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G = C22×C4⋊C4order 64 = 26

Direct product of C22 and C4⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C22×C4⋊C4, C23.59D4, C23.11Q8, C22.6C24, C24.36C22, C23.69C23, (C22×C4)⋊9C4, C42(C22×C4), (C23×C4).8C2, C2.2(C23×C4), C2.2(C22×D4), C2.1(C22×Q8), C23.40(C2×C4), (C2×C4).48C23, C22.58(C2×D4), C22.16(C2×Q8), C22.25(C22×C4), (C22×C4).97C22, (C2×C4)⋊10(C2×C4), SmallGroup(64,194)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22×C4⋊C4
C1C2C22C23C24C23×C4 — C22×C4⋊C4
C1C2 — C22×C4⋊C4
C1C24 — C22×C4⋊C4
C1C22 — C22×C4⋊C4

Generators and relations for C22×C4⋊C4
 G = < a,b,c,d | a2=b2=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 249 in 209 conjugacy classes, 169 normal (8 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×8], C22, C22 [×34], C2×C4 [×36], C2×C4 [×24], C23 [×15], C4⋊C4 [×16], C22×C4 [×26], C22×C4 [×8], C24, C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C22×C4⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4

Smallest permutation representation of C22×C4⋊C4
Regular action on 64 points
Generators in S64
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)(17 41)(18 42)(19 43)(20 44)(21 31)(22 32)(23 29)(24 30)(25 56)(26 53)(27 54)(28 55)(33 61)(34 62)(35 63)(36 64)(37 51)(38 52)(39 49)(40 50)(45 57)(46 58)(47 59)(48 60)
(1 44)(2 41)(3 42)(4 43)(5 22)(6 23)(7 24)(8 21)(9 30)(10 31)(11 32)(12 29)(13 18)(14 19)(15 20)(16 17)(25 61)(26 62)(27 63)(28 64)(33 56)(34 53)(35 54)(36 55)(37 60)(38 57)(39 58)(40 59)(45 52)(46 49)(47 50)(48 51)
(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 54 10 57)(2 53 11 60)(3 56 12 59)(4 55 9 58)(5 48 16 26)(6 47 13 25)(7 46 14 28)(8 45 15 27)(17 62 22 51)(18 61 23 50)(19 64 24 49)(20 63 21 52)(29 40 42 33)(30 39 43 36)(31 38 44 35)(32 37 41 34)

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

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

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

C22×C4⋊C4 is a maximal subgroup of
C24.625C23  C24.626C23  C24.5Q8  C24.631C23  C24.632C23  C24.634C23  C24.635C23  (C2×C4)⋊M4(2)  C24.152D4  (C22×C4).275D4  C23.36D8  C24.157D4  C23.37D8  C24.159D4  C24.160D4  C23.38D8  C24.524C23  C24.542C23  C23.195C24  C24.545C23  C23.199C24  C24.195C23  C23.226C24  C23.227C24  C23.234C24  C24.558C23  C24.215C23  C24.243C23  C23.313C24  C23.316C24  C24.252C23  C24.568C23  C24.569C23  C24.269C23  C23.349C24  C24.572C23  C24.573C23  C24.576C23  C24.299C23  C24.300C23  C23.401C24  C23.402C24  C24.579C23  C23.404C24  C23.479C24  C23.483C24  C23.491C24  C24.587C23  C24.589C23  C23.527C24  C24.183D4  D4×C22×C4  Q8×C22×C4  C22.81C25
C22×C4⋊C4 is a maximal quotient of
C23.167C24  C23.178C24  C24.91D4  C24.545C23  C23.199C24  C23.231C24  C23.233C24  C42.257C23  C42.674C23  C24.100D4  C4○D4.7Q8  C4○D4.8Q8  M4(2).29C23

40 conjugacy classes

class 1 2A···2O4A···4X
order12···24···4
size11···12···2

40 irreducible representations

dim111122
type++++-
imageC1C2C2C4D4Q8
kernelC22×C4⋊C4C2×C4⋊C4C23×C4C22×C4C23C23
# reps11231644

Matrix representation of C22×C4⋊C4 in GL5(𝔽5)

10000
01000
00400
00010
00001
,
10000
04000
00400
00040
00004
,
40000
01000
00100
00001
00040
,
20000
04000
00100
00002
00020

G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,2,0] >;

C22×C4⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes C_4
% in TeX

G:=Group("C2^2xC4:C4");
// GroupNames label

G:=SmallGroup(64,194);
// by ID

G=gap.SmallGroup(64,194);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103]);
// Polycyclic

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

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