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G = C24.124D4order 128 = 27

79th non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.124D4, C4.62+ 1+4, C87D44C2, D4⋊D42C2, C88D426C2, (C2×D8)⋊4C22, C2.D87C22, D4.7D42C2, C8.18D44C2, (C2×Q16)⋊4C22, C4.Q834C22, C4⋊C4.130C23, (C2×C8).153C23, (C2×C4).389C24, C22.4(C4○D8), (C22×C4).171D4, C23.273(C2×D4), D4⋊C444C22, Q8⋊C447C22, (C2×SD16)⋊41C22, (C2×D4).141C23, C23.20D42C2, C23.19D42C2, (C2×Q8).129C23, C22.19C2410C2, C4⋊D4.182C22, C2.70(C233D4), C22⋊C8.191C22, (C23×C4).569C22, (C22×C8).150C22, C22.649(C22×D4), C22⋊Q8.187C22, C2.50(D8⋊C22), (C22×C4).1067C23, C42⋊C2.151C22, C2.40(C2×C4○D8), (C2×C22⋊C8)⋊30C2, (C2×C4).706(C2×D4), (C2×C4○D4).162C22, SmallGroup(128,1923)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.124D4
C1C2C4C2×C4C22×C4C2×C4○D4C22.19C24 — C24.124D4
C1C2C2×C4 — C24.124D4
C1C22C23×C4 — C24.124D4
C1C2C2C2×C4 — C24.124D4

Generators and relations for C24.124D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, faf=ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Subgroups: 452 in 212 conjugacy classes, 86 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×19], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], D8, SD16 [×2], Q16, C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×4], C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C23×C4, C2×C4○D4 [×2], C2×C22⋊C8, D4⋊D4 [×2], D4.7D4 [×2], C88D4 [×2], C87D4, C8.18D4, C23.19D4 [×2], C23.20D4 [×2], C22.19C24 [×2], C24.124D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C4○D8 [×2], C22×D4, 2+ 1+4 [×2], C233D4, C2×C4○D8, D8⋊C22, C24.124D4

Smallest permutation representation of C24.124D4
On 32 points
Generators in S32
(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 27)(3 7)(4 29)(6 31)(8 25)(9 22)(11 24)(13 18)(15 20)(26 30)(28 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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)
(1 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)

G:=sub<Sym(32)| (9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,27)(3,7)(4,29)(6,31)(8,25)(9,22)(11,24)(13,18)(15,20)(26,30)(28,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28)>;

G:=Group( (9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,27)(3,7)(4,29)(6,31)(8,25)(9,22)(11,24)(13,18)(15,20)(26,30)(28,32), (1,30)(2,31)(3,32)(4,25)(5,26)(6,27)(7,28)(8,29)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (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), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,27)(18,26)(19,25)(20,32)(21,31)(22,30)(23,29)(24,28) );

G=PermutationGroup([(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,27),(3,7),(4,29),(6,31),(8,25),(9,22),(11,24),(13,18),(15,20),(26,30),(28,32)], [(1,30),(2,31),(3,32),(4,25),(5,26),(6,27),(7,28),(8,29),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(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)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,27),(18,26),(19,25),(20,32),(21,31),(22,30),(23,29),(24,28)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4N8A···8H
order12222222224···44···48···8
size11112244882···28···84···4

32 irreducible representations

dim111111111122244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D82+ 1+4D8⋊C22
kernelC24.124D4C2×C22⋊C8D4⋊D4D4.7D4C88D4C87D4C8.18D4C23.19D4C23.20D4C22.19C24C22×C4C24C22C4C2
# reps112221122231822

Matrix representation of C24.124D4 in GL6(𝔽17)

1600000
0160000
001000
0001600
0000160
000001
,
1600000
010000
001000
000100
0001160
0010016
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
800000
0150000
0010015
0001150
0001160
0010016
,
020000
900000
0001150
0010015
0000016
0000160

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,15,16,0,0,0,15,0,0,16],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,15,0,0,16,0,0,0,15,16,0] >;

C24.124D4 in GAP, Magma, Sage, TeX

C_2^4._{124}D_4
% in TeX

G:=Group("C2^4.124D4");
// GroupNames label

G:=SmallGroup(128,1923);
// by ID

G=gap.SmallGroup(128,1923);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,248,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^3>;
// generators/relations

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