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G = C42.211C23order 128 = 27

72nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.211C23, Q8⋊C818C2, D4⋊C8.8C2, C4⋊C4.33D4, C82Q83C2, (C2×D4).54D4, (C2×Q8).52D4, Q8⋊Q833C2, C4.39(C4○D8), (C4×C8).48C22, C4⋊Q8.31C22, C4.10D820C2, C4⋊C8.170C22, C4.68(C8⋊C22), D4.D4.4C2, (C4×D4).39C22, (C4×Q8).39C22, C2.24(D4⋊D4), C4.41(C8.C22), C22.177C22≀C2, C2.24(D4.7D4), C2.15(D4.10D4), C22.50C24.2C2, (C2×C4).968(C2×D4), SmallGroup(128,382)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C42.211C23
C1C2C22C2×C4C42C4×D4C22.50C24 — C42.211C23
C1C22C42 — C42.211C23
C1C22C42 — C42.211C23
C1C22C22C42 — C42.211C23

Generators and relations for C42.211C23
 G = < a,b,c,d,e | a4=b4=1, c2=d2=b2, e2=a2b2, ab=ba, cac-1=dad-1=a-1, eae-1=ab2, cbc-1=dbd-1=ebe-1=b-1, dcd-1=ac, ece-1=bc, de=ed >

Subgroups: 216 in 100 conjugacy classes, 34 normal (32 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×7], C22, C22 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×8], D4 [×2], Q8 [×6], C23, C42, C42 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×3], SD16 [×2], C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], C4×C8, Q8⋊C4 [×2], C4⋊C8 [×2], C4.Q8, C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C422C2 [×2], C4⋊Q8 [×2], C2×SD16, D4⋊C8, Q8⋊C8, C4.10D8, D4.D4, Q8⋊Q8, C82Q8, C22.50C24, C42.211C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, C4○D8 [×2], C8⋊C22, C8.C22, D4⋊D4, D4.7D4, D4.10D4, C42.211C23

Character table of C42.211C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F8G8H
 size 111182222444448881644448888
ρ111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-111111-1-1-1-1-11111111-1-1-1-1    linear of order 2
ρ41111-111111-1-1-1-1-111-1-1-1-1-11111    linear of order 2
ρ51111111111-11-11-1-1-1-11111-111-1    linear of order 2
ρ61111111111-11-11-1-1-11-1-1-1-11-1-11    linear of order 2
ρ71111-1111111-11-11-1-1-111111-1-11    linear of order 2
ρ81111-1111111-11-11-1-11-1-1-1-1-111-1    linear of order 2
ρ92222-2-2-222-20202000000000000    orthogonal lifted from D4
ρ1022220-2-2-2-22000002-2000000000    orthogonal lifted from D4
ρ1122220-2-2-2-2200000-22000000000    orthogonal lifted from D4
ρ122222022-2-2-22020-200000000000    orthogonal lifted from D4
ρ1322222-2-222-20-20-2000000000000    orthogonal lifted from D4
ρ142222022-2-2-2-20-20200000000000    orthogonal lifted from D4
ρ152-2-2202-20002i0-2i000002-2-22-200--2    complex lifted from C4○D8
ρ162-2-2202-2000-2i02i00000-222-2-200--2    complex lifted from C4○D8
ρ1722-2-2000-22002i0-2i000022-2-20-2--20    complex lifted from C4○D8
ρ1822-2-2000-22002i0-2i0000-2-2220--2-20    complex lifted from C4○D8
ρ1922-2-2000-2200-2i02i000022-2-20--2-20    complex lifted from C4○D8
ρ2022-2-2000-2200-2i02i0000-2-2220-2--20    complex lifted from C4○D8
ρ212-2-2202-2000-2i02i000002-2-22--200-2    complex lifted from C4○D8
ρ222-2-2202-20002i0-2i00000-222-2--200-2    complex lifted from C4○D8
ρ2344-4-40004-400000000000000000    orthogonal lifted from C8⋊C22
ρ244-44-400000000000000-22-220000    symplectic lifted from D4.10D4, Schur index 2
ρ254-4-440-440000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-44-4000000000000002-22-20000    symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of C42.211C23
On 64 points
Generators in S64
(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 17 15 5)(2 18 16 6)(3 19 13 7)(4 20 14 8)(9 62 57 54)(10 63 58 55)(11 64 59 56)(12 61 60 53)(21 32 34 28)(22 29 35 25)(23 30 36 26)(24 31 33 27)(37 48 52 44)(38 45 49 41)(39 46 50 42)(40 47 51 43)
(1 45 15 41)(2 48 16 44)(3 47 13 43)(4 46 14 42)(5 49 17 38)(6 52 18 37)(7 51 19 40)(8 50 20 39)(9 36 57 23)(10 35 58 22)(11 34 59 21)(12 33 60 24)(25 63 29 55)(26 62 30 54)(27 61 31 53)(28 64 32 56)
(1 29 15 25)(2 32 16 28)(3 31 13 27)(4 30 14 26)(5 35 17 22)(6 34 18 21)(7 33 19 24)(8 36 20 23)(9 49 57 38)(10 52 58 37)(11 51 59 40)(12 50 60 39)(41 62 45 54)(42 61 46 53)(43 64 47 56)(44 63 48 55)
(1 4 13 16)(2 15 14 3)(5 20 19 6)(7 18 17 8)(9 53 59 63)(10 62 60 56)(11 55 57 61)(12 64 58 54)(21 22 36 33)(23 24 34 35)(25 26 31 32)(27 28 29 30)(37 41 50 47)(38 46 51 44)(39 43 52 45)(40 48 49 42)

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

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

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

Matrix representation of C42.211C23 in GL4(𝔽17) generated by

0100
16000
0040
00113
,
0100
16000
0010
0001
,
121200
12500
00913
0038
,
0400
4000
0042
00113
,
0100
1000
00130
00013
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,4,1,0,0,0,13],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,9,3,0,0,13,8],[0,4,0,0,4,0,0,0,0,0,4,1,0,0,2,13],[0,1,0,0,1,0,0,0,0,0,13,0,0,0,0,13] >;

C42.211C23 in GAP, Magma, Sage, TeX

C_4^2._{211}C_2^3
% in TeX

G:=Group("C4^2.211C2^3");
// GroupNames label

G:=SmallGroup(128,382);
// by ID

G=gap.SmallGroup(128,382);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,456,422,520,1123,570,521,136,2804,1411,718,172]);
// Polycyclic

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

Export

Character table of C42.211C23 in TeX

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