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G = C24.394C23order 128 = 27

234th non-split extension by C24 of C23 acting via C23/C2=C22

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

Aliases: C24.394C23, C23.587C24, C22.3612+ 1+4, C22.2682- 1+4, C22⋊C4.12D4, C23⋊Q843C2, C23.210(C2×D4), C2.92(D45D4), C23.7Q884C2, C23.170(C4○D4), C23.11D479C2, C23.34D446C2, (C23×C4).453C22, (C22×C4).870C23, (C2×C42).642C22, C23.8Q8101C2, C22.396(C22×D4), C23.23D4.50C2, C23.10D4.40C2, (C22×D4).226C22, (C22×Q8).180C22, C24.C22125C2, C23.81C2379C2, C23.83C2374C2, C23.67C2377C2, C2.C42.294C22, C2.80(C22.46C24), C2.70(C22.36C24), C2.12(C22.56C24), C2.56(C23.38C23), C2.59(C22.33C24), (C2×C4).93(C2×D4), (C2×C22⋊Q8)⋊38C2, (C2×C4).419(C4○D4), (C2×C4⋊C4).401C22, C22.449(C2×C4○D4), (C2×C22⋊C4).254C22, (C2×C22.D4).24C2, SmallGroup(128,1419)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.394C23
C1C2C22C23C24C23×C4C23.34D4 — C24.394C23
C1C23 — C24.394C23
C1C23 — C24.394C23
C1C23 — C24.394C23

Generators and relations for C24.394C23
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=f2=a, e2=b, ab=ba, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg=abd, fg=gf >

Subgroups: 516 in 256 conjugacy classes, 96 normal (82 characteristic)
C1, C2 [×7], C2 [×4], C4 [×16], C22 [×7], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×4], Q8 [×4], C23, C23 [×4], C23 [×12], C42, C22⋊C4 [×4], C22⋊C4 [×14], C4⋊C4 [×13], C22×C4 [×13], C22×C4 [×10], C2×D4 [×5], C2×Q8 [×5], C24 [×2], C2.C42 [×12], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×8], C22⋊Q8 [×4], C22.D4 [×4], C23×C4 [×2], C22×D4, C22×Q8, C23.7Q8, C23.34D4, C23.8Q8, C23.23D4, C24.C22 [×2], C23.67C23, C23⋊Q8, C23.10D4, C23.11D4, C23.81C23 [×2], C23.83C23, C2×C22⋊Q8, C2×C22.D4, C24.394C23
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], 2- 1+4 [×2], C23.38C23, C22.33C24, C22.36C24, D45D4 [×2], C22.46C24, C22.56C24, C24.394C23

Smallest permutation representation of C24.394C23
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 33)(2 34)(3 35)(4 36)(5 43)(6 44)(7 41)(8 42)(9 13)(10 14)(11 15)(12 16)(17 58)(18 59)(19 60)(20 57)(21 54)(22 55)(23 56)(24 53)(25 52)(26 49)(27 50)(28 51)(29 40)(30 37)(31 38)(32 39)(45 62)(46 63)(47 64)(48 61)
(1 59)(2 60)(3 57)(4 58)(5 30)(6 31)(7 32)(8 29)(9 56)(10 53)(11 54)(12 55)(13 23)(14 24)(15 21)(16 22)(17 36)(18 33)(19 34)(20 35)(25 46)(26 47)(27 48)(28 45)(37 43)(38 44)(39 41)(40 42)(49 64)(50 61)(51 62)(52 63)
(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 33 9)(2 16 34 12)(3 15 35 11)(4 14 36 10)(5 52 43 25)(6 51 44 28)(7 50 41 27)(8 49 42 26)(17 53 58 24)(18 56 59 23)(19 55 60 22)(20 54 57 21)(29 64 40 47)(30 63 37 46)(31 62 38 45)(32 61 39 48)
(1 44 3 42)(2 7 4 5)(6 35 8 33)(9 62 11 64)(10 46 12 48)(13 45 15 47)(14 63 16 61)(17 37 19 39)(18 31 20 29)(21 26 23 28)(22 50 24 52)(25 55 27 53)(30 60 32 58)(34 41 36 43)(38 57 40 59)(49 56 51 54)
(1 16)(2 11)(3 14)(4 9)(5 62)(6 48)(7 64)(8 46)(10 35)(12 33)(13 36)(15 34)(17 23)(18 55)(19 21)(20 53)(22 59)(24 57)(25 29)(26 39)(27 31)(28 37)(30 51)(32 49)(38 50)(40 52)(41 47)(42 63)(43 45)(44 61)(54 60)(56 58)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4N4O···4T
order12···222224···44···4
size11···144444···48···8

32 irreducible representations

dim1111111111111122244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+42- 1+4
kernelC24.394C23C23.7Q8C23.34D4C23.8Q8C23.23D4C24.C22C23.67C23C23⋊Q8C23.10D4C23.11D4C23.81C23C23.83C23C2×C22⋊Q8C2×C22.D4C22⋊C4C2×C4C23C22C22
# reps1111121111211144422

Matrix representation of C24.394C23 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
100000
010000
004000
000400
000010
000001
,
040000
100000
004000
000400
000002
000030
,
400000
010000
004000
000100
000020
000002
,
200000
020000
000100
001000
000004
000040
,
010000
100000
001000
000100
000001
000010

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.394C23 in GAP, Magma, Sage, TeX

C_2^4._{394}C_2^3
% in TeX

G:=Group("C2^4.394C2^3");
// GroupNames label

G:=SmallGroup(128,1419);
// by ID

G=gap.SmallGroup(128,1419);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,723,100,1571,346]);
// Polycyclic

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

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