385C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
030000
300000
001200
000400
000040
000004
,
010000
400000
001000
000100
000033
000042
,
100000
010000
001000
004400
000010
000034
,
200000
030000
004000
001100
000040
000004

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

C24.385C23 in GAP, Magma, Sage, TeX

C_2^4._{385}C_2^3
% in TeX

G:=Group("C2^4.385C2^3");
// GroupNames label

G:=SmallGroup(128,1409);
// by ID

G=gap.SmallGroup(128,1409);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,1571,346,80]);
// Polycyclic

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

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