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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

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 ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 C8 D4 M4(2) C23⋊C4 C4.10D4 kernel C2×C22.M4(2) C22.M4(2) C2×C22⋊C8 C22×C4⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C22×C4 C23 C22 C22 # reps 1 4 2 1 4 4 16 4 4 2 2

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

 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 0 0 0 0 0 0 0 1 0 0 0 0 0 0 11 7 16 0 0 0 0 0 10 11 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 3 0 0 0 0 0 0 14 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 7 0 8 0 0 0 0 10 11 13 16 0 0 0 0 4 9 3 15 0 0 0 0 0 6 12 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 9 0 0 0 0 0 0 13 1 0 0 0 0 0 0 9 7 0 13 0 0 0 0 12 5 13 0

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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