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G = C2×C22.M4(2)  order 128 = 27

Direct product of C2 and C22.M4(2)

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

Aliases: C2×C22.M4(2), C23.28M4(2), (C22×C4)⋊3C8, C23.29(C2×C8), (C23×C4).13C4, C24.105(C2×C4), (C22×C4).649D4, C22.7(C22×C8), (C23×C4).16C22, C22⋊C8.152C22, C22.26(C22⋊C8), C22.44(C23⋊C4), (C22×C4).421C23, C23.158(C22×C4), C22.10(C2×M4(2)), C23.164(C22⋊C4), C22.17(C4.10D4), (C2×C4)⋊6(C2×C8), (C2×C4⋊C4).28C4, C2.5(C2×C22⋊C8), C2.2(C2×C23⋊C4), (C22×C4⋊C4).7C2, (C2×C22⋊C8).8C2, (C2×C4).1118(C2×D4), C2.1(C2×C4.10D4), (C2×C4⋊C4).731C22, (C22×C4).103(C2×C4), C22.89(C2×C22⋊C4), (C2×C4).162(C22⋊C4), SmallGroup(128,189)

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

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

Generators and relations for C2×C22.M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd5 >

Subgroups: 308 in 164 conjugacy classes, 68 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C4⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C24, C22⋊C8, C22⋊C8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C23×C4, C22.M4(2), C2×C22⋊C8, C22×C4⋊C4, C2×C22.M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, C23⋊C4, C4.10D4, C2×C22⋊C4, C22×C8, C2×M4(2), C22.M4(2), C2×C22⋊C8, C2×C23⋊C4, C2×C4.10D4, C2×C22.M4(2)

Smallest permutation representation of C2×C22.M4(2)
►On 64 points
Generators in S64
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim11111112244
type++++++-
imageC1C2C2C2C4C4C8D4M4(2)C23⋊C4C4.10D4
kernelC2×C22.M4(2)C22.M4(2)C2×C22⋊C8C22×C4⋊C4C2×C4⋊C4C23×C4C22×C4C22×C4C23C22C22
# reps142144164422

Matrix representation of C2×C22.M4(2) ►in GL8(𝔽17)

160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
160000000
016000000
00100000
00010000
00001000
00000100
0000117160
00001011016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
123000000
145000000
00900000
00080000
000011708
000010111316
000049315
000006129
,
016000000
160000000
000160000
001600000
000016900
000013100
000097013
0000125130

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,11,10,0,0,0,0,0,1,7,11,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[12,14,0,0,0,0,0,0,3,5,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,11,10,4,0,0,0,0,0,7,11,9,6,0,0,0,0,0,13,3,12,0,0,0,0,8,16,15,9],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,13,9,12,0,0,0,0,9,1,7,5,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0] >;
 

C2×C22.M4(2) in GAP, Magma, Sage, TeX 

C_2\times C_2^2.M_4(2)
 
% in TeX
 
G:=Group("C2xC2^2.M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,189);
 
// by ID
 
G=gap.SmallGroup(128,189);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,851,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^5>;
 
// generators/relations
 

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