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G = C42.31D4order 128 = 27

13rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.31D4, C42.12Q8, C23.10C42, C22⋊C8.5C4, C42.6C4.12C2, (C2×C42).140C22, C2.15(M4(2)⋊4C4), C22.58(C2.C42), (C2×C4).26(C4⋊C4), (C22×C4).162(C2×C4), (C2×C4).307(C22⋊C4), SmallGroup(128,40)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.31D4
C1C2C22C2×C4C42C2×C42C42.6C4 — C42.31D4
C1C22C23 — C42.31D4
C1C22C2×C42 — C42.31D4
C1C22C22C2×C42 — C42.31D4

Generators and relations for C42.31D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=a-1b-1c3 >

Subgroups: 112 in 61 conjugacy classes, 30 normal (6 characteristic)
C1, C2 [×3], C2, C4 [×6], C22, C22 [×3], C8 [×6], C2×C4 [×6], C2×C4 [×3], C23, C42, C42 [×3], C2×C8 [×6], C22×C4 [×3], C8⋊C4 [×3], C22⋊C8 [×6], C4⋊C8 [×3], C2×C42, C42.6C4 [×3], C42.31D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, M4(2)⋊4C4 [×3], C42.31D4

Character table of C42.31D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ2111111111111111-1-1-1-1-1-11-11-11    linear of order 2
ρ311111111111111-111-1-1-11-1-1-11-1    linear of order 2
ρ411111111111111-1-1-1111-1-11-1-1-1    linear of order 2
ρ511111-11-11-1-1-1-111-i-ii-i-ii-1i-1i1    linear of order 4
ρ6111111-11-1-1-1-11-1-ii-i-1-11-i-i1iii    linear of order 4
ρ711111-11-11-1-1-1-11-1-i-i-iiii1-i1i-1    linear of order 4
ρ811111-1-1-1-1111-1-1-i-11i-ii-1i-i-i1i    linear of order 4
ρ9111111-11-1-1-1-11-1-i-ii11-1i-i-1i-ii    linear of order 4
ρ1011111-1-1-1-1111-1-1i-11-ii-i-1-iii1-i    linear of order 4
ρ1111111-1-1-1-1111-1-1i1-1i-ii1-i-ii-1-i    linear of order 4
ρ1211111-1-1-1-1111-1-1-i1-1-ii-i1ii-i-1i    linear of order 4
ρ13111111-11-1-1-1-11-1ii-i11-1-ii-1-ii-i    linear of order 4
ρ1411111-11-11-1-1-1-11-1iii-i-i-i1i1-i-1    linear of order 4
ρ15111111-11-1-1-1-11-1i-ii-1-11ii1-i-i-i    linear of order 4
ρ1611111-11-11-1-1-1-111ii-iii-i-1-i-1-i1    linear of order 4
ρ172222-22-22-222-2-22000000000000    orthogonal lifted from D4
ρ182222-22222-2-22-2-2000000000000    orthogonal lifted from D4
ρ192222-2-22-2222-22-2000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2-2-2-2222000000000000    symplectic lifted from Q8, Schur index 2
ρ214-44-4000004i-4i000000000000000    complex lifted from M4(2)⋊4C4
ρ2244-4-4004i0-4i00000000000000000    complex lifted from M4(2)⋊4C4
ρ234-44-400000-4i4i000000000000000    complex lifted from M4(2)⋊4C4
ρ244-4-440-4i04i000000000000000000    complex lifted from M4(2)⋊4C4
ρ254-4-4404i0-4i000000000000000000    complex lifted from M4(2)⋊4C4
ρ2644-4-400-4i04i00000000000000000    complex lifted from M4(2)⋊4C4

Smallest permutation representation of C42.31D4
On 64 points
Generators in S64
(1 38 60 56)(2 53 61 35)(3 40 62 50)(4 55 63 37)(5 34 64 52)(6 49 57 39)(7 36 58 54)(8 51 59 33)(9 44 17 29)(10 26 18 41)(11 46 19 31)(12 28 20 43)(13 48 21 25)(14 30 22 45)(15 42 23 27)(16 32 24 47)
(1 54 64 40)(2 51 57 37)(3 56 58 34)(4 53 59 39)(5 50 60 36)(6 55 61 33)(7 52 62 38)(8 49 63 35)(9 46 21 27)(10 43 22 32)(11 48 23 29)(12 45 24 26)(13 42 17 31)(14 47 18 28)(15 44 19 25)(16 41 20 30)
(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 46 38 19 60 31 56 11)(2 43 53 12 61 28 35 20)(3 25 40 13 62 48 50 21)(4 30 55 22 63 45 37 14)(5 42 34 23 64 27 52 15)(6 47 49 16 57 32 39 24)(7 29 36 9 58 44 54 17)(8 26 51 18 59 41 33 10)

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

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

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

Matrix representation of C42.31D4 in GL8(𝔽17)

013000000
40000000
07040000
1001300000
0000131500
00000400
000069115
000088116
,
01000000
160000000
100010000
071600000
00004000
00000400
00000040
00000004
,
107000000
77000000
697100000
9710100000
0000941416
0000216414
00006151016
00002121016
,
77330000
1163140000
451060000
5310110000
000021610
0000141061
000061041
000001141

G:=sub<GL(8,GF(17))| [0,4,0,10,0,0,0,0,13,0,7,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,6,8,0,0,0,0,15,4,9,8,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,16,10,0,0,0,0,0,1,0,0,7,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[10,7,6,9,0,0,0,0,7,7,9,7,0,0,0,0,0,0,7,10,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,9,2,6,2,0,0,0,0,4,16,15,12,0,0,0,0,14,4,10,10,0,0,0,0,16,14,16,16],[7,11,4,5,0,0,0,0,7,6,5,3,0,0,0,0,3,3,10,10,0,0,0,0,3,14,6,11,0,0,0,0,0,0,0,0,2,14,6,0,0,0,0,0,16,10,10,11,0,0,0,0,1,6,4,4,0,0,0,0,0,1,1,1] >;

C42.31D4 in GAP, Magma, Sage, TeX

C_4^2._{31}D_4
% in TeX

G:=Group("C4^2.31D4");
// GroupNames label

G:=SmallGroup(128,40);
// by ID

G=gap.SmallGroup(128,40);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,360,2804,102]);
// Polycyclic

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

Export

Character table of C42.31D4 in TeX

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