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G = C24.46D4order 128 = 27

1st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.46D4, C22⋊C81C4, (C2×C42)⋊1C4, C22.9C4≀C2, (C2×C4).28C42, (C22×C4).16Q8, C23.11(C4⋊C4), C2.C421C4, (C22×C4).118D4, C2.4(C426C4), C2.3(C4.9C42), C24.4C4.4C2, C22.11(C23⋊C4), (C23×C4).189C22, C23.34D4.2C2, C2.3(C23.9D4), C23.136(C22⋊C4), C22.34(C2.C42), (C2×C4).13(C4⋊C4), (C4×C22⋊C4).8C2, (C22×C4).145(C2×C4), (C2×C4).292(C22⋊C4), SmallGroup(128,16)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.46D4
C1C2C22C23C24C23×C4C4×C22⋊C4 — C24.46D4
C1C2C2×C4 — C24.46D4
C1C22C23×C4 — C24.46D4
C1C2C22C23×C4 — C24.46D4

Generators and relations for C24.46D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=dc=cd, f2=a, ab=ba, ac=ca, eae-1=ad=da, af=fa, bc=cb, bd=db, ebe-1=fbf-1=bcd, ce=ec, cf=fc, de=ed, df=fd, fef-1=abcde3 >

Subgroups: 272 in 113 conjugacy classes, 34 normal (24 characteristic)
C1, C2 [×3], C2 [×5], C4 [×10], C22, C22 [×4], C22 [×11], C8 [×2], C2×C4 [×4], C2×C4 [×22], C23 [×3], C23 [×5], C42 [×2], C22⋊C4 [×6], C2×C8 [×2], M4(2) [×2], C22×C4 [×6], C22×C4 [×6], C24, C2.C42 [×2], C2.C42 [×2], C22⋊C8 [×2], C22⋊C8, C2×C42 [×2], C2×C22⋊C4 [×2], C2×M4(2), C23×C4, C4×C22⋊C4, C23.34D4, C24.4C4, C24.46D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C4≀C2 [×2], C4.9C42, C426C4, C23.9D4, C24.46D4

Smallest permutation representation of C24.46D4
On 32 points
Generators in S32
(2 25)(4 27)(6 29)(8 31)(10 19)(12 21)(14 23)(16 17)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 31)(18 28)(19 25)(20 30)(21 27)(22 32)(23 29)(24 26)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(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)
(2 21 25 12)(3 26)(4 14 27 23)(6 17 29 16)(7 30)(8 10 31 19)(9 13)(11 24)(15 20)(18 22)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G···4O4P4Q4R4S8A8B8C8D
order1222222224···44···444448888
size1111222242···24···488888888

32 irreducible representations

dim1111111222244
type+++++-++
imageC1C2C2C2C4C4C4D4Q8D4C4≀C2C23⋊C4C4.9C42
kernelC24.46D4C4×C22⋊C4C23.34D4C24.4C4C2.C42C22⋊C8C2×C42C22×C4C22×C4C24C22C22C2
# reps1111444211822

Matrix representation of C24.46D4 in GL6(𝔽17)

1600000
0160000
001040
000100
0000160
0000016
,
010000
100000
0016000
002140
0000160
0000161
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1170000
1060000
0013131113
000004
002240
002140
,
400000
0130000
0011101
0001600
0000138
000004

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,4,0,16,0,0,0,0,0,0,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,16,2,0,0,0,0,0,1,0,0,0,0,0,4,16,16,0,0,0,0,0,1],[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],[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],[11,10,0,0,0,0,7,6,0,0,0,0,0,0,13,0,2,2,0,0,13,0,2,1,0,0,11,0,4,4,0,0,13,4,0,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,0,10,0,13,0,0,0,1,0,8,4] >;

C24.46D4 in GAP, Magma, Sage, TeX

C_2^4._{46}D_4
% in TeX

G:=Group("C2^4.46D4");
// GroupNames label

G:=SmallGroup(128,16);
// by ID

G=gap.SmallGroup(128,16);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924]);
// Polycyclic

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

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