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G = C23.392C24order 128 = 27

109th central stem extension by C23 of C24

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

Aliases: C23.392C24, C24.303C23, C22.1922+ 1+4, C22.1442- 1+4, C2.19(D4×Q8), C4⋊C4.230D4, C22⋊C415Q8, C23.17(C2×Q8), C2.63(D45D4), C23⋊Q8.6C2, C2.19(D43Q8), C4.44(C4.4D4), C22.85(C22×Q8), (C2×C42).520C22, (C23×C4).377C22, (C22×C4).827C23, C22.272(C22×D4), C23.7Q8.43C2, (C22×Q8).117C22, C23.67C2350C2, C23.83C2320C2, C24.C22.19C2, C2.C42.144C22, C2.21(C22.50C24), C2.29(C22.36C24), C2.19(C23.37C23), (C4×C4⋊C4)⋊70C2, (C2×C4⋊Q8)⋊11C2, (C2×C4).63(C2×D4), (C2×C4).38(C2×Q8), (C4×C22⋊C4).46C2, C2.18(C2×C4.4D4), (C2×C22⋊Q8).30C2, (C2×C4).123(C4○D4), (C2×C4⋊C4).262C22, C22.269(C2×C4○D4), (C2×C22⋊C4).157C22, SmallGroup(128,1224)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C23.392C24
C1C2C22C23C22×C4C23×C4C4×C22⋊C4 — C23.392C24
C1C23 — C23.392C24
C1C23 — C23.392C24
C1C23 — C23.392C24

Generators and relations for C23.392C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=db=bd, f2=cb=bc, g2=b, ab=ba, eae-1=ac=ca, faf-1=ad=da, ag=ga, fef-1=geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 468 in 254 conjugacy classes, 112 normal (42 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×18], C22 [×7], C22 [×10], C2×C4 [×16], C2×C4 [×38], Q8 [×12], C23, C23 [×2], C23 [×6], C42 [×6], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×6], C22×C4 [×8], C22×C4 [×6], C2×Q8 [×16], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×4], C22⋊Q8 [×4], C4⋊Q8 [×4], C23×C4, C22×Q8, C22×Q8 [×2], C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C24.C22 [×2], C23.67C23 [×2], C23.67C23 [×2], C23⋊Q8 [×2], C23.83C23 [×2], C2×C22⋊Q8, C2×C4⋊Q8, C23.392C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C4.4D4 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ 1+4, 2- 1+4, C2×C4.4D4, C23.37C23, C22.36C24, D45D4, D4×Q8, D43Q8, C22.50C24, C23.392C24

Smallest permutation representation of C23.392C24
On 64 points
Generators in S64
(2 59)(4 57)(5 61)(6 43)(7 63)(8 41)(9 21)(10 38)(11 23)(12 40)(14 32)(16 30)(17 27)(19 25)(22 35)(24 33)(34 37)(36 39)(42 52)(44 50)(46 56)(48 54)(49 62)(51 64)
(1 45)(2 46)(3 47)(4 48)(5 63)(6 64)(7 61)(8 62)(9 23)(10 24)(11 21)(12 22)(13 20)(14 17)(15 18)(16 19)(25 30)(26 31)(27 32)(28 29)(33 38)(34 39)(35 40)(36 37)(41 49)(42 50)(43 51)(44 52)(53 60)(54 57)(55 58)(56 59)
(1 58)(2 59)(3 60)(4 57)(5 52)(6 49)(7 50)(8 51)(9 34)(10 35)(11 36)(12 33)(13 31)(14 32)(15 29)(16 30)(17 27)(18 28)(19 25)(20 26)(21 37)(22 38)(23 39)(24 40)(41 64)(42 61)(43 62)(44 63)(45 55)(46 56)(47 53)(48 54)
(1 47)(2 48)(3 45)(4 46)(5 61)(6 62)(7 63)(8 64)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)(33 40)(34 37)(35 38)(36 39)(41 51)(42 52)(43 49)(44 50)(53 58)(54 59)(55 60)(56 57)
(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 63 55 52)(2 6 56 41)(3 61 53 50)(4 8 54 43)(5 58 44 45)(7 60 42 47)(9 18 39 29)(10 16 40 25)(11 20 37 31)(12 14 38 27)(13 36 26 21)(15 34 28 23)(17 33 32 22)(19 35 30 24)(46 64 59 49)(48 62 57 51)
(1 20 45 13)(2 14 46 17)(3 18 47 15)(4 16 48 19)(5 36 63 37)(6 38 64 33)(7 34 61 39)(8 40 62 35)(9 42 23 50)(10 51 24 43)(11 44 21 52)(12 49 22 41)(25 57 30 54)(26 55 31 58)(27 59 32 56)(28 53 29 60)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4X4Y4Z4AA4AB
order12···2224···44···44444
size11···1442···24···48888

38 irreducible representations

dim111111111122244
type++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2Q8D4C4○D42+ 1+42- 1+4
kernelC23.392C24C4×C22⋊C4C4×C4⋊C4C23.7Q8C24.C22C23.67C23C23⋊Q8C23.83C23C2×C22⋊Q8C2×C4⋊Q8C22⋊C4C4⋊C4C2×C4C22C22
# reps1111242211441211

Matrix representation of C23.392C24 in GL6(𝔽5)

100000
040000
001000
000100
000010
000014
,
400000
040000
004000
000400
000010
000001
,
400000
040000
001000
000100
000010
000001
,
100000
010000
001000
000100
000040
000004
,
010000
400000
002000
000300
000030
000003
,
100000
040000
000100
004000
000013
000004
,
200000
030000
000100
004000
000010
000001

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

C23.392C24 in GAP, Magma, Sage, TeX

C_2^3._{392}C_2^4
% in TeX

G:=Group("C2^3.392C2^4");
// GroupNames label

G:=SmallGroup(128,1224);
// by ID

G=gap.SmallGroup(128,1224);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,560,253,344,758,723,675,80]);
// Polycyclic

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

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