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G = C4233D4order 128 = 27

27th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4233D4, C23.718C24, C24.101C23, C22.4912+ 1+4, C425C437C2, C232D448C2, (C2×C42).730C22, (C22×C4).229C23, C22.450(C22×D4), C23.10D4109C2, (C22×D4).295C22, C2.70(C22.29C24), C2.45(C22.54C24), C2.C42.421C22, (C2×C41D4)⋊11C2, (C2×C4).435(C2×D4), (C2×C422C2)⋊27C2, (C2×C4⋊C4).527C22, (C2×C22⋊C4).337C22, SmallGroup(128,1550)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4233D4
C1C2C22C23C22×C4C22×D4C232D4 — C4233D4
C1C23 — C4233D4
C1C23 — C4233D4
C1C23 — C4233D4

Generators and relations for C4233D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 804 in 315 conjugacy classes, 92 normal (9 characteristic)
C1, C2 [×7], C2 [×5], C4 [×13], C22, C22 [×6], C22 [×35], C2×C4 [×6], C2×C4 [×27], D4 [×40], C23, C23 [×35], C42 [×4], C22⋊C4 [×18], C4⋊C4 [×6], C22×C4 [×10], C2×D4 [×36], C24, C24 [×4], C2.C42 [×6], C2×C42, C2×C22⋊C4 [×15], C2×C4⋊C4 [×3], C422C2 [×4], C41D4 [×4], C22×D4 [×10], C425C4, C232D4 [×6], C23.10D4 [×6], C2×C422C2, C2×C41D4, C4233D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×6], C22.29C24 [×3], C22.54C24 [×4], C4233D4

Character table of C4233D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L4M
 size 11111111888884444448888888
ρ111111111111111111111111111    trivial
ρ211111111-111-1-1-1-1-11-1111-1-111-1    linear of order 2
ρ311111111-11111111111-1-1-1-1-1-1-1    linear of order 2
ρ411111111111-1-1-1-1-11-11-1-111-1-11    linear of order 2
ρ51111111111-1-111-11-1-1-1-111-1-11-1    linear of order 2
ρ611111111-11-11-1-11-1-11-1-11-11-111    linear of order 2
ρ711111111-11-1-111-11-1-1-11-1-111-11    linear of order 2
ρ81111111111-11-1-11-1-11-11-11-11-1-1    linear of order 2
ρ911111111-1-1-111-1-1-11-111111-1-1-1    linear of order 2
ρ10111111111-1-1-1-111111111-1-1-1-11    linear of order 2
ρ11111111111-1-111-1-1-11-11-1-1-1-1111    linear of order 2
ρ1211111111-1-1-1-1-1111111-1-11111-1    linear of order 2
ρ1311111111-1-11-11-11-1-11-1-111-11-11    linear of order 2
ρ14111111111-111-11-11-1-1-1-11-111-1-1    linear of order 2
ρ15111111111-11-11-11-1-11-11-1-11-11-1    linear of order 2
ρ1611111111-1-111-11-11-1-1-11-11-1-111    linear of order 2
ρ172-22-22-22-200000-222-2-220000000    orthogonal lifted from D4
ρ182-22-22-22-20000022-22-2-20000000    orthogonal lifted from D4
ρ192-22-22-22-200000-2-2222-20000000    orthogonal lifted from D4
ρ202-22-22-22-2000002-2-2-2220000000    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
ρ234-444-4-4-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ24444-4-44-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ254-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-4-4-444000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of C4233D4
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 43 22 10)(2 44 23 11)(3 41 24 12)(4 42 21 9)(5 25 38 54)(6 26 39 55)(7 27 40 56)(8 28 37 53)(13 64 46 36)(14 61 47 33)(15 62 48 34)(16 63 45 35)(17 31 50 60)(18 32 51 57)(19 29 52 58)(20 30 49 59)
(1 18 6 63)(2 50 7 34)(3 20 8 61)(4 52 5 36)(9 31 54 48)(10 59 55 14)(11 29 56 46)(12 57 53 16)(13 44 58 27)(15 42 60 25)(17 40 62 23)(19 38 64 21)(22 51 39 35)(24 49 37 33)(26 47 43 30)(28 45 41 32)
(1 59)(2 31)(3 57)(4 29)(5 46)(6 14)(7 48)(8 16)(9 50)(10 18)(11 52)(12 20)(13 38)(15 40)(17 42)(19 44)(21 58)(22 30)(23 60)(24 32)(25 62)(26 35)(27 64)(28 33)(34 54)(36 56)(37 45)(39 47)(41 49)(43 51)(53 61)(55 63)

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

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

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

Matrix representation of C4233D4 in GL10(ℤ)

1000000000
0100000000
0010000000
0001000000
0000-100000
00000-10000
000000-1020
0000000-102
000000-1010
0000000-101
,
-1000000000
0-100000000
00-1-2000000
0011000000
0000120000
0000-1-10000
000000-1000
0000000100
00000000-10
0000000001
,
0100000000
-1000000000
0000-1-20000
0000010000
00-1-2000000
0001000000
0000000100
000000-1000
000000010-1
000000-1010
,
0100000000
1000000000
0000100000
0000010000
0010000000
0001000000
0000000100
0000001000
0000000001
0000000010

G:=sub<GL(10,Integers())| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,2,0,1],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1],[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,-2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C4233D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{33}D_4
% in TeX

G:=Group("C4^2:33D4");
// GroupNames label

G:=SmallGroup(128,1550);
// by ID

G=gap.SmallGroup(128,1550);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,794,185,80]);
// Polycyclic

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

Export

Character table of C4233D4 in TeX

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