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G = C42.112D4  order 128 = 27

94th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.112D4, (C2×C8)⋊25D4, C4.14(C4×D4), C4⋊1D4⋊13C4, C42⋊8C4⋊6C2, C2.3(C8⋊3D4), C4.77(C4⋊D4), C42.156(C2×C4), (C22×C4).298D4, C23.800(C2×D4), C22.36(C4⋊1D4), C22.95(C8⋊C22), (C2×C42).317C22, (C22×C8).405C22, (C22×D4).44C22, (C22×C4).1405C23, C22.67(C4.4D4), C2.22(C23.37D4), C2.3(C42.29C22), C2.13(C24.3C22), (C2×C8⋊C4)⋊28C2, (C2×C4).739(C2×D4), (C2×C4⋊1D4).6C2, (C2×D4⋊C4)⋊46C2, (C2×D4).106(C2×C4), (C2×C4⋊C4).88C22, (C2×C4).596(C4○D4), (C2×C4).419(C22×C4), (C2×C4).138(C22⋊C4), C22.283(C2×C22⋊C4), SmallGroup(128,693)

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

Derived series
C1 — C2×C4 — C42.112D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C8⋊C4 — C42.112D4
Lower central
C1 — C2 — C2×C4 — C42.112D4
Upper central
C1 — C23 — C2×C42 — C42.112D4
Jennings
C1 — C2 — C2 — C22×C4 — C42.112D4

Generators and relations for C42.112D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, cbc-1=dbd=b-1, dcd=bc-1 >

Subgroups: 548 in 206 conjugacy classes, 64 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C8⋊C4, D4⋊C4, C2×C42, C2×C4⋊C4, C4⋊1D4, C4⋊1D4, C22×C8, C22×D4, C22×D4, C42⋊8C4, C2×C8⋊C4, C2×D4⋊C4, C2×C4⋊1D4, C42.112D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C8⋊C22, C24.3C22, C23.37D4, C42.29C22, C8⋊3D4, C42.112D4

Smallest permutation representation of C42.112D4
►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 52 12 38)(2 49 9 39)(3 50 10 40)(4 51 11 37)(5 48 23 44)(6 45 24 41)(7 46 21 42)(8 47 22 43)(13 29 25 35)(14 30 26 36)(15 31 27 33)(16 32 28 34)(17 54 62 57)(18 55 63 58)(19 56 64 59)(20 53 61 60)

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L8A···8H
order12···222224444444444448···8
size11···188882222444488884···4

32 irreducible representations

dim11111122224
type+++++++++
imageC1C2C2C2C2C4D4D4D4C4○D4C8⋊C22
kernelC42.112D4C42⋊8C4C2×C8⋊C4C2×D4⋊C4C2×C4⋊1D4C4⋊1D4C42C2×C8C22×C4C2×C4C22
# reps11141824244

Matrix representation of C42.112D4 ►in GL8(𝔽17)

160000000
016000000
00010000
001600000
00000010
00000001
00001000
00000100
,
10000000
01000000
001600000
000160000
000001600
00001000
000000016
00000010
,
415000000
013000000
00100000
000160000
00000033
000000314
0000141400
000014300
,
160000000
131000000
000160000
001600000
00000010
000000016
00001000
000001600

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[4,0,0,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,0,0,0,0,3,14,0,0],[16,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0] >;
 

C42.112D4 in GAP, Magma, Sage, TeX 

C_4^2._{112}D_4
 
% in TeX
 
G:=Group("C4^2.112D4");
 
// GroupNames label
 
G:=SmallGroup(128,693);
 
// by ID
 
G=gap.SmallGroup(128,693);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,723,436,2019,248]);
 
// 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^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=b*c^-1>;
 
// generators/relations
 

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