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G = C42.199D4  order 128 = 27

181st non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C42.199D4, C23.720C24, C24.103C23, C22.4932+ (1+4), C22.3762- (1+4), C42⋊8C4⋊73C2, C23.Q8⋊93C2, C23.4Q8⋊66C2, (C2×C42).732C22, (C22×C4).231C23, C22.452(C22×D4), C23.11D4⋊129C2, C23.10D4.72C2, (C22×D4).297C22, (C22×Q8).235C22, C23.78C23⋊64C2, C2.72(C22.29C24), C2.C42.423C22, C2.52(C22.56C24), C2.69(C23.38C23), C2.59(C22.31C24), C2.59(C22.57C24), (C2×C4).437(C2×D4), (C2×C42.C2)⋊28C2, (C2×C4⋊C4).529C22, (C2×C4.4D4).35C2, (C2×C22⋊C4).339C22, SmallGroup(128,1552)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C42.199D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×D4 — C23.10D4 — C42.199D4
Lower central
C1 — C23 — C42.199D4
Upper central
C1 — C23 — C42.199D4
Jennings
C1 — C23 — C42.199D4

Subgroups: 484 in 232 conjugacy classes, 92 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×16], C22 [×3], C22 [×4], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], Q8 [×4], C23, C23 [×14], C42 [×4], C22⋊C4 [×16], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×10], C2×D4 [×4], C2×Q8 [×4], C24 [×2], C2.C42 [×8], C2×C42, C2×C22⋊C4 [×12], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C4.4D4 [×4], C42.C2 [×4], C22×D4, C22×Q8, C42⋊8C4, C23.10D4 [×2], C23.78C23 [×2], C23.Q8 [×4], C23.11D4 [×2], C23.4Q8 [×2], C2×C4.4D4, C2×C42.C2, C42.199D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×3], 2- (1+4) [×3], C22.29C24, C23.38C23, C22.31C24, C22.56C24 [×2], C22.57C24 [×2], C42.199D4

Generators and relations
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=b2c-1 >

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

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

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

Matrix representation ►G ⊆ GL10(ℤ)

1000000000
0100000000
0000100000
0000010000
0010000000
0001000000
000000000-1
0000000010
0000000-100
0000001000
,
-1000000000
0-100000000
000-1000000
0010000000
00000-10000
0000100000
0000000100
0000001000
000000000-1
00000000-10
,
0100000000
-1000000000
0000010000
0000100000
0001000000
0010000000
0000000001
0000000010
0000000100
0000001000
,
0100000000
1000000000
0000100000
0000010000
00-10000000
000-1000000
0000000010
0000000001
0000001000
0000000100

G:=sub<GL(10,Integers())| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0],[0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

Character table of C42.199D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11111111884444448888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-111-1-1-1-11111-1-111    linear of order 2
ρ311111111-1-1111111-1-111-1-111-1-1    linear of order 2
ρ41111111111-1-111-1-11111-1-1-1-1-1-1    linear of order 2
ρ511111111-1-1-1-111-1-111-1-11111-1-1    linear of order 2
ρ61111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ71111111111-1-111-1-1-1-1-1-1-1-11111    linear of order 2
ρ811111111-1-111111111-1-1-1-1-1-111    linear of order 2
ρ911111111-1111-1-1-1-1-11-11-111-11-1    linear of order 2
ρ10111111111-1-1-1-1-1111-1-11-11-111-1    linear of order 2
ρ11111111111-111-1-1-1-11-1-111-11-1-11    linear of order 2
ρ1211111111-11-1-1-1-111-11-111-1-11-11    linear of order 2
ρ13111111111-1-1-1-1-111-111-1-111-1-11    linear of order 2
ρ1411111111-1111-1-1-1-11-11-1-11-11-11    linear of order 2
ρ1511111111-11-1-1-1-1111-11-11-11-11-1    linear of order 2
ρ16111111111-111-1-1-1-1-111-11-1-111-1    linear of order 2
ρ172-22-22-22-200-222-2-220000000000    orthogonal lifted from D4
ρ182-22-22-22-200-22-222-20000000000    orthogonal lifted from D4
ρ192-22-22-22-2002-2-22-220000000000    orthogonal lifted from D4
ρ202-22-22-22-2002-22-22-20000000000    orthogonal lifted from D4
ρ214-4-44-444-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-4-4-444-44000000000000000000    orthogonal lifted from 2+ (1+4)
ρ244-444-4-4-44000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ25444-4-44-4-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{199}D_4
 
% in TeX
 
G:=Group("C4^2.199D4");
 
// GroupNames label
 
G:=SmallGroup(128,1552);
 
// by ID
 
G=gap.SmallGroup(128,1552);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723,794,185,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

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