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G = C23.46D4order 64 = 26

17th non-split extension by C23 of D4 acting via D4/C22=C2

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

Aliases: C23.46D4, C22.8SD16, C4.Q89C2, (C2×C4).38D4, C22⋊C810C2, C4⋊D4.6C2, D4⋊C412C2, C4.30(C4○D4), C4⋊C4.64C22, (C2×C8).37C22, C2.12(C2×SD16), C2.17(C8⋊C22), (C2×C4).106C23, (C2×D4).22C22, C22.102(C2×D4), (C22×C4).52C22, C2.12(C22.D4), (C2×C4⋊C4)⋊12C2, SmallGroup(64,162)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.46D4
C1C2C4C2×C4C4⋊C4C2×C4⋊C4 — C23.46D4
C1C2C2×C4 — C23.46D4
C1C22C22×C4 — C23.46D4
C1C2C2C2×C4 — C23.46D4

Generators and relations for C23.46D4
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d4=c, dad-1=eae=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd3 >

Subgroups: 113 in 57 conjugacy classes, 27 normal (15 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C22⋊C8, D4⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C23.46D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, C4○D4 [×2], C22.D4, C2×SD16, C8⋊C22, C23.46D4

Character table of C23.46D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111228224444484444
ρ11111111111111111111    trivial
ρ2111111-11111111-1-1-1-1-1    linear of order 2
ρ3111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ4111111-111-1-1-1-11-11111    linear of order 2
ρ51111-1-111111-1-1-1-1-1-111    linear of order 2
ρ61111-1-1-11111-1-1-1111-1-1    linear of order 2
ρ71111-1-1111-1-111-1-111-1-1    linear of order 2
ρ81111-1-1-111-1-111-11-1-111    linear of order 2
ρ92222-2-20-2-20000200000    orthogonal lifted from D4
ρ102222220-2-20000-200000    orthogonal lifted from D4
ρ112-22-2000-22-2i2i00000000    complex lifted from C4○D4
ρ122-22-20002-200-2i2i000000    complex lifted from C4○D4
ρ132-22-20002-2002i-2i000000    complex lifted from C4○D4
ρ142-22-2000-222i-2i00000000    complex lifted from C4○D4
ρ152-2-22-22000000000-2--2-2--2    complex lifted from SD16
ρ162-2-222-2000000000--2-2-2--2    complex lifted from SD16
ρ172-2-222-2000000000-2--2--2-2    complex lifted from SD16
ρ182-2-22-22000000000--2-2--2-2    complex lifted from SD16
ρ1944-4-4000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C23.46D4
On 32 points
Generators in S32
(1 11)(2 29)(3 13)(4 31)(5 15)(6 25)(7 9)(8 27)(10 22)(12 24)(14 18)(16 20)(17 30)(19 32)(21 26)(23 28)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(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)
(2 18)(3 7)(4 24)(6 22)(8 20)(9 30)(10 16)(11 28)(12 14)(13 26)(15 32)(17 21)(25 27)(29 31)

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

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

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

C23.46D4 is a maximal subgroup of
C24.115D4  C24.183D4  C24.117D4  (C2×D4).301D4  (C2×D4).304D4  C42.225D4  C42.227D4  C42.235D4  C234SD16  C24.121D4  C24.125D4  C24.129D4  C4.2+ 1+4  C4.192+ 1+4  C42.284D4  C42.286D4  C42.291D4
 C4⋊C4.D2p: C24.16D4  C4⋊C4.18D4  C4⋊C4.19D4  C24.18D4  C42.353C23  C42.359C23  C42.423C23  C42.426C23 ...
 C2p.(C2×SD16): C42.223D4  C42.279D4  C23.43D12  C23.38D20  C23.38D28 ...
C23.46D4 is a maximal quotient of
C24.157D4  C4.Q89C4  (C2×C4).24D8  C24.89D4  (C2×C8).169D4  (C2×C4).23Q16  M5(2).C22  C23.10SD16
 C23.D4p: C23.38D8  C23.43D12  C23.38D20  C23.38D28 ...
 C4⋊C4.D2p: C24.159D4  C4.67(C4×D4)  C24.84D4  C2.(C83Q8)  D6.SD16  D6.4SD16  C4⋊C4.228D6  C4⋊D4.S3 ...

Matrix representation of C23.46D4 in GL4(𝔽17) generated by

0100
1000
0010
0001
,
16000
01600
0010
0001
,
1000
0100
00160
00016
,
4000
01300
00125
001212
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,13,0,0,0,0,12,12,0,0,5,12],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C23.46D4 in GAP, Magma, Sage, TeX

C_2^3._{46}D_4
% in TeX

G:=Group("C2^3.46D4");
// GroupNames label

G:=SmallGroup(64,162);
// by ID

G=gap.SmallGroup(64,162);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,194,1444,376,88]);
// Polycyclic

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

Export

Character table of C23.46D4 in TeX

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