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G = C2×M4(2)⋊C4  order 128 = 27

Direct product of C2 and M4(2)⋊C4

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

Aliases: C2×M4(2)⋊C4, C24.172D4, C8⋊2(C22×C4), C4.4(C22×Q8), (C2×M4(2))⋊7C4, C4.48(C23×C4), C2.D8⋊62C22, C4.Q8⋊42C22, (C22×C4).64Q8, C4⋊C4.349C23, C23.76(C4⋊C4), M4(2)⋊14(C2×C4), (C2×C4).186C24, (C2×C8).243C23, (C22×C4).784D4, C23.642(C2×D4), (C22×C4).905C23, (C22×C8).242C22, (C23×C4).518C22, C22.133(C22×D4), (C22×M4(2)).3C2, C22.106(C8⋊C22), C22.95(C8.C22), C42⋊C2.286C22, (C2×M4(2)).256C22, (C2×C8)⋊7(C2×C4), C4.65(C2×C4⋊C4), (C2×C4.Q8)⋊5C2, (C2×C2.D8)⋊36C2, C2.3(C2×C8⋊C22), (C2×C4).61(C4⋊C4), C22.36(C2×C4⋊C4), C2.25(C22×C4⋊C4), (C2×C4).141(C2×Q8), C2.3(C2×C8.C22), (C2×C4).1409(C2×D4), (C22×C4⋊C4).44C2, (C2×C4⋊C4).904C22, (C22×C4).330(C2×C4), (C2×C4).574(C22×C4), (C2×C42⋊C2).54C2, SmallGroup(128,1642)

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

Derived series
C1 — C4 — C2×M4(2)⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×M4(2)⋊C4
Lower central
C1 — C2 — C4 — C2×M4(2)⋊C4
Upper central
C1 — C23 — C23×C4 — C2×M4(2)⋊C4
Jennings
C1 — C2 — C2 — C2×C4 — C2×M4(2)⋊C4

Generators and relations for C2×M4(2)⋊C4
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, cd=dc >

Subgroups: 412 in 268 conjugacy classes, 180 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C4.Q8, C2.D8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C23×C4, C23×C4, C2×C4.Q8, C2×C2.D8, M4(2)⋊C4, C22×C4⋊C4, C2×C42⋊C2, C22×M4(2), C2×M4(2)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C8⋊C22, C8.C22, C23×C4, C22×D4, C22×Q8, M4(2)⋊C4, C22×C4⋊C4, C2×C8⋊C22, C2×C8.C22, C2×M4(2)⋊C4

Smallest permutation representation of C2×M4(2)⋊C4
►On 64 points
Generators in S64
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4X8A···8H
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim1111111122244
type++++++++-++-
imageC1C2C2C2C2C2C2C4D4Q8D4C8⋊C22C8.C22
kernelC2×M4(2)⋊C4C2×C4.Q8C2×C2.D8M4(2)⋊C4C22×C4⋊C4C2×C42⋊C2C22×M4(2)C2×M4(2)C22×C4C22×C4C24C22C22
# reps12281111634122

Matrix representation of C2×M4(2)⋊C4 ►in GL8(𝔽17)

160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
162000000
161000000
001620000
001610000
000000160
000000016
00000100
000016000
,
160000000
016000000
00100000
00010000
00001000
00000100
000000160
000000016
,
115000000
016000000
00490000
000130000
000011400
00004600
0000001311
000000114

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

C2×M4(2)⋊C4 in GAP, Magma, Sage, TeX 

C_2\times M_4(2)\rtimes C_4
 
% in TeX
 
G:=Group("C2xM4(2):C4");
 
// GroupNames label
 
G:=SmallGroup(128,1642);
 
// by ID
 
G=gap.SmallGroup(128,1642);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2804,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,c*d=d*c>;
 
// generators/relations
 

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