C2xHe3:3C4 - GroupNames

G = C₂×He₃⋊₃C₄ order 216 = 2³·3³

Direct product of C₂ and He₃⋊₃C₄

direct product, non-abelian, supersoluble, monomial

Aliases: C₂×He₃⋊₃C₄, C₆².₄S₃, He₃⋊₇(C₂×C₄), (C₂×He₃)⋊₃C₄, (C₃×C₆)⋊₂Dic₃, (C₃×C₆).₁₉D₆, C₆.₅(C₃⋊Dic₃), C₃²⋊₃(C₂×Dic₃), C₂².(He₃⋊C₂), (C₂²×He₃).₂C₂, (C₂×He₃).₁₄C₂², C₆.₂₀(C₂×C₃⋊S₃), (C₂×C₆).₉(C₃⋊S₃), C₃.₂(C₂×C₃⋊Dic₃), C₂.₂(C₂×He₃⋊C₂), SmallGroup(216,71)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — He₃ — C₂×He₃⋊₃C₄

Chief series

C₁ — C₃ — C₃² — He₃ — C₂×He₃ — He₃⋊₃C₄ — C₂×He₃⋊₃C₄

Lower central

He₃ — C₂×He₃⋊₃C₄

Upper central

C₁ — C₂×C₆

Generators and relations for C₂×He₃⋊₃C₄
G = < a,b,c,d,e | a²=b³=c³=d³=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=bc^-1, ebe^-1=b^-1, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 224 in 88 conjugacy classes, 38 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₂×C₄, C₃², Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₃×C₆, C₂×Dic₃, C₂×C₁₂, He₃, C₃×Dic₃, C₆², C₂×He₃, C₂×He₃, C₆×Dic₃, He₃⋊₃C₄, C₂²×He₃, C₂×He₃⋊₃C₄
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, Dic₃, D₆, C₃⋊S₃, C₂×Dic₃, C₃⋊Dic₃, C₂×C₃⋊S₃, He₃⋊C₂, C₂×C₃⋊Dic₃, He₃⋊₃C₄, C₂×He₃⋊C₂, C₂×He₃⋊₃C₄

Smallest permutation representation of C₂×He₃⋊₃C₄
►On 72 points

Generators in S₇₂

(1 40)(2 37)(3 38)(4 39)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 35)(14 36)(15 33)(16 34)(21 25)(22 26)(23 27)(24 28)(29 58)(30 59)(31 60)(32 57)(41 62)(42 63)(43 64)(44 61)(49 71)(50 72)(51 69)(52 70)(53 66)(54 67)(55 68)(56 65)

(1 62 29)(2 30 63)(3 64 31)(4 32 61)(5 27 52)(6 49 28)(7 25 50)(8 51 26)(9 33 56)(10 53 34)(11 35 54)(12 55 36)(13 67 20)(14 17 68)(15 65 18)(16 19 66)(21 72 48)(22 45 69)(23 70 46)(24 47 71)(37 59 42)(38 43 60)(39 57 44)(40 41 58)

(1 26 15)(2 27 16)(3 28 13)(4 25 14)(5 66 63)(6 67 64)(7 68 61)(8 65 62)(9 58 69)(10 59 70)(11 60 71)(12 57 72)(17 32 50)(18 29 51)(19 30 52)(20 31 49)(21 36 39)(22 33 40)(23 34 37)(24 35 38)(41 45 56)(42 46 53)(43 47 54)(44 48 55)

(1 29 65)(2 66 30)(3 31 67)(4 68 32)(5 19 16)(6 13 20)(7 17 14)(8 15 18)(9 45 33)(10 34 46)(11 47 35)(12 36 48)(21 44 72)(22 69 41)(23 42 70)(24 71 43)(25 61 50)(26 51 62)(27 63 52)(28 49 64)(37 53 59)(38 60 54)(39 55 57)(40 58 56)

(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)(65 66 67 68)(69 70 71 72)

G:=sub<Sym(72)| (1,40)(2,37)(3,38)(4,39)(5,46)(6,47)(7,48)(8,45)(9,18)(10,19)(11,20)(12,17)(13,35)(14,36)(15,33)(16,34)(21,25)(22,26)(23,27)(24,28)(29,58)(30,59)(31,60)(32,57)(41,62)(42,63)(43,64)(44,61)(49,71)(50,72)(51,69)(52,70)(53,66)(54,67)(55,68)(56,65), (1,62,29)(2,30,63)(3,64,31)(4,32,61)(5,27,52)(6,49,28)(7,25,50)(8,51,26)(9,33,56)(10,53,34)(11,35,54)(12,55,36)(13,67,20)(14,17,68)(15,65,18)(16,19,66)(21,72,48)(22,45,69)(23,70,46)(24,47,71)(37,59,42)(38,43,60)(39,57,44)(40,41,58), (1,26,15)(2,27,16)(3,28,13)(4,25,14)(5,66,63)(6,67,64)(7,68,61)(8,65,62)(9,58,69)(10,59,70)(11,60,71)(12,57,72)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(21,36,39)(22,33,40)(23,34,37)(24,35,38)(41,45,56)(42,46,53)(43,47,54)(44,48,55), (1,29,65)(2,66,30)(3,31,67)(4,68,32)(5,19,16)(6,13,20)(7,17,14)(8,15,18)(9,45,33)(10,34,46)(11,47,35)(12,36,48)(21,44,72)(22,69,41)(23,42,70)(24,71,43)(25,61,50)(26,51,62)(27,63,52)(28,49,64)(37,53,59)(38,60,54)(39,55,57)(40,58,56), (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)(65,66,67,68)(69,70,71,72)>;

G:=Group( (1,40)(2,37)(3,38)(4,39)(5,46)(6,47)(7,48)(8,45)(9,18)(10,19)(11,20)(12,17)(13,35)(14,36)(15,33)(16,34)(21,25)(22,26)(23,27)(24,28)(29,58)(30,59)(31,60)(32,57)(41,62)(42,63)(43,64)(44,61)(49,71)(50,72)(51,69)(52,70)(53,66)(54,67)(55,68)(56,65), (1,62,29)(2,30,63)(3,64,31)(4,32,61)(5,27,52)(6,49,28)(7,25,50)(8,51,26)(9,33,56)(10,53,34)(11,35,54)(12,55,36)(13,67,20)(14,17,68)(15,65,18)(16,19,66)(21,72,48)(22,45,69)(23,70,46)(24,47,71)(37,59,42)(38,43,60)(39,57,44)(40,41,58), (1,26,15)(2,27,16)(3,28,13)(4,25,14)(5,66,63)(6,67,64)(7,68,61)(8,65,62)(9,58,69)(10,59,70)(11,60,71)(12,57,72)(17,32,50)(18,29,51)(19,30,52)(20,31,49)(21,36,39)(22,33,40)(23,34,37)(24,35,38)(41,45,56)(42,46,53)(43,47,54)(44,48,55), (1,29,65)(2,66,30)(3,31,67)(4,68,32)(5,19,16)(6,13,20)(7,17,14)(8,15,18)(9,45,33)(10,34,46)(11,47,35)(12,36,48)(21,44,72)(22,69,41)(23,42,70)(24,71,43)(25,61,50)(26,51,62)(27,63,52)(28,49,64)(37,53,59)(38,60,54)(39,55,57)(40,58,56), (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)(65,66,67,68)(69,70,71,72) );

G=PermutationGroup([[(1,40),(2,37),(3,38),(4,39),(5,46),(6,47),(7,48),(8,45),(9,18),(10,19),(11,20),(12,17),(13,35),(14,36),(15,33),(16,34),(21,25),(22,26),(23,27),(24,28),(29,58),(30,59),(31,60),(32,57),(41,62),(42,63),(43,64),(44,61),(49,71),(50,72),(51,69),(52,70),(53,66),(54,67),(55,68),(56,65)], [(1,62,29),(2,30,63),(3,64,31),(4,32,61),(5,27,52),(6,49,28),(7,25,50),(8,51,26),(9,33,56),(10,53,34),(11,35,54),(12,55,36),(13,67,20),(14,17,68),(15,65,18),(16,19,66),(21,72,48),(22,45,69),(23,70,46),(24,47,71),(37,59,42),(38,43,60),(39,57,44),(40,41,58)], [(1,26,15),(2,27,16),(3,28,13),(4,25,14),(5,66,63),(6,67,64),(7,68,61),(8,65,62),(9,58,69),(10,59,70),(11,60,71),(12,57,72),(17,32,50),(18,29,51),(19,30,52),(20,31,49),(21,36,39),(22,33,40),(23,34,37),(24,35,38),(41,45,56),(42,46,53),(43,47,54),(44,48,55)], [(1,29,65),(2,66,30),(3,31,67),(4,68,32),(5,19,16),(6,13,20),(7,17,14),(8,15,18),(9,45,33),(10,34,46),(11,47,35),(12,36,48),(21,44,72),(22,69,41),(23,42,70),(24,71,43),(25,61,50),(26,51,62),(27,63,52),(28,49,64),(37,53,59),(38,60,54),(39,55,57),(40,58,56)], [(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),(65,66,67,68),(69,70,71,72)]])

C₂×He₃⋊₃C₄ is a maximal subgroup of
He₃⋊C₄² C₆².D₆ C₆².₃D₆ C₆².₄D₆ C₆².₂₉D₆ C₆².₃₀D₆ C₆².₃₁D₆ C₆²⋊₄Dic₃ He₃⋊₄M₄(2) C₆².₉D₆ C₂×C₄×He₃⋊C₂ C₆².₁₆D₆
C₂×He₃⋊₃C₄ is a maximal quotient of
He₃⋊₈M₄(2) C₆².₃₀D₆ C₆²⋊₄Dic₃

40 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	4A	4B	4C	4D	6A	···	6F	6G	···	6R	12A	···	12H
order	1	2	2	2	3	3	3	3	3	3	4	4	4	4	6	···	6	6	···	6	12	···	12
size	1	1	1	1	1	1	6	6	6	6	9	9	9	9	1	···	1	6	···	6	9	···	9

40 irreducible representations

dim	1	1	1	1	2	2	2	3	3	3
type	+	+	+		+	-	+
image	C₁	C₂	C₂	C₄	S₃	Dic₃	D₆	He₃⋊C₂	He₃⋊₃C₄	C₂×He₃⋊C₂
kernel	C₂×He₃⋊₃C₄	He₃⋊₃C₄	C₂²×He₃	C₂×He₃	C₆²	C₃×C₆	C₃×C₆	C₂²	C₂	C₂
# reps	1	2	1	4	4	8	4	4	8	4

Matrix representation of C₂×He₃⋊₃C₄ ►in GL₅(𝔽₁₃)

1	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

0	12	0	0	0
1	12	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

1	0	0	0	0
0	1	0	0	0
0	0	3	0	0
0	0	0	3	0
0	0	0	0	3

0	12	0	0	0
1	12	0	0	0
0	0	0	0	3
0	0	1	0	0
0	0	0	9	0

0	5	0	0	0
5	0	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	0	0	1

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[0,1,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[0,1,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,9,0,0,3,0,0],[0,5,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C₂×He₃⋊₃C₄ in GAP, Magma, Sage, TeX

C_2\times {\rm He}_3\rtimes_3C_4

% in TeX

G:=Group("C2xHe3:3C4");

// GroupNames label

G:=SmallGroup(216,71);

// by ID

G=gap.SmallGroup(216,71);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,387,1444,382]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C2×He3⋊3C4 order 216 = 23·33

Direct product of C2 and He3⋊3C4

G = C₂×He₃⋊₃C₄ order 216 = 2³·3³

Direct product of C₂ and He₃⋊₃C₄